Asset Pricing and Portfolio Choice Theory (Solutions Manual) [1 ed.]

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Solutions Manual: Asset Pricing and Portfolio Choice Theory Kerry Back

Contents

Part I Single-Period Models 1

Utility Functions and Risk Aversion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2

Portfolio Choice and Stochastic Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3

Equilibrium and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

4

Arbitrage and Stochastic Discount Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

5

Mean-Variance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

6

Beta Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

7

Representative Investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Part II Dynamic Models 8

Dynamic Securities Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

9

Portfolio Choice by Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

10 Conditional Beta Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

11 Some Dynamic Equilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

12 Brownian Motion and Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

13 Continuous-Time Securities Markets and SDF Processes . . . . . . . . . . . . . . . . . . . . . 111

4

Contents

14 Continuous-Time Portfolio Choice and Beta Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 127 Part III Derivative Securities 15 Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 16 Forwards, Futures, and More Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 17 Term Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Part IV Topics 18 Heterogeneous Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 19 Asymmetric Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 20 Alternative Preferences in Single-Period Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 21 Alternative Preferences in Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 22 Production Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Part I

Single-Period Models

1 Utility Functions and Risk Aversion Coefficients

1.1. Calculate the risk tolerance of each of the five special utility functions in Section 1.7 to verify the formulas given in the text.

↵w

)

u0 (w) = ↵e

u(w) = log w

)

u0 (w) =

u(w) =

u(w) =

e

1 1

⇢

w1

u(w) = log(w u(w) =

⇢ 1

⇢

✓

w

⇣ ⇢

◆1

⇢

)

⇣)

)

⇢

)

↵w

,

u00 (w) =

↵2 e

↵w

,

u0 (w) 1 = . 00 u (w) ↵

u0 (w) = w. u00 (w) u0 (w) w u0 (w) = w ⇢ , u00 (w) = ⇢w ⇢ 1 , = . 00 u (w) ⇢ 0 1 1 u (w) u0 (w) = , u00 (w) = , =w w ⇣ (w ⇣)2 u00 (w) ✓ ◆ ⇢ ✓ ◆ ⇢ 1 w ⇣ w ⇣ 0 00 u (w) = , u (w) = , ⇢ ⇢ u0 (w) w ⇣ = . 00 u (w) ⇢ 1 , w

u00 (w) =

1 , w2

⇣.

1.2. Let "˜ be a random variable with zero mean and variance equal to 1. Let ⇡( ) be the risk premium for the gamble "˜ at wealth w, meaning u(w

⇡( )) = E [u(w + "˜)] .

Assuming ⇡ is a sufficiently di↵erentiable function, we have the Taylor series approximation 1 ⇡( ) ⇡ ⇡(0) + ⇡ 0 (0) + ⇡ 00 (0) 2 for small

2

. Obviously, ⇡(0) = 0. Assuming di↵erentiation and expectation can be interchanged,

di↵erentiate both sides of (1.13) to show that ⇡ 0 (0) = 0 and ⇡ 00 (0) is the coefficient of absolute risk aversion.

4

1 Utility Functions and Risk Aversion Coefficients

Taking first derivatives gives ⇥ ⇤ ⇡( ))⇡ 0 ( ) = E u0 (w + "˜)˜ " .

u0 (w At

= 0, this implies

u0 (w)⇡ 0 (0) = u0 (w)E[˜ "] = 0 . Thus, ⇡ 0 (0) = 0. Taking second derivatives gives u00 (w At

⇡( ))[⇡ 0 ( )]2

u0 (w

= 0, we obtain

⇥ ⇤ ⇡( ))⇡ 00 ( ) = E u00 (w + "˜)˜ "2 .

u0 (w)⇡ 00 (0) = u00 (w)E[˜ "2 ] = u00 (w) . Hence, ⇡ 00 (0) =

u00 (w)/u0 (w).

1.3. Consider the five special utility functions in Section 1.7 (the utility functions with linear risk tolerance). Which of these utility functions, for some parameter values, have decreasing absolute risk aversion and increasing relative risk aversion? Which of these utility functions are monotone increasing and bounded on the domain w

0?

Shifted log and shifted power utility functions have absolute risk aversion ⇢/(w risk aversion ⇢w/(w

⇣) and relative

⇣), with ⇢ = 1 being shifted log. Absolute risk aversion is decreasing in w.

Relative risk aversion is increasing in w if ⇣ < 0. The shifted power utility function is monotone increasing and bounded on the domain w

0 if ⇣ < 0 and ⇢ > 1.

1.4. Consider a person with constant relative risk aversion ⇢. (a) Verify that the fraction of wealth he will pay to avoid a gamble that is proportional to wealth is independent of initial wealth (i.e., show that ⇡ defined in (1.10) is independent of w for logarithmic and power utility). (b) Consider a gamble "˜. Assume 1 + "˜ is lognormally distributed; specifically, assume 1 + "˜ = ez˜, 2

where z˜ is normally distributed with variance

and mean

2 /2.

Note that by the rule for

means of exponentials of normals, E[˜ "] = 0. Show that ⇡ defined in (1.10) equals 1

e

⇢

2 /2

.

1 Utility Functions and Risk Aversion Coefficients

5

Note: This is consistent with the approximation (1.4), because a first-order Taylor series expansion of the exponential function ex around x = 0 shows that ex ⇡ 1 + x when |x| is small. (a) For log utility, the left-hand side of (1.10) is log((1

⇡)w) = log(1

⇡) + log w ,

and the right-hand side is E[log((1 + "˜)w)] = E[log(1 + "˜)] + log w , so (1.10) is equivalent to log(1

,

⇡) = E[log(1 + "˜)]

⇡=1

exp (E[log(1 + "˜)]) .

Hence, ⇡ does not depend on w. For power utility, the left-hand side of (1.10) is 1 1

⇢

⇡)w)1

((1

and the right-hand side is  1 E ((1 + "˜)w)1 1 ⇢

⇢

⇢

1

=

1

⇢

1

=

1

⇢

w1

⇢

w1

⇢

so (1.10) is equivalent to (1

⇡)1

⇢

⇥ = E (1 + "˜)1

⇢

Hence, ⇡ does not depend on w. (b) We have E[log(1 + "˜)] = E[˜ z] =

2 /2,

⇤

,

⇡=1

(1

⇡)1

⇢

,

⇥ E (1 + "˜)1

⇢

⇥ E (1 + "˜)1

⇤

⇢

,

⇤

1 1 ⇢

.

so the proportional risk premium for log utility is ⇡=1

2 /2

e

.

For ⇢ 6= 1, ⇥ E (1 + "˜)1

⇢

⇤

h = E e(1

⇢)˜ z

i

=e

(1 ⇢)

2 /2+(1

Therefore, the proportional risk premium is ⇡=1

e

⇢

2 /2

.

⇢)2

2 /2

=e

⇢(1 ⇢)

2 /2

.

6

1 Utility Functions and Risk Aversion Coefficients

1.5. Consider a person with constant relative risk aversion ⇢. (a) Suppose the person has wealth of $100,000 and faces a gamble in which he wins or loses x with equal probabilities. Calculate the amount he would pay to avoid the gamble, for various values of ⇢ (say, between 0.5 and 40), and for x = $100, x = $1,000, x = $10,000, and x = $25,000. For large gambles, do large values of ⇢ seem reasonable? What about small gambles? (b) Suppose ⇢ > 1 and the person has wealth w. Suppose he is o↵ered a gamble in which he loses x or wins y with equal probabilities. Show that he will reject the gamble no matter how large y is if x w

0.51/(⇢

1

1)

,

⇢

log(0.5) + log(1 x/w) . log(1 x/w)

For example, if w is $100,000, then the person would reject a gamble in which he loses $10,000 or wins 1 trillion dollars with equal probabilities when ⇢ satisfies this inequality for x/w = 0.1. What values of ⇢ (if any) seem reasonable? (a) Set " = x/100000. From the solution to the previous exercise, the amount the person would pay is 100000⇡, where, for log utility, ⇡=1

e0.5 log(1+")+0.5 log(1

")

,

and, for power utility, ⇡=1 This implies the following:

⇥

0.5(1 + ")1

⇢

+ 0.5(1

")1

⇢

⇤

1 1 ⇢

⇢

$100

$1,000

$10,000

$25,000

0.5

$0.03

$2.50

$251

$1,588

1

$0.05

$5

$501

$3,175

2

$0.10

$10

$1,000

$6,250

5

$0.25

$25

$2,434

$13,486

10

$0.50

$50

$4,424

$19,086

15

$0.75

$75

$5,826

$21,198

20

$1.00

$99

$6,763

$22,214

30

$1.50

$148

$7,832

$23,186

40

$2.00

$195

$8,387

$23,655

.

1 Utility Functions and Risk Aversion Coefficients

7

For the largest gamble, ⇢ > 5 (or, perhaps ⇢ > 2) would seem unreasonable. But, for ⇢  5, the premium for the $100 gamble is $0.25 or less, which may be too small. (b) Given 1

⇢ < 0, the person rejects the gamble if w1

⇢

< 0.5(w

x)1

⇢

+ 0.5(w + y)1

⇢

.

This is true for all y > 0 if w1

⇢

 0.5(w

x)1

⇢

,

w

,

0.5 1

0.5 1

,

x w

,

0.5 ⇢

,

⇢

1 ⇢

⇢

,

⇢

,

⇢

x

0.5 ⇢

1

1

1

1 1

7.6, 25% losses if ⇢

1

1

x w

log(0.5)

log(1

x/w)

log(1 x/w) 1 log(0.5) log(0.5) 1 log(1 x/w) log(0.5) + log(1 x/w) log(1 x/w) 

Thus, all gambles involving 1% losses are rejected if ⇢ ⇢

(w x) h i 1 0.5 1 ⇢ 1 w

⇢

1

1

,

1

3.5, and 50% losses if ⇢

70, 2% losses if ⇢

36, 10% losses if

2. Surely, there should be some possible

gain that would compensate someone for a 50% chance of a 10% loss, implying ⇢ < 7.6. One could obviously argue for even smaller ⇢. 1.6. This exercise is a very simple version of a model of the bid-ask spread presented by Stoll (1978). Consider a person with constant absolute risk aversion ↵. Starting from a random wealth w, ˜ (a) Compute the maximum amount the person would pay to obtain a random payo↵ x ˜; i.e., compute BID satisfying E[u(w)] ˜ = E[u(w ˜+x ˜

BID)] .

(b) Compute the minimum amount the person would require to accept the payo↵ ASK satisfying E[u(w)] ˜ = E[u(w ˜

x ˜ + ASK)] .

x ˜; i.e., compute

8

1 Utility Functions and Risk Aversion Coefficients

Note that E[u(w)] ˜ =

exp

✓

1 ↵E[w] ˜ + ↵2 var(w) ˜ 2

◆

.

(a) We have E[u(w ˜+x ˜

BID)] =

exp

✓

↵E[w] ˜

1 ↵E[˜ x] + ↵BID + ↵2 [var(w) ˜ + 2 cov(˜ x, w) ˜ + var(˜ x)] 2

◆

.

◆

.

Thus, BID satisfies 1 = exp

✓

1 ↵E[˜ x] + ↵BID + ↵2 [2 cov(˜ x, w) ˜ + var(˜ x)] 2

◆

.

This implies BID = E[˜ x]

↵ cov(˜ x, w) ˜

1 ↵ var(˜ x) . 2

(b) We have E[u(w ˜

x ˜ + ASK)] =

exp

✓

↵E[w] ˜ + ↵E[˜ x]

1 ↵ASK + ↵2 [var(w) ˜ 2

2 cov(˜ x, w) ˜ + var(˜ x)]

Thus, ASK satisfies ✓

1 = exp ↵E[˜ x]

1 ↵ASK + ↵2 [ 2 cov(˜ x, w) ˜ + var(˜ x)] 2

◆

.

This implies ASK = E[˜ x]

1 ↵ cov(˜ x, w) ˜ + ↵ var(˜ x) . 2

1.7. Show that condition (ii) in the discussion of second-order stochastic dominance in the endof-chapter notes implies condition (i); i.e., assume y˜ = x ˜ + z˜ + "˜ where z˜ is a nonpositive random variable and E[˜ "|˜ x + z˜] = 0 and show that E[u(˜ x)]

E[u(˜ y )] for every monotone concave function u.

Note: The statement of (ii) is that y˜ has the same distribution as x ˜ + z˜ + "˜, which is a weaker condition than y˜ = x ˜ + z˜ + "˜, but if y˜ has the same distribution as x ˜ + z˜ + "˜ and y˜0 = x ˜ + z˜ + "˜, then E[u(˜ y )] = E[u(˜ y 0 )] so one can without loss of generality take y˜ = x ˜ + z˜ + "˜ (though this is not true for the reverse implication (i) ) (ii)). y˜ equals x ˜ + z˜ plus mean-independent noise, so by concavity and Jensen’s inequality, E[u(˜ x + z˜)] E[u(˜ y )], as shown in Section 1.8. Because z˜ is nonpositive and u is monotone, E[u(˜ x)] Therefore, E[u(˜ x)]

E[u(˜ y )].

E[u(˜ x + z˜)].

1 Utility Functions and Risk Aversion Coefficients

9

1.8. Use the law of iterated expectations to show that if E[˜ "|˜ y ] = 0 then cov(˜ y , "˜) = 0 (thus mean-independence implies uncorrelated). By iterated expectations and mean-independence, E[˜ y "˜] = E[˜ y E[˜ "|˜ y ]] = 0 . Furthermore, E[˜ "] = E[E[˜ "|˜ y ]] = 0 . Therefore, cov(˜ y , "˜) = E[˜ y "˜]

E[˜ y ]E[˜ "] = 0 .

1.9. Show that any monotone utility function with linear risk tolerance is a monotone affine transform of one of the five utility functions: negative exponential, log, power, shifted log, or shifted power. Hint: Consider first the special cases (i) risk tolerance = A and (ii) risk tolerance = Bw. In case (i) use the fact that u00 (w) d log u0 (w) = u0 (w) dw and in case (ii) use the fact that wu00 (w) d log u0 (w) = u0 (w) d log w to derive formulas for log u0 (w) and hence u0 (w) and hence u(w). For the case A 6= 0 and B 6= 0, define v(w) = u

✓

w

A B

◆

,

show that the risk tolerance of v is Bw, apply the results from case (ii) to v, and then derive the form of u. In case (i), set ↵ = 1/A. For any constant y, 0

Z

w

d log u0 (x) dx dx y Z w = log u0 (y) + ↵ dx 0

log u (w) = log u (y) +

y

0

= log u (y) Hence,

↵(w

y) .

10

1 Utility Functions and Risk Aversion Coefficients

u0 (w) = u0 (y)e

= u0 (y)e↵y e

↵(w y)

↵w

.

This implies Z

w

u0 (x) dx Z w = u(y) + u0 (y)e↵y e

u(w) = u(y) +

y

↵x

dx

y

= u(y) + u0 (y)e↵y This is an affine transform of of

e

e

↵w .

1⇥ e ↵

↵y

e

↵w

⇤

.

For u to be monotone, it must be a monotone affine transform

↵w .

In case (ii), set ⇢ = 1/B. For any constant y > 0 and any w > 0, Z w d log u0 (x) 0 0 log u (w) = log u (y) + d log x d log x y Z w = log u0 (y) ⇢ d log x y

= log u0 (y)

⇢(log w

log y) .

Hence, u0 (w) = u0 (y)e

⇢(log w log y)

= u0 (y)y ⇢ w

⇢

.

This implies Z

w

u0 (x) dx Z w 0 ⇢ = u(y) + u (y)y x

u(w) = u(y) +

y

⇢

dx .

y

If ⇢ = 1, then u(w) = u(y) + u0 (y)y(log w

log y) ,

which is a monotone affine transform of log w. If ⇢ 6= 1, then u(w) = u(y) + u0 (y)y ⇢ which is an affine transform of w1 transform of w1

⇢ /(1

⇢ /(1

1 1

⇢

w1

⇢

y1

⇢

.

⇢). For u to be monotone, it must be a monotone affine

⇢).

For the case A 6= 0 and B 6= 0, set v(x) = u

✓

x

A B

◆

1 Utility Functions and Risk Aversion Coefficients

11

for x > 0. This implies x A B 00 xB A u B

u0

v 0 (x) = v 00 (x)



=B A+B

✓

x

A B

◆

= Bx . Therefore, from case (ii), on the region x > 0, either v(x) = log x if B = 1, or v(x) = x1

⇢ /(1

⇢)

for ⇢ = 1/B, up to an affine transform. Moreover, u(w) = v(A + Bw) . Hence, for w such that A + Bw > 0, either u(w) = log(A + Bw) if B = 1, or u(w) = (A + Bw)1

⇢ /(1

⇢) for ⇢ = 1/B, up to an affine transform. Setting ⇣ =

transform, u(w) = log(w

on the region (w

A/B, we have, up to an affine

⇣) on the region w > ⇣ if B = 1, or ✓ ◆ 1 w ⇣ 1 ⇢ u(w) = , 1 ⇢ ⇢

⇣)/⇢ > 0. Monotonicity of u in the case B 6= 1 requires that u be a monotone

affine transform of ⇢ 1

⇢

✓

w

⇣ ⇢

◆1

⇢

.

1.10. Suppose a person has log utility: u(w) = log w for each w > 0. (a) Construct a gamble w ˜ such that E[u(w)] ˜ = 1. Verify that E[w] ˜ = 1. (b) Construct a gamble w ˜ such that w ˜ > 0 in each state of the world and E[u(w)] ˜ =

1.

(c) Given a constant wealth w, construct a gamble "˜ with w + "˜ > 0 in each state of the world, E[˜ "] = 0 and E[u(w + "˜)] =

1. n

(a) Consider flipping a sequence of coins and having wealth e2 if the first heads appears on the n–th toss. The probability of the first heads appearing on the n–th toss is 2 utility is

1 X

2

n

log e2

n

n=1

(b) Consider flipping coins and having wealth e expected utility is

=

1 X

n=1 2n

n,

so the expected

1 = 1.

if the first heads appears on the n–th toss. The

12

1 Utility Functions and Risk Aversion Coefficients 1 X

2

n

2n

log e

=

n=1

1 X

1.

1=

n=1

(c) Obviously, there are many ways to do this. Here is one. Let 0 < w>

1 X

1+

Define p = /(1 + ). With probability 1 "˜ =

e

2n

n

e

2n

1 X

2

n

2

n

2

< 1 be such that

.

n=1

p, let 1 X

w

n=1

On this event, we have w + "˜ = (1 + )w

!

e

.

2n

> 0.

n=1

For n = 1, 2, . . ., let 2n

"˜ = e with probability p2

w

n.

Then w + "˜ > 0 in each state of the world, and ! "1 1 X X n n 2n E[˜ "] = (1 p) w 2 e +p 2 n e 2 n=1

= [(1

p)

p] w

1 X

2

n

e

2n

n=1

!

w

n=1

#

= 0. Moreover, E[u(w + "˜)] = (1

1 X

p) log (1 + )w

2

n

e

2n

n=1

!

+p

"

1 X

n=1

2

n

log e

2n

#

=

1.

1.11. Show that risk neutrality [u(w) = w for all w] can be regarded as a limiting case of negative exponential utility as ↵ ! 0 by showing that there are monotone affine transforms of negative exponential utility that converges to w as ↵ ! 0. Hint: Take an exact first-order Taylor series expansion of negative exponential utility, expanding in ↵ around ↵ = 0. Writing the expansion as c0 + c1 ↵, show that e

↵w

↵ as ↵ ! 0.

c0

!w

1 Utility Functions and Risk Aversion Coefficients

Set f (↵) =

e

↵w .

13

We have f (↵) = f (0) + f 0 (ˆ ↵)↵

for some 0 < ↵ ˆ < ↵, and f 0 (↵) = we

↵w .

Thus, ↵w

e

=

1 + we

↵w ˆ

↵.

This implies e

↵w

↵

+1

= we

↵w ˆ

! w,

as ↵ ! 0. 1.12. The notation and concepts in this exercise are from Appendix A. Suppose there are three possible states of the world which are equally likely, so ⌦ = {!1 , !2 , !3 } with P({!1 }) = P({!2 }) = P({!3 }) = 1/3. Let G be the collection of all subsets of ⌦:

G = {;, {!1 }, {!2 }, {!3 }, {!1 , !2 }, {!1 , !3 }, {!2 , !3 }, ⌦} . Let x ˜ and y˜ be random variables, and set ai = x ˜(!i ) for i = 1, 2, 3. Suppose y˜(!1 ) = b1 and y˜(!2 ) = y˜(!3 ) = b2 6= b1 . (a) What is prob(˜ x = aj | y˜ = bi ) for i = 1, 2 and j = 1, 2, 3 ? (b) What is E[˜ x | y˜ = bi ] for i = 1, 2 ? (c) What is the –field generated by y˜ ? (a) prob(˜ x = a1 | y˜ = b1 ) = 1 , prob(˜ x = a2 | y˜ = b1 ) = 0 , prob(˜ x = a3 | y˜ = b1 ) = 0 , prob(˜ x = a1 | y˜ = b2 ) = 0 , prob(˜ x = a2 | y˜ = b2 ) = 1/2 , prob(˜ x = a3 | y˜ = b2 ) = 1/2 . (b) E[˜ x | y˜ = b1 ] = a1 , E[˜ x | y˜ = b2 ] = (a2 + a3 )/2 .

14

1 Utility Functions and Risk Aversion Coefficients

(c) The –field generated by y˜ is {;, {!1 }, {!2 , !3 }, ⌦} . 1.13. Let y˜ = ex˜ , where x ˜ is normally distributed with mean µ and variance stdev(˜ y) p = e E[˜ y]

We have E[˜ y ] = eµ+

2 /2

2

1.

, and var(˜ y ) = E[˜ y 2 ] E[˜ y ]2 ⇥ ⇤ = E e2˜x e2(µ+ 2

= e2µ+2 = e2µ+

2

e2µ+

⇣

⇣ = E[˜ y ]2 e so stdev(˜ y ) = E[˜ y]

e

2

2

p e

1 ⌘

2 /2) 2

⌘

1 ,

2

1.

2.

Show that

2 Portfolio Choice and Stochastic Discount Factors

2.1. Consider the portfolio choice problem of a CARA investor with n risky assets having normally distributed returns studied in Section 2.5, but suppose there is no risk-free asset, so the budget constraint is 10 = w0 . Show that the optimal portfolio is ✓ ◆ 1 ↵w0 10 ⌃ 1 µ 1 = ⌃ µ+ ⌃ ↵ ↵10 ⌃ 1 1 Note: As will be seen in Section 5.1, the two vectors ⌃

1µ

1

1.

and ⌃

11

play an important role in

mean-variance analysis even without the CARA/normal assumption. The expected payo↵ of a portfolio

0µ

and the variance is 0 ⌃ . The expected utility is ✓ ◆ 1 2 0 0 exp ↵ µ+ ↵ ⌃ . 2 is

Maximizing this is equivalent to maximizing 0

Let

1 0 ↵ ⌃ . 2

µ

denote the Lagrange multiplier for the constraint 10 = w0 . The Lagrangean is 0

(µ

1 0 ↵ ⌃ , 2

1)

and the first-order condition is µ

1

↵⌃ = 0 ,

which is solved by = Imposing the constraint 10 = w0 yields

1 ⌃ ↵

1

(µ

1) .

16

2 Portfolio Choice and Stochastic Discount Factors

1 0 1⌃ ↵

1

µ

↵

10 ⌃

1

1 = w0 .

Therefore, = and 1 ⌃ ↵

=

1

µ+

10 ⌃ 1 µ ↵w0 , 10 ⌃ 1 1 ✓

↵w0 10 ⌃ 1 µ ↵10 ⌃ 1 1

◆

⌃

1

1.

2.2. Suppose there is a risk-free asset and n risky assets. Adopt the notation of Section 2.5, but do not assume the risky asset returns are normally distributed. Consider an investor with quadratic utility who seeks to maximize ⇣E[w] ˜

1 E[w] ˜2 2

1 var(w) ˜ . 2

Show that the optimal portfolio for the investor is =

1 (⇣ 1 + 2

w0 Rf )⌃

1

(µ

Rf 1) ,

where 2 = (µ

Rf 1)0 ⌃

1

(µ

Rf 1) .

It is shown in Chapter 5 that  is the maximum Sharpe ratio of any portfolio. Hint: In the first-order conditions, define

= (µ

Rf 1)0 , solve for

The expected payo↵ of a portfolio

in terms of , and then compute .

of risky assets is w0 Rf + 0 (µ Rf 1), and the variance is

0⌃

The expected utility is ⇣[w0 Rf +

0

(µ

= ⇣[w0 Rf +

0

1 1 0 [w0 Rf + 0 (µ Rf 1)]2 ⌃ 2 2 1 2 2 1 0 Rf 1)] w0 Rf w0 Rf 0 (µ Rf 1) (µ 2 2

Rf 1)]

(µ

Rf 1)(µ

Rf 1)0

The first-order condition for maximizing this is ⇣(µ Setting

= (µ

Rf 1)

Rf 1)

(µ

Rf 1)(µ

Rf 1)0

⌃ = 0.

Rf 1)0 , we have ⇣(µ

with solution

w0 Rf (µ

Rf 1)

w0 Rf (µ

Rf 1)

(µ

Rf 1)

⌃ = 0,

1 0 ⌃ . 2

.

2 Portfolio Choice and Stochastic Discount Factors

= (⇣

w 0 Rf

1

)⌃

(µ

17

Rf 1) .

Thus, = (⇣

w 0 Rf

)(µ

= (⇣

w 0 Rf

)2 ,

Rf 1)0 ⌃

1

(µ

Rf 1)

implying =

2 (⇣ 1 + 2

w 0 Rf ) ,

and =

1 (⇣ 1 + 2

w0 Rf )⌃

1

(µ

Rf 1) .

2.3. This exercise provides another illustration of the absence of wealth e↵ects for CARA utility. The investor chooses how much to consume at date 0 and how much to invest, but the investment amount does not a↵ect the optimal portfolio of risky assets. Consider the portfolio choice problem in which there is consumption at date 0 and date 1. Suppose there is a risk-free asset with return Rf and n risky assets the returns of which are joint normally distributed with mean vector µ and nonsingular covariance matrix ⌃. Consider an investor who has time-additive utility and CARA utility for date–1 consumption: u1 (c) =

e

↵c

.

Show that (2.27) is the investor’s optimal portfolio of risky assets. The investor chooses date–0 consumption c0 and a portfolio of risky assets to maximize ✓ ◆ 1 2 0 0 u(c0 ) exp ↵(w0 c0 )Rf ↵ (µ Rf 1) + ↵ ⌃ . 2 The optimal portfolio

is the portfolio that maximizes 0

(µ

Rf 1)

1 0 ↵ ⌃ , 2

just as in the problem with only date–1 consumption. Hence, (2.27) is the optimal portfolio. 2.4. This exercise repeats the previous one, but using asset payo↵s and prices instead of returns and solving for the optimal number of shares to hold of each asset instead of the optimal amount to invest.

18

2 Portfolio Choice and Stochastic Discount Factors

Suppose there is a risk-free asset with return Rf and n risky assets with payo↵s x ˜i and prices pi . Assume the vector x ˜ = (˜ x1 · · · x ˜n )0 is normally distributed with mean µx and nonsingular covariance matrix ⌃x . Let p = (p1 · · · pn )0 . Suppose there is consumption at date 0 and consider an investor with initial wealth w0 and CARA utility at date 1: u1 (c) =

e

↵c

.

Let ✓i denote the number of shares the investor considers holding of asset i and set ✓ = (✓1 · · · ✓n )0 . The investor chooses consumption c0 at date 0 and a portfolio ✓, producing wealth (w0

c0

✓0 p)Rf + ✓0 x ˜ at date 1. Show that the optimal vector of share holdings is ✓=

1 ⌃ 1 (µx ↵ x

Rf p) .

The investor chooses date–0 consumption c0 and a portfolio ✓ of risky assets to maximize ✓ ◆ 1 2 0 0 0 u(c0 ) exp ↵(w0 c0 ✓ p)Rf ↵✓ µx + ↵ ✓ ⌃x ✓ . 2 The optimal portfolio ✓ is the portfolio that maximizes 1 0 ↵✓ ⌃x ✓ . 2

Rf p0 ✓ + µ0x ✓ The first-order condition is Rf p + µ x

↵⌃x ✓ = 0 ,

with solution ✓=

1 ⌃ 1 (µx ↵ x

Rf p) .

2.5. Consider a utility function v(c0 , c1 ). The marginal rate of substitution is defined to be the negative of the slope of an indi↵erence curve and is equal to MRS(c0 , c1 ) =

@v(c0 , c1 )/@c0 . @v(c0 , c1 )/@c1

The elasticity of intertemporal substitution is defined as d log(c1 /c0 ) , d log MRS(c0 , c1 ) where the marginal rate of substitution is varied holding utility constant. Show that, if

2 Portfolio Choice and Stochastic Discount Factors

v(c0 , c1 ) =

1 1

⇢

c10

⇢

+

1

c11

⇢

⇢

19

,

then the elasticity of intertemporal substitution is 1/⇢. Holding utility constant implies c0 ⇢ dc0 + c1 ⇢ , dc1 = 0 , so dc1 1 = dc0

✓

c0 c1

◆

⇢

.

This is the marginal rate of substitution. Setting x = c1 /c0 , we have log MRS =

log + ⇢ log x .

Hence, d log MRS = ⇢. d log x The elasticity of intertemporal substitution is the reciprocal 1/⇢.

2.6. This exercise shows that an improvement in the investment opportunity set leads to higher saving (the substitution e↵ect dominates) when the elasticity of intertemporal substitution is high and higher consumption (the wealth e↵ect dominates) when the elasticity of intertemporal substitution is low. Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Assume the investor has time-additive power utility, so he solves max

1 1

⇢

c10

⇢

+

1 1

⇢

c11

⇢

subject to c0 +

1 c1 = w0 . Rf

Show that the optimal consumption-to-wealth ratio c0 /w0 is a decreasing function of Rf if ⇢ < 1 and an increasing function of Rf if ⇢ > 1. Substituting the budget constraint, the objective function is 1 1

⇢

c10

⇢

+

1 1

⇢

Rf1

⇢

(w0

c0 )1

and the first-order condition is c0 ⇢

Rf1

⇢

(w0

c0 )

⇢

= 0.

⇢

,

20

2 Portfolio Choice and Stochastic Discount Factors

This implies 1/⇢

c0 = so c0 =

1+ is an increasing function of Rf if 1

(w0

c0 ) ,

1/⇢ R1 1/⇢ f w0 . 1/⇢ R1 1/⇢ f

1+

The factor

1 1/⇢

Rf

1/⇢ R1 1/⇢ f 1/⇢ R1 1/⇢ f

1/⇢ > 0 and a decreasing function of Rf if 1

1/⇢ < 0.

2.7. Each part of this exercise illustrates the absence of wealth e↵ects for CARA utility and is ˜ + "˜ where "˜ not true for general utility functions. The assumption in Part (c) that y˜ = aRf + bR ˜ is without loss of generality (even without the normality has zero mean and is uncorrelated with R ˜ var(R), ˜ a = (E[˜ assumption): define b = cov(˜ y , R)/ y]

˜ bE[R])/R ˜ = y˜ f and "

aRf

˜ This is a bR.

special case of an orthogonal projection (linear regression), which is discussed in more generality in Section 4.5. The optimal portfolio in Part (c) can be interpreted as the optimal portfolio in the absence of an endowment plus a hedge ( b) for y˜. ˜ Consider an Suppose there is a risk-free asset with return Rf and a risky asset with return R. investor who maximizes expected end-of-period utility of wealth and who has CARA utility and invests w0 . Suppose the investor has a random endowment y˜ at the end of the period, so his endof-period wealth is

f Rf

˜ + y˜, where + R

f

denotes the investment in the risk-free asset and

the investment in the risky asset. ˜ have a joint normal distribution. Derive the optimal portfolio. Show that (a) Suppose y˜ and R ˜ are uncorrelated, then the optimal if y˜ and R

is the same as if there were no end-of-period

endowment. ˜ are independent, then the optimal (b) Show that if y˜ and R

is the same as if there were no

˜ are normally distributed. Hint: Use end-of-period endowment, regardless of whether y˜ and R the law of iterated expectations as in Section 1.8 and the fact that if v˜ and x ˜ are independent random variables then E[˜ vx ˜] = E[˜ v ]E[˜ x].

2 Portfolio Choice and Stochastic Discount Factors

21

˜ have a joint normal distribution and y˜ = aRf +bR+ ˜ "˜ for constants a and b and (c) Suppose y˜ and R ˜ Show that the optimal some "˜ that has zero mean and is uncorrelated with R. ⇤

is

⇤

b, where

denotes the optimal investment in the risky asset when there is no end-of-period endowment.

(a) The investment in the risk-free asset is f = w0 . The expected end-of-period wealth is ⇣ ⌘ ˜ w 0 Rf + E[R] Rf + E[˜ y ] and the variance of end-of-period wealth is 2

˜ + 2 cov(R, ˜ y˜) + var(˜ var(R) y) =

The expected utility is ✓ ⇣ ⇣ ˜ exp ↵ w 0 Rf + E[R] Maximizing this over

Rf

⌘

2

˜ + var(˜ var(R) y) .

⌘

1 ⇣ + E[˜ y ] + ↵2 2

2

⌘◆ ˜ var(R) + var(˜ y .

is equivalent to maximizing ⇣

˜ E[R]

Rf

for which the solution is =

⌘

1 ↵ 2

2

˜ , var(R)

˜ E[R]

Rf . ˜ ↵ var(R)

(b) The expected utility is h ⇣ E exp ↵w0 Rf

↵

⇣

˜ R

Rf

⌘

⌘i ↵˜ y =

h E e

↵w0 Rf ↵

(R˜

Rf )

e

↵˜ y

By independence, this equals h E e and maximizing this over

↵w0 Rf ↵

(R˜

Rf )

i ⇥ E e

↵˜ y

is equivalent to maximizing h E e

↵w0 Rf ↵

(R˜

Rf )

i

⇤

,

,

which is the same as if y˜ = 0. (c) The expected end-of-period wealth is ˜ w0 Rf + (E[R]

Rf + E[˜ y ] = (w0 + a

˜ , )Rf + ( + b)E[R]

and the variance of end-of-period wealth is 2

˜ + 2 cov(R, ˜ y˜) + var(˜ var(R) y) = (

2

˜ + var(˜ + 2 b + b2 ) var(R) ") .

i

.

22

2 Portfolio Choice and Stochastic Discount Factors

The expected utility is ✓ ⇣ exp ↵ (w0 + a Maximizing this over

⌘ ⇣ ˜ + 1 ↵2 ( )Rf + ( + b)E[R] 2

2

˜ + var(˜ + 2 b + b ) var(R) ") 2

⌘◆

.

is equivalent to maximizing ˜ (E[R]

1 ↵( 2

Rf )

for which the solution is =

˜ E[R]

2

˜ , + 2 b) var(R)

Rf ˜ ↵ var(R)

b.

2.8. This exercise illustrates the concept of “precautionary savings”—the risk imposed by y˜ results in higher savings w0

c0 .

Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Suppose the investor has time-additive utility with u0 = u and u1 = u for a common function u and discount factor . Suppose the investor has a random endowment y˜ at the end of the period, so he chooses c0 to maximize u(c0 ) + E[u((w0

c0 )Rf + y˜)] .

Suppose the investor has convex marginal utility (u000 > 0) and suppose that E[˜ y ] = 0. Show that the optimal c0 is smaller than if y˜ = 0. The first-order condition is that u0 (c0 ) = E[u0 ((w0

c0 )Rf + y˜)] .

By Jensen’s inequality and the convexity of u0 , E[u0 ((w0

c0 )Rf + y˜)] > u0 (E[(w0

c0 )Rf + y˜]) = u0 ((w0

c0 )Rf ) .

Thus, u0 (c0 ) > u0 ((w0

c0 )Rf ) .

The first-order condition if y˜ = 0 is for these to be equal. Because the left-hand side is decreasing in c0 and the right-hand side increasing in c0 , equality requires that c0 be increased. Thus, the optimal c0 would be larger if y˜ = 0.

2 Portfolio Choice and Stochastic Discount Factors

23

2.9. Letting c⇤0 denote optimal consumption in the previous problem, define the “precautionary premium” ⇡ by u0 ((w0

⇡

c⇤0 )Rf ) = E[u0 ((w0

c⇤0 )Rf + y˜)] .

(a) Show that c⇤0 would be the optimal consumption of the investor if he had no end-of-period endowment and had initial wealth w0

⇡.

(b) Assume the investor has CARA utility. Show that the precautionary premium is independent of initial wealth (again, no wealth e↵ects with CARA utility). (a) The first-order condition is that u0 (c⇤0 ) = E[u0 ((w0

c⇤0 )Rf + y˜)] .

By the definition of the precautionary premium, this implies u0 (c⇤0 ) = u0 ((w0

c⇤0 )Rf ) .

⇡

This is the first-order condition for initial wealth w0 (b) With CARA utility

e

↵w ,

the marginal utility is ↵e

is ⇡ satisfying ↵e Multiplying by e↵(w0

with solution

c⇤0 )Rf /↵

↵(w0 ⇡ c⇤0 )Rf

yields

h = ↵E e

⇥ e↵⇡Rf = E e ⇡=

⇡ when y˜ = 0. ↵w .

Therefore the precautionary premium

↵((w0 c⇤0 )Rf +˜ y)

↵˜ y

⇥ 1 log E e ↵Rf

⇤

i

.

,

↵˜ y

⇤

.

2.10. The assumption in this exercise is a weak form of market completeness. The conclusion follows in a complete market from the formulation (2.35) of the portfolio choice problem. Suppose there is a stochastic discount factor m ˜ with the property that for every function g there exists a portfolio ✓ (depending on g) such that n X

✓i x ˜i = g(m) ˜ .

i=1

Consider an investor with no labor income y˜. Show that his optimal wealth is a function of m. ˜ Hint: For any feasible w, ˜ define w ˜ ⇤ = E[w| ˜ m], ˜ and use the result of Section 1.8.

24

2 Portfolio Choice and Stochastic Discount Factors

Setting w ˜ ⇤ = E[w| ˜ m] ˜ and "˜ = w ˜

w ˜ ⇤ , we have that w ˜ is w ˜ ⇤ + "˜. Moreover,

E[˜ " | m] ˜ = E[w ˜ | m] ˜

E[w ˜ ⇤ | m] ˜ =w ˜⇤

w ˜⇤ = 0 .

Also, because w ˜ ⇤ is a function of m, ˜ E[˜ "|w ˜ ⇤ ] = E [E[˜ " | m] ˜ |w ˜⇤] = 0 . Thus, "˜ is mean-independent noise. Hence w ˜ ⇤ is preferred to w. ˜ Because w ˜ ⇤ is a function of m, ˜ there exists by assumption a portfolio ✓˜ with payo↵ equal to w ˜ ⇤ . The cost of the portfolio is E[m ˜w ˜ ⇤ ] = E[mE[ ˜ w ˜ | m]] ˜ = E[m ˜ w] ˜ , by iterated expectations. Hence, the feasibility of w ˜ implies the feasibility of w ˜⇤.

3 Equilibrium and Efficiency

3.1. Suppose each investor h has a concave utility function, and suppose a feasible allocation (w ˜1 , . . . , w ˜m ) of market wealth w ˜m satisfies the first-order condition u0h (w ˜h ) =

˜ hm

for each investor h, where m ˜ is a stochastic discount factor and is the same for each investor. Show that the allocation solves the social planner’s problem (3.2) with weights

h

= 1/

h.

Note: The

first-order condition holds with the stochastic discount factor being the same for each investor in a competitive equilibrium of a complete market, because there is a unique stochastic discount factor in a complete market. Recall that

h

in the first-order condition is the Lagrange multiplier for the

investor’s budget constraint (see Section 2.1) and hence is the marginal value of beginning-of-period wealth. Thus, the weights in the social planner’s problem can be taken to be the reciprocals of the marginal values of wealth. Other things equal, investors with high wealth have low marginal values of wealth and hence have high weights in the social planner’s problem. By concavity and the first-order conditions, the allocation maximizes H X 1 h=1

h

uh ( w ˜h )

H X

m ˜w ˜h

h=1

0 ) is any in each state of the world, over all allocations (w ˜1 , . . . , w ˜H ), feasible or not. If (w ˜10 , . . . , w ˜H

other feasible allocation, then H X h=1

so

w ˜h0 = w ˜m =

H X h=1

w ˜h ,

26

3 Equilibrium and Efficiency

H X 1 h=1

h

H X

uh ( w ˜h )

H X 1

m ˜w ˜h

h=1

h=1 H X

=

h=1

h

1 h

H X

uh ( w ˜h0 )

m ˜w ˜h0

h=1

H X

uh ( w ˜h0 )

m ˜w ˜h .

h=1

Hence, H X 1

h

h=1

H X 1

uh ( w ˜h )

h=1

h

uh ( w ˜h0 ) .

3.2. Suppose there are two investors, the first having constant relative risk aversion ⇢ > 0 and the second having constant relative risk aversion 2⇢. (a) Show that the Pareto optimal sharing rules are w ˜1 = w ˜m + ⌘

p

⌘ 2 + 2⌘ w ˜m ,

and

w ˜2 =

p ⌘ 2 + 2⌘ w ˜m

⌘,

for ⌘ > 0. Hint: Use the first-order condition and the quadratic formula. Because ⌘ is arbitrary in (0, 1), there are many equivalent ways to write the sharing rules. (b) Suppose the market is complete and satisfies the law of one price. Show that the stochastic discount factor in a competitive equilibrium is ⇣p ⌘ 2 + 2⌘ w ˜m

m ˜ = for positive constants

2⇢ .

⌘

2⇢

and ⌘.

(a) The marginal utility of the first investor is w is w

⌘

⇢,

and the marginal utility of the second investor

The first-order condition is ˜1 1w

⇢

=

˜2 2w

2⇢

=

˜m 2 (w

w ˜1 )

This implies w ˜ 1 = (w ˜m where

=(

1/ 2)

1/⇢ .

w ˜ 1 )2 ,

Thus w ˜12

2 (2w ˜ m + )w ˜1 + w ˜m = 0.

Applying the quadratic formula yields

2⇢

.

3 Equilibrium and Efficiency

p (2w ˜ m + )2 w ˜1 = 2 p 2 =w ˜m + ⌘ ± ⌘ + 2⌘ w ˜m , 2w ˜m +

±

27

2 4w ˜m

where ⌘ = /2. This implies w ˜2 = w ˜m

w ˜1 =

To obtain w ˜2 > 0, we must have w ˜2 =

⌘+

⌘±

p ⌘ 2 + 2⌘ w ˜m .

p ⌘ 2 + 2⌘ w ˜m and w ˜1 = w ˜m + ⌘

(b) Pareto optimality of the competitive equilibrium implies w ˜2 =

p ⌘ 2 + 2⌘ w ˜m

p

⌘ 2 + 2⌘ w ˜m .

⌘

for some ⌘ > 0. Thus, the second investor’s marginal utility at a competitive equilibrium equals ⇣p

⌘ 2 + 2⌘ w ˜m

⌘

⌘

2⇢

.

The first-order condition for portfolio choice implies ⇣p ⌘ 2 + 2⌘ w ˜m is a stochastic discount factor for some

⌘

⌘

2⇢

> 0. This is the unique stochastic discount factor in

a complete market.

3.3. Suppose there are n risky assets with payo↵s x ˜i and no risk-free asset. Assume there is consumption only at date 1. Let µ denote the mean and ⌃ the covariance matrix of the vector ˜ = (˜ ˜ has a normal distribution, and assume ⌃ is nonsingular. X x1 · · · x ˜n )0 of asset payo↵s. Assume X Let ✓¯ = (✓¯1 · · · ✓¯n )0 denote the vector of asset supplies. Assume all investors have CARA utility and no endowments y˜h . Define ↵ to be the aggregate absolute risk aversion as in Section 1.3. Show that the vector p= is an equilibrium price vector for any

µ

↵⌃ ✓¯

¯ explaining > 0. Interpret the risk adjustment vector ↵⌃ ✓,

in economic terms why a large element of this vector implies an asset has a low price relative to its expected payo↵. Note: When

< 0, this is also an equilibrium price vector, but each investor

has a negative marginal value of wealth. In this model, investors are forced to hold assets because there is no date–0 consumption. When

< 0, they are forced to invest in undesirable assets and

28

3 Equilibrium and Efficiency

would be better o↵ if they had less wealth. Including consumption at date 0 or changing the budget constraint to p0 ✓  p0 ✓¯h instead of p0 ✓ = p0 ✓¯h (i.e., allowing free disposal of wealth) eliminates the equilibria with

< 0.

Investor h chooses ✓h to maximize the certainty equivalent 1 0 ↵✓ ⌃✓ 2

✓0 µ

subject to the budget constraint p0 ✓ = wh0 . The optimum is ✓h = where

h

1 ⌃ ↵h

1

(µ

h p)

is the Lagrange multiplier for the budget constraint. Thus, ! H H X X 1 h ✓h = ⌃ 1 µ ⌃ 1p . ↵ ↵h h=1

h=1

To check if markets clear, we need to compute

h,

which is determined by the budget equation of

investor h: wh0 = p0 ✓h = Instead of computing

h,

1 0 p⌃ ↵h

1

h 0

µ

↵h

1 p0 ✓¯ = p0 ⌃ ↵

1

H X h=1

H X

µ

implying

µ

p.

¯ it suffices to sum this over h, noting that aggregate initial wealth is p0 ✓.

This yields

For p =

1

p⌃

1 = ↵h ↵ h

h=1

✓

p0 ⌃

h

↵h

!

p0 ⌃

↵p0 ✓¯ p0 ⌃ 1 p 1µ

1

◆

p,

.

↵⌃ ✓¯ , the expression in parentheses in the last displayed equation is 1/ . Thus, for

such prices, aggregate demand is 1 ⌃ ↵

1

µ

1 ⌃ ↵

1

p=

1 ⌃ ↵

1

µ

1 ⌃ ↵

1

⇥

µ

↵⌃ ✓¯

⇤

= ✓¯ .

3.4. Reconsider the previous problem assuming there is a risk-free asset in zero net supply (meaning investors can borrow from and lend to each other) and assuming there is consumption at date 0. Both the price vector p of the risky assets and the risk-free return Rf are determined endogenously in equilibrium. Suppose the utility functions of investor h are

3 Equilibrium and Efficiency

u0 (c) =

↵h c

e

Let c¯0 denote the aggregate endowment

and

PH

h=1 yh0

log = PH

h=1 ⌧h .

he

↵h c

.

at date 0 and define

H X ⌧h h=1

where ⌧h = 1/↵h and ⌧ =

u1 (c) =

⌧

log

29

by

h,

Using the result of Exercise 2.4 on the optimal demands for

the risky assets, show that the equilibrium risk-free return and price vector p are given by ✓ ◆ 1 1 2 ¯0 ¯ Rf = exp ↵ ✓¯0 µ c¯0 ↵ ✓ ⌃✓ , 2 1 ¯ . p= (µ ↵⌃ ✓) Rf Explain in economic terms why the risk-free return is higher when ✓¯0 µ is higher and lower when , c¯0 , or ✓¯0 ⌃ ✓¯ is higher. From Exercise 2.4, the optimal portfolio of investor h is ✓h =

1 ⌃ ↵h

1

(µ

Rf p) .

Thus, the aggregate demand for risky assets is ! H X 1 1 ⌃ 1 (µ Rf p) = ⌃ ↵h ↵

1

(µ

Rf p) .

h=1

Market clearing implies 1 ✓¯ = ⌃ ↵

1

(µ

)

Rf p)

p=

1 (µ Rf

¯ . ↵⌃ ✓)

To derive Rf we need to clear the market for date–0 consumption (or the market for the risk-free asset). Investor h chooses c0 and ✓ to maximize exp( ↵h c0 )

hE



exp =

↵h [(wh0 exp( ↵h c0 )

c0

p0 ✓)Rf + ✓0 x ˜] h exp

Substituting the optimal portfolio ✓ = ✓h yields  ✓ 0 E exp ↵h ✓ x ˜ = exp (µ Rf p)0 ⌃ and substituting p =

1 Rf (µ

¯ yields ↵⌃ ✓)

↵h (wh0

1

1 µ + (µ 2

c0



0

↵h ✓ 0 x ˜

p ✓)Rf E exp

0

Rf p) ⌃

1

(µ

Rf p)

◆

,

.

30

3 Equilibrium and Efficiency

 E exp

↵h ✓ 0 x ˜

= exp

✓

↵2 ¯0 ✓ ⌃✓ 2

◆

p ✓h )Rf exp

✓

↵✓¯0 µ +

,

Thus, the first-order condition for ch0 is ↵h exp( ↵h ch0 ) =

h ↵ h Rf

exp

↵h (wh0

0

ch0

↵2 ¯0 ↵✓¯0 µ + ✓ ⌃✓ 2

◆

.

Dividing by ↵h , taking logs and rearranging yields ch0 =

log h ↵h

log Rf + (wh0 ↵h

p0 ✓h )Rf +

ch0

↵2 ¯0 ¯ ✓ ⌃✓ . 2↵h

↵ ¯0 ✓µ ↵h

By the definition of , log ↵

=

H X log h=1

h

↵h

.

Thus, aggregate demand for the consumption good can be expressed as H X

ch0 =

h=1

log ↵

H

X log Rf + Rf (wh0 ↵

p0 ✓h ) + ✓¯0 µ

ch0

h=1

By market clearing for the risky assets, Thus, each of these is equivalent to

h=1 ch0

The solution of this is 1

= c¯ if and only if

log Rf + ✓¯0 µ ↵

log ↵

c¯0 =

Rf =

PH

⇢

exp ↵ ✓¯0 µ

c¯0

PH

↵ ¯0 ¯ ✓ ⌃✓ . 2

h=1 wh0

ch0

p0 ✓h = 0.

↵ ¯0 ¯ ✓ ⌃✓ . 2 1 2 ¯0 ¯ ↵ ✓ ⌃✓ . 2

✓¯0 µ is expected aggregate date–1 consumption. Investors prefer to smooth consumption over time, so when ✓¯0 µ is larger, they wish to borrow to consume more at date 0. The risk-free return must rise to o↵set this inclination to borrow. The reverse is true when c¯0 is larger. When

is

higher, investors do not discount the future as much, and hence wish to save to finance date–1 consumption. The risk-free return must fall to o↵set this inclination to save. ✓¯0 ⌃ ✓¯ is the variance of aggregate date–1 consumption. When it is larger, there is more risk, and investors expected date–1 utilities are smaller. They wish to transfer wealth from date 0 to date 1 in this circumstance, and the risk-free return must fall to o↵set that desire.

3.5. Suppose the payo↵ of the market portfolio w ˜m has k possible values. Denote these possible values by a1 < · · · < ak . For convenience, suppose ai

ai

1

is the same number

for each i.

3 Equilibrium and Efficiency

Suppose there is a risk-free asset with payo↵ equal to 1. Suppose there are k

31

1 call options on the

market portfolio, with the exercise price of the i–th option being ai . The payo↵ of the i–th option is max(0, w ˜m

ai ).

(a) Show for each i = 1, . . . , k unit of option i + 1 pays

2 that a portfolio that is long one unit of option i and short one if w ˜m

ai+1 and 0 otherwise. This portfolio of options is a bull

spread. (b) Consider the following k portfolios. Show that the payo↵ of portfolio i is 1 when w ˜m = ai and 0 otherwise. Thus, these are Arrow securities for the events on which w ˜m is constant. (i) i = 1: long one unit of the risk-free asset, short 1/

units of option 1, and long 1/

units

of option 2. This portfolio of options is a short bull spread. (ii) 1 < i < k: long 1/

units of option i

1, short 2/

units of option i, and long 1/

units

of option i + 1. These portfolios are butterfly spreads. (iii) i = k

1: long 1/

(iv) i = k: long 1/

units of option k

units of option k

2 and short 2/

units of option k

1.

1.

(c) Given any function f , define z˜ = f (w ˜m ). Show that there is a portfolio of the risk-free asset and the call options with payo↵ equal to z˜. (a) When w ˜m

ai+1 , a long position in option i pays w ˜m

has cash flow ai+1

w ˜m . The sum of these is ai+1

(b) (i) Being long 1/

units of option 2 and short 1/

ai , and a short position in option i + 1

ai =

.

units of option 1 pays

1 when w ˜m

a2

and 0 otherwise. Combining this with a payo↵ of 1 yields 1 when w ˜m = a1 and 0 otherwise. (ii) Being long 1/

units of option i

and 0 otherwise. Being short 1/ when w ˜m

1 and short 1/

units of option i pays 1 when w ˜m

units of option i and long 1/

units of option i+1 pays

ai 1

ai+1 and 0 otherwise. The sum of these pays 1 when w ˜m = ai and 0 otherwise.

(iii) Being long 1/

units of option k

w ˜m = ak . Being short 2/

2 produces a payo↵ of 1 when w ˜ m = ak

units of option k

The sum of the two pays 1 when w ˜ m = ak (iv) Being long 1/

units of option k

1

1 produces a payo↵ of

1

and 2 when

2 when w ˜ m = ak .

and 0 otherwise.

1 produces a payo↵ of 1 when w ˜m = ak and 0 otherwise.

(c) Let zi denote the value of z˜ when w ˜m = ai . The portfolio consisting of z1 units of the risk-free asset, (z2 z1 )/

units of option 1, and (zi+1 2zi +zi

has payo↵ equal to z˜.

1 )/

units of option i for i = 2, . . . , k 1

32

3 Equilibrium and Efficiency

3.6. Suppose all investors have CARA utility. Consider an allocation w ˜ h = a h + bh w ˜m where bh = ⌧h /⌧ and

PH

h=1 ah

= 0. Show that the allocation is Pareto optimal. Hint: Show that it

solves the social planner’s problem with weights

defined as

h

h

= ⌧h eah /⌧h .

The first-order condition for the social planner’s problem is (8 h) Setting w ˜ h = a h + bh w ˜m and condition holds for ⌘˜ = e

⌧w ˜m .

h

↵h

he

↵h w ˜h

= ⌘˜ .

= ⌧h eah /⌧h , the left-hand side is e

⌧w ˜m .

Thus, the first-order

By concavity, the first-order condition is sufficient for optimality.

3.7. Suppose all investors have shifted CRRA utility with the same coefficient ⇢ > 0. Suppose w ˜m > ⇣. Consider an allocation w ˜ h = ⇣ h + bh ( w ˜m where

PH

h=1 bh

⇣)

= 1. Show that the allocation is Pareto optimal. Hint: Show that it solves the social

planner’s problem with weights

h

defined as

h

= b⇢h .

The first-order condition for the social planner’s problem is (8 h) Setting w ˜ h = ⇣ h + bh ( w ˜m

⇣) and

order condition holds for ⌘˜ = (w ˜m

h

˜h hw

⇢

= ⌘˜ .

= b⇢h , the left-hand side is (w ˜m

⇣)

⇢.

⇣)

⇢.

Thus, the first-

By concavity, the first-order condition is sufficient for

optimality.

3.8. Show that if each investor has shifted CRRA utility with the same coefficient ⇢ > 0 and shift ⇣h , then, as asserted in Section 3.6, any Pareto optimal allocation involves an affine sharing rule. Pareto optimality implies the first-order condition (8 h) for some weights investors yields

h

˜h h (w

⇣h )

⇢

= ⌘˜

and Lagrange multiplier ⌘˜. This implies w ˜h

⇣h =

1/⇢ ˜ 1/⇢ h ⌘

and adding over

3 Equilibrium and Efficiency

w ˜m

⇣=

H X

1/⇢ h

h=1

so

H X

⌘˜ =

h=1

1/⇢ h

!⇢

!

1/⇢

⌘˜

(w ˜m

⇣)

33

,

⇢

.

Substituting this into the first-order condition yields 0 1⇢ H X 1/⇢ A ˜ h ⇣h ) ⇢ = @ (w ˜m h (w j

⇣)

⇢

,

j=1

which implies

w ˜h

⇣h = P H

1/⇢ h 1/⇢ j

j=1

(w ˜m

⇣) .

3.9. Consider an economy with date–0 consumption as in Section 3.7. Assume the investors have time-additive utility and the date–1 allocation solves the social planning problem (3.1). Using the first-order condition (2.390 ), show that the equilibrium allocation is Pareto optimal. Hint: Using the first-order condition (3.4) with ⌘˜ = ⌘˜1 , show that 0 ch0 ) h uh0 (˜

(8 h)

By (2.390 ),

u0h1 (˜ ch1 ) = 0 uh0 (ch0 )

= Rf E[˜ ⌘1 ] .

⌘˜1 0 h uh0 (ch0 )

is a stochastic discount factor. Thus E



⌘˜1 0 h uh0 (ch0 )

=

1 . Rf

0 ch0 ) h uh0 (˜

= ⌘0 ,

This implies

defining ⌘0 = Rf E[˜ ⌘1 ]. Thus, the first-order conditions for the social planner’s problem are satisfied. By concavity, this implies Pareto optimality.

4 Arbitrage and Stochastic Discount Factors

4.1. Assume there are two possible states of the world: !1 and !2 . There are two assets, a risk-free asset returning Rf in each state, and a risky asset with initial price equal to 1 and date–1 payo↵ x ˜. Let Rd = x ˜(!1 ) and Ru = x ˜(!2 ). Assume without loss of generality that Ru > Rd . (a) What conditions on Rf , Rd and Ru are equivalent to the absence of arbitrage opportunities? (b) Assuming the conditions from the previous part hold, compute the unique vector of state prices, and compute the unique risk neutral probabilities of states !1 and !2 . (c) Suppose another asset is introduced into the market that pays max(˜ x

K, 0) for some constant

K. Compute the price at which this asset should trade, assuming the conditions from part (a) hold. ˜ (a) The payo↵ of a zero-cost portfolio is (R

Rf ) for some . For this to be nonnegative in both

states and positive in one state, we must have either (i) and Rf

> 0 and Ru > Rd

Rf or (ii)

Rd . Thus, a necessary and sufficient condition for the absence of arbitrage

opportunities is that Ru > Rf > Rd . (b) Let qd denote the state price of state !1 and qu the state price of state !2 . The state prices satisfy q d Rf + q u Rf = 1 , q d Rd + q u Ru = 1 . The unique solution to this system of equations is qd =

Ru Rf , Rf (Ru Rd )

and

qu =

Rf Rd . Rf (Ru Rd )

36

4 Arbitrage and Stochastic Discount Factors

The risk neutral probabilities are qd Rf and qu Rf . (c) The asset should trade at qu max(xu

K, 0) + qd max(xd

K, 0), where xd denotes the value of

x ˜ in state 1 and xu the value of x ˜ in state 2.

4.2. Show that, if there is a strictly positive stochastic discount factor, then there are no arbitrage opportunities. If x ˜ is a nonnegative marketed payo↵, then its price is E[m˜ ˜ x]

0, and E[m˜ ˜ x] = 0 if and only if

x ˜ = 0 with probability one. Therefore, there are no arbitrage opportunities.

4.3. Show by example that the law of one price can hold but there can still be arbitrage opportunities. Suppose there are two possible states of the world, and the market consists of the two Arrow securities having prices pi . Then the market is complete, and each payo↵ x ˜ = (x1 , x2 ) has a unique cost p1 x1 + p2 x2 . If p1 < 0, then buying the first asset is an arbitrage opportunity.

4.4. Suppose there is no risk-free asset. For what value of ⌫ = E[m] ˜ does the projection m ˜ ⌫p equal the projection m ˜ p? In the absence of a risk-free asset, m ˜ ⌫p belongs to the span of the assets if and only if the constant in (4.9) is zero, i.e., if and only if ⌫

(p0

˜ 0 ])0 ⌃ 1 E[X ˜0] = 0 . ⌫E[X x

˜ 0 ] and p = p0 . Then, Set µ = E[X ⌫=

p0 ⌃ x 1 µ . 1 + µ0 ⌃ x 1 µ

Because m ˜ p is the unique stochastic discount factor in the span of the assets, we must have m ˜ ⌫p = m ˜ p.

4.5. Assume there are three possible states of the world: !1 , !2 , and !3 . Assume there are two assets: a risk-free asset returning Rf in each state, and a risky asset with return R1 in state !1 , R2 in state !2 , and R3 in state !3 . Assume the probabilities are 1/4 for state !1 , 1/2 for state !2 , and 1/4 for state !3 . Assume Rf = 1.0, and R1 = 1.1, R2 = 1.0, and R3 = 0.9.

4 Arbitrage and Stochastic Discount Factors

37

(a) Prove that there are no arbitrage opportunities. (b) Describe the one-dimensional family of state-price vectors (q1 , q2 , q3 ). (c) Describe the one-dimensional family of stochastic discount factors m ˜ = (m1 , m2 , m3 ) , where mi denotes the value of the stochastic discount factor in state !i . Verify that m1 = 4, m2 =

2, m3 = 4 is a stochastic discount factor.

(d) Consider the formula ˜ y˜)0 ⌃ 1 (X ˜ y˜p = E[˜ y ] + Cov(X, x

˜ E[X])

for the projection of a random variable y˜ onto the linear span of a constant and a random vector ˜ When the vector X ˜ has only one component x X. ˜ (is a scalar), the formula simplifies to y˜p = E[˜ y ] + (˜ x

E[˜ x]) ,

where =

cov(˜ x, y˜) . var(˜ x)

Apply this formula with y˜ being the stochastic discount factor m1 = 4, m2 =

2, m3 = 4 and

x ˜ being the risky asset return R1 = 1.1, R2 = 1.0, R3 = 0.9 to compute the projection of the stochastic discount factor onto the span of the risk-free and risky assets. (e) The projection in part (d) is by definition the payo↵ of some portfolio. What is the portfolio? ˜ denote the risky asset return. A zero-cost portfolio has payo↵ (R ˜ (a) Let R equals 0.1 in state 1, 0 in state 2, and ˜ (R

Rf ) for some . This

0.1 in state 3. Obviously, there is no

such that

Rf ) is nonnegative in all states and positive in some state.

(b) State prices must satisfy q1 + q2 + q3 = 1 1.1q1 + q2 + 0.9q3 = 1 . Subtracting the top from the bottom shows that q3 = q1 and substituting this into the first shows that q2 = 1

2q1 . q1 is arbitrary.

(c) Stochastic discount factors are given by m1 = q1 /(1/4) = 4q1 ,

m2 = q2 /(1/2) = 2

4q1 ,

m3 = q3 /(1/4) = 4q1 ,

38

4 Arbitrage and Stochastic Discount Factors

with q1 being arbitrary. Taking q1 = 1 yields m1 = 4, m2 =

2, m3 = 4.

˜ = 1 and E[m] (d) We have E[R] ˜ = 1 and ˜ y˜) = 1 (0.1)(3) + 1 (0)( 3) + 1 ( 0.1)(3) = 0 . cov(R, 4 2 4 Thus, the projection is m ˜ p = E[m] ˜ = 1. (e) m ˜ p is the payo↵ of holding the risk-free asset. 4.6. Suppose there is a risk-free asset and the risky asset returns have a joint normal distribution. Use the reasoning in Exercise 2.10 and the formula (5.31) for m ˜ p to show that the optimal portfolio of risky assets for an investor with zero labor income is ⇡ = ⌃

1 (µ

Rf 1) for some real number

, where ⌃ denotes the covariance matrix of the risky asset returns. Hint: if w ˜ and m ˜ are joint normal, then E[w| ˜ m] ˜ is the orthogonal projection of w ˜ on a constant and m—i.e., ˜ E[w| ˜ m] ˜ = E[w] ˜ +

cov(w, ˜ m) ˜ (m ˜ var(m) ˜

E[m]) ˜ .

For any feasible w, ˜ let w ˜ ⇤ = E[w ˜|m ˜ p ]. Then w ˜ equals w ˜ ⇤ plus mean-independent noise, so w ˜ ⇤ is preferred to w. ˜ From (5.31), m ˜p

E[m ˜ p] =

Hence, ⇤

w ˜ = E[w] ˜

1 Rf

✓

1 (µ Rf

cov(w, ˜ m ˜ p) var(m ˜ p)

Rf 1)0 ⌃ ◆

(µ

1

˜ vec (R

Rf 1)0 ⌃

1

µ) .

˜ vec (R

µ) .

This is in the span of the assets and budget feasible, because E[m ˜ pw ˜ ⇤ ] = E[m ˜ p w] by iterated expectations. Thus, w ˜ ⇤ is feasible. The portfolio of risky assets producing w ˜ ⇤ is ⌃ ✓ ◆ ✓ ◆ cov(w, ˜ m ˜ p) cov(w ˜⇤, m ˜ p) 1 1 = = , Rf var(m ˜ p) Rf var(m ˜ p)

1 (µ

Rf 1) for

the second equality following from iterated expectations. 4.7. Assume there is a finite number of assets, and the payo↵ of each asset has a finite variance. Assume the law of one price holds. Apply facts stated in Section 4.8 to show that there is a unique stochastic discount factor m ˜ p in the span of the asset payo↵s. Show that the orthogonal projection of any other stochastic discount factor onto the span of the asset payo↵s equals m ˜ p.

4 Arbitrage and Stochastic Discount Factors

39

The span of the assets is a finite-dimensional subspace of L2 . The law of one price states that there is a unique price C[˜ x] for each x ˜ in the span of the payo↵s. The function C[·] is linear. Therefore, it has a Riesz representation C[˜ x] = E[˜ xm ˜ p ] for a unique m ˜ p in the span of the assets. Given any stochastic discount factor m, ˜ we have m ˜ =m ˜ ⇤ + "˜, where the orthogonal projection m ˜ ⇤ is in the span of the assets and "˜ is orthogonal to the span of the assets. Hence, C[˜ x] = E[m˜ ˜ x] = E[˜ xm ˜ ⇤ ] for all x ˜ in the span of the assets. Thus, m ˜ ⇤ is also in the span of the assets and represents the price function. By the uniqueness of the Riesz representation, it must be that m ˜⇤ = m ˜ p.

5 Mean-Variance Analysis

5.1. Derive the minimum global variance portfolio directly by solving the problem: minimize ⇡ 0 ⌃⇡ subject to 10 ⇡ = 1. The first-order condition is 2⌃⇡ = 1. Thus, ⇡ = ( /2)⌃

1 1.

Imposing the constraint 10 ⇡ = 1

implies ⇡=

1 ⌃ 10 ⌃ 1 1

1

1.

5.2. Assume there is a risk-free asset. Consider an investor with quadratic utility, who seeks to maximize ⇣E[w] ˜

1 E[w] ˜2 2

1 var(w) ˜ . 2

(a) Show that the investor will choose a portfolio on the mean-variance frontier. (b) Assume (5.14) holds, so the tangency portfolio is efficient. Under what circumstances will the investor choose a mean-variance efficient portfolio? Explain the economics of the condition you derive. Hint: Compare Exercise 2.2. (a) From Exercise 2.2, the optimal portfolio is = This is proportional to ⌃

1 (µ

2 (⇣ 1 + 2

w0 Rf )⌃

1

(µ

Rf 1) .

Rf 1) and hence is on the mean-variance frontier.

(b) When the tangency portfolio is efficient, it is a positive proportion of ⌃ of the optimal portfolio’s proportion of ⌃

1 (µ

Rf 1) depends on ⇣

1 (µ

Rf 1). The sign

w0 Rf . When ⇣ > w0 Rf ,

the optimal portfolio is on the efficient part of the frontier, and when ⇣ < w0 Rf , the optimal portfolio is on the inefficient part of the frontier. ⇣ is the bliss level of wealth for the quadratic

42

5 Mean-Variance Analysis

utility function. When ⇣ < w0 Rf , the investor can exceed the bliss level by simply holding the risk-free asset. To avoid this, he holds an inefficient portfolio of risky assets.

5.3. Suppose that the risk-free return is equal to the expected return of the global minimum variance portfolio (Rf = B/C). Show that there is no tangency portfolio. Hint: Show there is no and

satisfying 1

⌃

(µ

Rf 1) = ⇡µ + (1

)⇡1 .

Recall that we are assuming µ is not proportional to 1. The mean-variance frontier considering only the risky assets is the set ⇡µ + (1 , and the mean-variance frontier including the risk-free asset is the set ⌃

)⇡1 for some

1 (µ

Rf 1) for some

. For the frontiers to intersect, we must have 1

⌃

(µ

Rf 1) = ⇡µ + (1

)⇡1 .

This is equivalent to ✓

10 ⌃

and premultiplying by ⌃ gives ✓

1µ

◆

10 ⌃

1

⌃

1µ

◆

µ=

✓

1 Rf + 0 1⌃

µ=

✓

Rf +

11

1 10 ⌃

11

◆

⌃

◆

1.

1

1,

Because µ is not proportional to 1, this equation can hold only if 10 ⌃ 1 µ

= Rf +

1 10 ⌃

11

= 0.

This implies 10 ⌃ 1 µ and substituting Rf = B/C = 10 ⌃

Rf +

1 µ/10 ⌃ 1 1

1 10 ⌃

11

= 0,

yields

1 = 0, 10 ⌃ 1 1 which is impossible.

5.4. Consider the problem of choosing a portfolio ⇡ of risky assets, a proportion and a proportion

`

0 to lend to maximize the expected return ⇡ 0 µ +

` R`

b

0 to borrow

b Rb

subject to the

5 Mean-Variance Analysis

constraints (1/2)⇡ 0 ⌃⇡  k and 10 ⇡ +

`

43

= 1. Assume B/C > Rb > R` , where B and C are

b

defined in (5.6). Define ⇡b = ⇡` =

1 10 ⌃ 1 (µ

Rb 1)

1 10 ⌃

1 (µ

R` 1)

⌃

1

(µ

Rb 1) ,

⌃

1

(µ

R` 1) .

Using the Kuhn-Tucker conditions, show that the solution is either (i) ⇡ = (1 (ii) ⇡ = ⇡` + (1

)⇡b for 0 

 1, or (iii) ⇡ = (1 +

b )⇡b

for

` )⇡`

for 0 

`

 1,

0.

b

The Kuhn-Tucker conditions are µ

⌃⇡

1 = 0,

R`

+ ⌘` = 0 ,

Rb +

+ ⌘b = 0 ,

`,

b , ⌘` , ⌘b ,

0,

1 0 ⇡ ⌃⇡  k , 2

⌘` If

`

> 0, then ⌘` = 0,

`

= ⌘b

b

10 ⇡ + ` b = 1, ✓ ◆ 1 0 = ⇡ ⌃⇡ k = 0 . 2

= R` , and 1 ⇡= ⌃

Also, If

1

= R` implies ⌘b = Rb R` > 0. Hence, b

> 0, then ⌘b = 0,

=

If

= `

=

`

1

This implies ⇡ = (1

(µ

` )⇡` .

Rb 1) .

= 0. This implies 10 ⇡ = 1+

b.

Hence, ⇡ = (1+

b )⇡b .

= 0, then 1 ⇡= ⌃

where

`.

Rb , and

Rb implies ⌘` = Rb R` > 0, so b

R` 1) .

= 0, and 10 ⇡ = 1

b

1 ⇡= ⌃ Also,

(µ

= R` + ⌘ `

R` and

= Rb

1

(µ

⌘b  Rb . Thus,

From 10 ⇡ = 1, it follows that ⇡ = ⇡` + (1

1) , = R` + (1

)⇡b .

5.5. Establish the properties claimed for the risk-free return proxies:

)Rb for some 0 

 1.

44

5 Mean-Variance Analysis

˜ (a) Show that var(R)

˜ p + bm e˜p ) for every return R. ˜ var(R

˜p, R ˜ p + bz e˜p ) = 0. (b) Show that cov(R ˜ p + bc e˜p represents the constant bc times the expectation operator (c) Prove (5.18), showing that R on the space of returns. ˜ p + b˜ (a) By Fact 15, the minimum variance return is R ep for some b. Using Fact 8, we have ˜ p + b˜ ˜p) var(R ep ) = var(R

˜ p ]E[˜ 2bE[R ep ] + var(˜ ep ) ,

and by Fact 17, this equals ⇣ ˜ p ) + b2 (1 var(R

⌘ ˜ p ] E[˜ 2bE[R ep ] .

E[˜ ep ])

By Fact 16, E[˜ ep ] > 0, so the minimum variance return is found by minimizing (b2 (1

E[˜ ep ])

˜ p ] in b, with solution b = bm . 2bE[R ˜p, R ˜ p + bz e˜p ) = var(R ˜p) (b) Using Fact 8, we have cov(R

˜ p ]E[˜ bz E[R ep ] = 0.

(c) Using Fact 11 and the definition of bc , we have ˜ = bc E[R ˜ p + b˜ ˜ p2 ] + bbc E[˜ bc E[R] ep + "˜] = E[R ep ] . From Facts 2, 8, 11, and 16, ˜ R ˜ p + bc e˜p )] = E[(R ˜ p + b˜ ˜ p + bc e˜p )] E[R( ep + "˜)(R ˜ 2 ] + bbc E[˜ = E[R e2p ] p ˜ 2 ] + bbc E[˜ = E[R ep ] . p Thus, ˜ = E[R( ˜ R ˜ p + bc e˜p )] . bc E[R]

5.6. Show that x ˜=

1 (1 ˜p] E[R

e˜p )

˜ as R ˜ p + (R ˜ is a stochastic discount factor. Hint: Write any return R 1

˜ p ) and use the fact that R

e˜p is orthogonal to excess returns (because e˜p represents the expectation operator on the space

of excess returns). When there is a risk-free asset, x ˜, being spanned by a constant and an excess return, is in the span of the returns and hence must equal m ˜ p . Use this fact to demonstrate (5.20).

5 Mean-Variance Analysis

45

We have ˜ = E[˜ xR]

1 E[(1 ˜p] E[R

˜p] + e˜p )R

=

1 E[(1 ˜p] E[R

˜p] e˜p )R

1 E[(1 ˜p] E[R

˜ e˜p )(R

˜ p )] R

= 1, ˜ using E[R

˜ p ] = E[˜ ˜ R ep (R

˜ p ] for the second equality and Fact 8 for the third. Thus, x R ˜ is a

stochastic discount factor. This implies 1 E[˜ ep ] 1 = E[˜ x] = . ˜ Rf E[Rp ] Moreover, x ˜=m ˜ p implies ˜p = R

x ˜ , E[˜ x2 ]

and E[˜ x2 ] =

1 (1 ˜ E[Rp ]2

2E[˜ ep ] + E[˜ e2p ]) =

1 E[˜ ep ] 1 = , 2 ˜ ˜p] E[Rp ] Rf E[R

using Fact 16 for the second equality. Thus, ˜ p = Rf E[R ˜p] R

˜2] 5.7. Show that E[R

1 (1 ˜p] E[R

e˜p )

!

= Rf (1

e˜p ) .

˜ 2 ] for every return R ˜ (thus, R ˜ p is the “minimum second-moment E[R p

˜ 2 ] = a, which return”). The returns having a given second moment a are the returns satisfying E[R is equivalent to ˜ + E[R] ˜ 2 = a; var(R) ˜p thus, they plot on the circle x2 + y 2 = a in (standard deviation, mean) space. Use the fact that R ˜ p must be on the inefficient is the minimum second-moment return to illustrate graphically that R ˜ p ] > 0 in the absence of a part of the frontier, with and without a risk-free asset (assuming E[R risk-free asset). Using Facts 1, 2 and 8, ˜ 2 ] = E[(R˜p + b˜ ˜ p2 ] + b2 E[˜ E[R ep + "˜)2 ] = E[R e2p ] + E[˜ "2 ]

˜ p2 ] . E[R

With a risk-free asset, the cone intersects the vertical axis at Rf > 0, and the point on the cone ˜p] > 0 closest to the origin is on the lower part. In the absence of a risk-free asset, the assumption E[R

46

5 Mean-Variance Analysis

implies that global minimum variance portfolio has a positive expected return (use the definition of bm and Facts 16 and 17 — which imply 1

E[˜ ep ] > 0 — to deduce this). Thus, the point on the

hyperbola closest to the origin must be on the lower part of the hyperbola. ˜ p , e˜p and "˜ are joint normally distributed 5.8. If all returns are joint normally distributed, then R ˜=R ˜ p + b˜ ˜ (because R ˜ p is a return and e˜p in the orthogonal decomposition R ep + "˜ of any return R and "˜ are excess returns). Assuming all returns are joint normally distributed, use the orthogonal decomposition to compute the optimal return for a CARA investor. ˜ that maximizes When returns are normally distributed, a CARA investor chooses the return R ˜ E[R]

1 ˜ . ↵w0 var(R) 2

˜=R ˜ p + b˜ Given R ep + "˜ and Facts 11 and 15, the objective function is ˜ + b˜ E[R ep ]

1 ˜ p + b˜ ↵w0 [var(R ep ) + var(˜ ")] , 2

so it is optimal to choose "˜ = 0. The investor chooses b to maximize bE[˜ ep ]

1 ˜ p , e˜p ) + b2 var(˜ ↵w0 [2b cov(R ep )] , 2

and the optimum satisfies E[˜ ep ]

˜ p , e˜p ) ↵w0 cov(R

implying b=

E[˜ ep ] ↵w0 var(˜ ep )

↵w0 var(˜ ep )b = 0 ,

˜ p , e˜p ) cov(R . var(˜ ep )

Using Facts 8 and 17, we can simplify this further to b=

˜p] 1 + ↵w0 E[R . ↵w0 (1 E[˜ ep ])

˜ p is on the mean-variance frontier. Hence, when there is a risk-free asset, it must 5.9. The return R be the return of a weighted average of the tangency portfolio and the risk-free asset. The purpose of this exercise is to compute the weighted average. When there is a risk-free asset, the projection m ˜ p defined in Section 4.4 is the same as the projection m ˜ ⌫p defined in Section 4.5. Furthermore, the ˜ vec of vector of asset payo↵s in the formula (4.10) for m ˜ ⌫p can always be replaced by the vector R

5 Mean-Variance Analysis

47

returns whenever the asset prices are positive, because the linear span of the returns is the same as the linear span of the payo↵s. Substituting the returns for the payo↵s and substituting E[m] ˜ = 1/Rf ˜ vec ] = 1 in (4.10), we have and E[m ˜R ✓ 1 m ˜p = + 1 Rf

1 µ Rf

◆0

˜p = ⇡0 R ˜ vec + (1 Using this formula, show that R ⇤

1

⌃

˜ vec (R

)Rf for some

µ) . (i.e., calculate ).

We have m ˜p = Hence

1 ⇥ 1 + (µ Rf

Rf 1)0 ⌃

1

µ

⇤

var(m ˜ p) = where 2 = (µ Rf 1)0 ⌃

1 (µ

1 (µ Rf

Rf 1)0 ⌃

1

˜ vec . R

2 , Rf2

Rf 1) is the squared maximum Sharpe ratio. Because E[m ˜ p ] = 1/Rf ,

this implies E[m ˜ 2p ] =

1 + 2 . Rf2

Therefore ⇤ Rf ⇥ Rf 1 + (µ Rf 1)0 ⌃ 1 µ (µ Rf 1)0 ⌃ 2 1+ 1 + 2 Rf (1 + A Rf B) Rf (B Rf C) 0 ˜ vec = ⇡⇤ R , 1 + 2 1 + 2

˜p = R

1

˜ vec R

in the notation of Section 5.3. Setting = we have 1 because 2 = A

=

Rf (B Rf C) , 1 + 2

1 + 2 + R f B 1+

Rf2 C

2

=

1 + A Rf B , 1 + 2

2Rf B + Rf2 C. Thus, ˜ p = (1 R

˜ vec . )Rf + ⇡⇤0 R

5.10. Assuming there is a risk-free asset, show that

in Exercise 5.9 is negative when Rf
B/C. (This verifies that the portfolio generating R tangency portfolio when the tangency portfolio is efficient and long the tangency portfolio when it is inefficient.)

48

5 Mean-Variance Analysis

=

Rf (B Rf C) Rf C and positive when B < Rf C.

6 Beta Pricing Models

6.1. Suppose there is a risk-free asset. Use the formula (5.9) for frontier portfolios to show that ˜ ⇤ is on the mean-variance frontier. the beta-pricing model (6.5) implies the return R ˜ ⇤ = Rf + ⇡ 0 (Rvec Let R

Rf 1). Stacking the formula (6.5) for assets 1, . . . , n gives µ = Rf 1 +

˜⇤] E[R

Rf ⌃⇡ , ⇤ ˜ var(R )

so ⇡= ⌃ where =

1

(µ

Rf 1) ,

˜⇤) var(R ⇡ 0 ⌃⇡ = 0 . ˜ ⇤ ] Rf ⇡ (µ Rf 1) E[R

6.2. Suppose there is no risk-free asset. Use the formula (5.1) for frontier portfolios to show that ˜ ⇤ being on the mean-variance frontier and a beta-pricing model (6.5) is equivalent to the return R not equal to the global minimum variance return. ˜ ⇤ = ⇡ 0 Rvec . Stacking the formula (6.5) for assets 1, . . . , n gives Let R µ = ↵1 +

˜⇤] ↵ E[R ⌃⇡ , ˜⇤) var(R

so ⇡= ⌃ where = To have ⇡ = ⇡1 , we must have

˜⇤) var(R ˜ ⇤ ] Rf E[R

1

µ+ ⌃

and

= 0, which implies

=

1

1,

↵

˜⇤) var(R . ˜ ⇤ ] Rf E[R

= 0 and ⇡ = 0, contradicting 10 ⇡ = 1.

50

6 Beta Pricing Models

6.3. Suppose investors can borrow and lend at di↵erent rates. Let Rb denote the return on borrowing and R` the return on lending. Suppose B/C > Rb > R` , where B and C are defined in (5.6). Suppose each investor chooses a mean-variance efficient portfolio, as described in Exercise 5.4. Show that the CAPM holds with R`  Rz  Rb . From Exercise 5.4, the optimal portfolio of risky assets for each investor h is h [ h ⇡`

where 0 

h

 1 and

h

+ (1

h )⇡b ] ,

0 is the fraction of investor h’s wealth that is invested in risky assets. It

follows that the market portfolio of risky assets is ⇡` + (1

)⇡b for some 0 

 1. The vector

of covariances of the risky asset returns with the market return is therefore ˜ vec , R ˜ m ) = ⌃[ ⇡` + (1 Cov(R Define ✓b = /(B

)⇡b ] =

CRb ) and ✓` = (1 µ=

B

)/(B

CRb

(µ

Rb 1) +

1 B

CR`

(µ

R` 1) .

CR` ). Then, we have

1 ˜ vec , R ˜ m ) + ✓ b Rb + ✓ ` R` 1 . Cov(R ✓b + ✓` ✓b + ✓`

Thus, Rz =

✓ b Rb + ✓ ` R ` , ✓b + ✓`

which is a convex combination of Rb and R` .

6.4. Assuming normally distributed returns, no end-of-period endowments, and investors with CARA utility, derive the CAPM from the portfolio formula (2.27), i.e., from h

=

1 ⌃ ↵h

1

(µ

Rf 1) ,

where ↵h denotes the absolute risk aversion of investor h. Show that the factor risk premium is ˜ m ), where ↵ is the aggregate absolute risk aversion defined in Section 1.3 and w0 = ↵w0 var(R P 10 H h=1 h is the market value of risky assets at date 0. Setting

=

PH

h=1

h

and adding the optimal portfolios over investors gives =

1 ⌃ ↵

1

(µ

Rf 1) ,

which in equilibrium is the vector of market values of risky assets at date 0. Thus,

6 Beta Pricing Models

51

µ = Rf 1 + ↵⌃ . ˜m = The market portfolio is ⇡ = (1/10 ) . The return on the market portfolio is R

0R ˜ vec / 0 1,

and

the vector of betas of the asset returns with respect to the market return is 1 ⇡ 0 ⌃⇡

⌃⇡ .

We have µ = Rf 1 + ↵(10 )⌃⇡ = ↵(10 )(⇡ 0 ⌃⇡)

1 ⇡ 0 ⌃⇡

⌃⇡ ,

so the factor risk premium is ˜m) . ↵(10 )(⇡ 0 ⌃⇡) = ↵w0 var(R ˜ ⇤ that is on the mean-variance frontier and is an affine function 6.5. Assume there exists a return R ˜ = a + b0 F˜ . Assume either (i) there is a risk-free asset and R ˜⇤ = of a vector F˜ ; i.e., R 6 Rf , or (ii) ˜ ⇤ is di↵erent from the global minimum variance return. Show that there is no risk-free asset and R there is a beta pricing model with factors F˜ . Note: In the context of a factor model with factors ˜ = a + b0 F˜ is called well diversified, because it has no idiosyncratic risk. If there is F˜ , a return R a finite number of assets satisfying a factor model, then there is no risky well diversified return, P P 2 because var( ⇡ "˜i ) = ⇡ var(˜ "i ) > 0 if ⇡ 6= 0. However, if there is an infinite number of assets, then one can take ⇡i = 1/n for i = 1, . . . , n and n ! 1 to obtain a well diversified limit return.

˜ ⇤ as the factor. Thus, for Under either assumption (i) or (ii), there is a beta pricing model with R ˜ every return R, ˜ = ↵ + cov(R ˜ ⇤ , R) ˜ E[R] ˜ = ↵ + b0 Cov(F˜ , R) ˜ , = ↵ + ˆ 0 ⌃F 1 Cov(F˜ , R) where we define ˆ = ⌃F b . ˜ i for i = 1, . . . , n satisfy 6.6. Assume the asset returns R ˜ i = E[R ˜ i ] + Cov(F˜ , R ˜ i )0 ⌃ 1 (F˜ R F

E[F˜ ]) + "˜i ,

52

6 Beta Pricing Models

where each "˜i is mean-independent of the factors F˜ , i.e., E[˜ "i |F˜ ] = 0 (note it is not being assumed that cov(˜ "i , "˜j ) = 0). Assume markets are complete, investors have strictly monotone preferences, and the market return is well diversified in the sense of having no idiosyncratic risk: ˜ m = E[R ˜ m ] + Cov(F˜ , R ˜ m )0 ⌃ 1 (F˜ R F

E[F˜ ]) .

Show that there is a beta pricing model with factors F˜ . Hint: Pareto optimality implies sharing rules w ˜ h = fh ( w ˜m ). Complete markets and strictly monotone preferences implies Pareto optimality. Given the sharing rules, the marginal utility of any investor depends only on w ˜m and hence depends only on F˜ . Thus, there is a stochastic discount factor that is proportional to an investor’s marginal utility and equal to m ˜ = g(F˜ ) for some function g. By the strict monotonicity of preferences, m ˜ is strictly positive. Hence, E[m] ˜ 6= 0. This implies a beta pricing model with m ˜ as the factor. This further implies an APT pricing formula with pricing errors

E[m˜ ˜ "i ]/E[m]. ˜ The pricing errors are zero, because,

E[m˜ ˜ "i ] = E[E[g(F˜ )˜ "i | F˜ ] = E[g(F˜ )E[˜ "i | F˜ ] = 0 . Thus, there is exact APT pricing, i.e., a beta pricing model with F˜ as the factors. 6.7. Suppose two assets satisfy a one-factor model: ˜ 1 = E[R ˜ 1 ] + f˜ + "˜1 , R ˜ 2 = E[R ˜2] R

f˜ + "˜2

where E[f˜] = E["˜1 ] = E[˜ "2 ] = 0, var(f˜) = 1, cov(f˜, "˜1 ) = cov(f˜, "˜2 ) = 0, and cov(˜ "1 , "˜2 ) = 0. Assume var(˜ "1 ) = var(˜ "2 ) =

2.

˜⇤ = R ˜ 1 and R ˜ ⇤ = ⇡R ˜ 1 + (1 Define R 1 2

˜ 2 with ⇡ = 1/(2 + ⇡)R

2 ).

˜ ⇤ and R ˜ ⇤ do not satisfy a one-factor model with factor f˜. (a) Show that R 1 2 ˜ ⇤ and R ˜ ⇤ satisfy a zero-factor model, i.e., (b) Show that R 1 2 ˜ ⇤ = E[R ˜ ⇤ ] + "˜⇤ , R 1 1 1 ˜ 2⇤ = E[R ˜ 2⇤ ] + "˜⇤2 , R where E["˜⇤1 ] = E[˜ "⇤2 ] = 0 and cov(˜ "⇤1 , "˜⇤2 ) = 0. (c) Assume exact APT pricing with nonzero risk premium ˜i] model, i.e., E[R

Rf =

for the two assets in the one-factor

˜ i , f˜) for i = 1, 2. Show that there cannot be exact APT pricing cov(R

˜ ⇤ and R ˜⇤. in the zero-factor model for R 1 2

6 Beta Pricing Models

1)f˜ + "˜⇤2 , where "˜⇤2 = ⇡ "˜1 + (1

˜ ⇤ = E[R ˜ ⇤ ] + (2⇡ (a) R 2 2 cov(˜ "1 , "˜⇤2 ) = ⇡

2

53

⇡)˜ "2 is uncorrelated with f˜. However,

6= 0.

(b) Define "˜⇤1 = f + "˜1 and "˜⇤2 = (2⇡

1)f + ⇡ "˜1 + (1

⇡)˜ "2 . Then

˜ 1⇤ = E[R ˜ 1⇤ ] + "˜⇤1 , R ˜ ⇤ = E[R ˜ ⇤ ] + "˜⇤ , R 2 2 2 and cov(˜ "⇤1 , "˜⇤2 ) = (2⇡

1) var(f˜) + ⇡

2

= 2⇡

1+⇡

2

= 0.

˜ ⇤ ] = E[R ˜ ⇤ ] = Rf , but E[R ˜ ⇤ ] = Rf + . (c) Exact APT pricing with zero factors means E[R 1 2 1

6.8. Assume there are H investors with CARA utility and the same absolute risk aversion ↵. Assume there is a risk-free asset. Assume there are two risky assets with payo↵s x ˜i that are joint normally distributed with mean vector µ and nonsingular covariance matrix ⌃. Assume HU < H investors are unaware of the second asset and invest only in the risk-free asset and the first risky asset. If all investors invested in both assets (HU = 0), then the equilibrium price vector would be p⇤ =

1 µ Rf

↵ ⌃ ✓¯ , HRf

where ✓¯ is the vector of supplies of the risky assets (see Exercise 3.4). Assume HU > 0, and set HI = H

HU .

(a) Show that the equilibrium price of the first asset is p1 = p⇤1 , and the equilibrium price of the second asset is p2 =

p⇤2

↵ HRf

(b) Show that there exist A > 0 and

✓

HU HI

◆✓ var(˜ x2 )

cov(˜ x1 , x ˜ 2 )2 var(˜ x1 )

◆

< p⇤2 .

such that

˜1, R ˜m) cov(R , ˜m) var(R ˜ ˜ ˜ 2 ] = A + Rf + cov(R2 , Rm ) , E[R ˜m) var(R ˜ 1 ] = Rf + E[R

˜m] = E[R

Rf

A⇡2 ,

where ⇡2 = p2 ✓¯2 /(p1 ✓¯1 + p2 ✓¯2 ) is the relative date–0 market capitalization of the second risky asset. Note that relative to .

is less than in the CAPM, and the second risky asset has a positive “alpha,”

54

6 Beta Pricing Models

(a) The optimal portfolio of investors who do not invest in the second risky asset is ✓U = (✓U 1 0)0 , where ✓U 1 =

E[˜ x 1 ] R f p1 . ↵ var(˜ x1 )

The optimal portfolio of investors who invest in both risky assets satisfies ↵⌃✓I = µ Let

ij

Rf p .

denote the (i, j)–th element of ⌃. The market clearing condition ✓¯ = HI ✓I + HU ✓U

implies 0

Thus,

1 1 0 HU @ A (µ ↵⌃ ✓¯ = HI ↵⌃✓I + HU ↵⌃✓U = HI (µ Rf p) + ⌃ 11 0 0 0 1 H 0 A (µ Rf p) . =@ HU 12 / 11 HI 0

Rf p = ↵ @

µ

0

H HU

0

12 / 11

1/H

= ↵@

HU HHI

HI 0

1

1

⌃ ✓¯

A

1

A ⌃ ✓¯

1/HI 0 1 0 13 1 0 0 0 1 H U @ A+ A5 ⌃ ✓¯ = ↵4 @ H 0 1 HHI 12 1 11 0 1 0 ↵ H U ↵ @0 A ✓¯ = ⌃ ✓¯ + 2 H HHI 0 12 2

12 11

22

Hence, p=

1 µ Rf

Note that p2 < p⇤2 because ˜i = x (b) Set R ˜i /pi and

↵ ⌃ ✓¯ HRf 11 12

>

↵ HRf

✓

HU HI

11

◆✓

22

2 12 11

◆

2 12 .

¯ ˜1 + ✓¯2 x ˜2 ˜ m = ✓1 x R . ¯ ¯ ✓ 1 p 1 + ✓ 2 p2

We have pi ˜i, R ˜m) = cov(R ( ¯ ✓1 p1 + ✓¯2 p2

¯ +

i1 ✓1

¯

i2 ✓2 ) .

0 1 0 @ A. ✓¯2

Rf p)

6 Beta Pricing Models

The formula for p in the previous part implies ↵ ↵ µ = Rf p + ⌃ ✓¯ + H H

✓

HU HI

◆✓

2 11 2

2 12 11

◆

Thus,

55

0 1 0 @ A. ✓¯2

↵ ¯ ˜1, R ˜m) (✓1 p1 + ✓¯2 p2 ) cov(R H ˜1, R ˜m) cov(R = Rf + , ˜m) var(R

˜ 1 ] = Rf + E[R

where =

↵ ¯ ˜m) . (✓1 p1 + ✓¯2 p2 ) var(R H

Also, ✓ ◆✓ ◆ 2 2 ↵ ¯ ↵ HU 11 2 12 ¯ ˜ ˜ ˜ E[R2 ] = Rf + (✓1 p1 + ✓2 p2 ) cov(R2 , Rm ) + ✓¯2 H Hp2 HI 11 ! ✓ ◆ ˜2, R ˜m) ˜ 1 ) var(R ˜ 2 ) cov(R ˜1, R ˜ 2 )2 cov(R ↵ HU var(R + ✓¯2 p2 = Rf + ˜m) ˜1) H HI var(R var(R = Rf +

˜2, R ˜m) cov(R + A, ˜m) var(R

where ↵ A= H

✓

HU HI

◆

˜ 1 ) var(R ˜ 2 ) cov(R ˜1, R ˜ 2 )2 var(R ˜1) var(R

!

✓¯2 p2 > 0 .

It follows that ˜ m ] = ⇡1 E[R ˜ 1 ] + ⇡2 E[R ˜ 2 ] = Rf + E[R

+ ⇡2 A .

Hence, ˜m] = E[R

Rf

˜m] ⇡2 A < E[R

Rf .

6.9. Suppose there is no risk-free asset and the minimum-variance return is di↵erent from the constant-mimicking return, i.e., bm 6= bc . From Section 6.2, we know there is a beta pricing model with the constant-mimicking return as the factor: ˜ =↵+ E[R]

˜ R ˜ p + bc e˜p ) cov(R,

˜ From Section 6.6, we can conclude there is a stochastic discount factor that is an for every return R. affine function of the constant-mimicking return unless ↵ = 0. However, the existence of a stochastic discount factor that is an affine function of the constant-mimicking return would contradict the result of Section 5.11. So, it must be that ↵ = 0 in (6.25). Calculate ↵ to demonstrate this.

56

6 Beta Pricing Models

We have ˜ R ˜ p + bc e˜p ) = E[R( ˜ R ˜ p + bc e˜p )] cov(R, ˜ = bc E[R]

˜ R ˜ p + bc e˜p ] E[R]E[

˜ R ˜ p + bc e˜p ] . E[R]E[

Thus, ˜ =↵+ E[R]

˜ R ˜ p + bc e˜p ) cov(R,

implies ˜ E[R]{1

˜ p + bc e˜p ]} = ↵ . bc + E[R

˜ only if This can be true for all R 1

˜ p + bc e˜p ] = 0 bc + E[R

and

↵ = 0.

Note that this implies =

bc

1 . ˜ E[Rp + bc e˜p ]

The denominator of this expression is nonzero because bc 6= bm . 6.10. Suppose there is no risk-free asset and the minimum-variance return is di↵erent from the constant-mimicking return, i.e., bm 6= bc . From Section 5.11, we know that there is a stochastic discount factor that is an affine function of the minimum-variance return: m ˜ = for some

and

˜ p + bm e˜p ) + (R

. From Section 6.6, we can conclude there is a beta pricing model with the

minimum-variance return as the factor unless E[m] ˜ = 0. However, this would contradict the result of Section 6.2. So it must be that E[m] ˜ = 0 for the stochastic discount factor m ˜ in (6.26). Calculate E[m] ˜ to demonstrate this. We have E[m] ˜ = = =

˜ p ] + bm E[˜ + E[R ep ] ˜ p ]E[˜ E[R ep ] 1 E[˜ ep ] ˜p] E[˜ ep ]) + E[R . 1 E[˜ ep ]

˜p] + + E[R (1

6 Beta Pricing Models

57

Because m ˜ is a stochastic discount factor and e˜p is an excess return, 0 = E[m˜ ˜ ep ] ˜ p e˜p ] + bm E[˜ = E[˜ ep ] + E[R e2p ] = E[˜ ep ] + bm E[˜ ep ] = E[˜ ep ] + = E[˜ ep ]

(1

˜ p ]E[˜ E[R ep ] 1 E[˜ ep ] ˜p] E[˜ ep ]) + E[R 1 E[˜ ep ]

= E[˜ ep ]E[m] ˜ , using Facts 8 and 16 for the third equality. Because E[˜ ep ] 6= 0 (by Fact 17), this implies E[m] ˜ = 0.

7 Representative Investors

7.1. Show that if uh0 and uh1 are concave for each h, then the social planner’s utility functions u ˆ0 and u ˆ1 are concave. Consider u ˆ0 . The argument is identical for u ˆ1 . For convenience, drop the subscript 0 from u ˆ0 and uh0 . Consider two aggregate consumption levels c1 and c2 . Let {ch1 | h = 1, . . . , H} and {ch2 | h = P PH 1, . . . , H} satisfy H h=1 ch1 = c1 and h=1 ch2 = c2 . Let 0 < < 1. We have u ˆ( c1 + (1

)c2 )

H X

h uh (

h=1 H X

ch1 + (1

h uh (ch1 )

+ (1

h=1

using the definition of u ˆ — and the fact that

)ch2 ) )

H X

h uh (ch1 ) ,

h=1

PH

ch1 + (1

h=1

)ch2 = c1 + (1

)c2 — for the first

inequality and concavity of the uh for the second. Because this is true for every {ch1 | h = 1, . . . , H} P PH satisfying H h=1 ch1 = c1 and for every and {ch2 | h = 1, . . . , H} satisfying h=1 ch2 = c2 , we have u ˆ( c1 + (1

)c2 )

u ˆ(c1 ) + (1

)ˆ u(c2 ) .

7.2. Use the results on affine sharing rules in Section 3.6 to establish (b0 ) in Section 7.2. The solution of the social planner’s problem in each period is ch = ⇣h + bh (c where

⇣) ,

1/⇢

bh = P h H

j=1

1/⇢ j

.

60

7 Representative Investors

For shifted log utility, we have u ˆ(c) =

=

H X

h log(bh (c

h=1 H X

h log bh

⇣)) H X

+

h=1

Thus, ignoring the constants dates 0 and 1 is

PH

H X

h

h=1

!

h=1

h bh ,

h=1

h

!

log(c

⇣) .

the representative investor’s utility for consumption at

log(c0

H X

⇣) +

h

h=1

which is an affine transform of

log(c0

!

⇣) + log(c1

log(c1

⇣) ,

⇣) .

For shifted power utility, we have u ˆ(c) =

=

H X

1 h

h=1 H X

1

⇢

1 ⇢ h bh

h=1

⇣))1

(bh (c !

1 1

⇢

(c

⇢

⇣)1

⇢

.

Thus, the representative investor’s utility for consumption at dates 0 and 1 is, up to an affine transform, 1 1

⇢

(c0

⇣)1

⇢

+

1

⇢

(c1

⇣)1

⇢

.

7.3. Suppose there is a representative investor with quadratic utility u(w) = E[w ˜m ] 6= ⇣. Show that

(⇣

w)2 . Assume

in the CAPM (6.12) equals var(w ˜m ) , E[⌧ (w ˜m )]

where ⌧ (w) denotes the coefficient of risk tolerance of the representative investor at wealth level w. Thus, the risk premium is higher when market wealth is riskier or when the representative investor is more risk averse. We have u0 (w) = 2(⇣

w), u00 (w) =

discount factor m ˜ = u0 ( w ˜m ) = 2 (⇣ E[w ˜m ] 6= ⇣. Thus,

2, and ⌧ (w) =

u0 (w)/u00 (w) = ⇣

w. There is a stochastic

w ˜m ) for some , and E[m] ˜ 6= 0 by virtue of the assumption

7 Representative Investors

1 ˜ cov(m, ˜ R) E[m] ˜ ˜ We have E[m] ˜ = for each return R. ˜ = 2 E[⌧ (w ˜m )] and cov(m, ˜ R)

61

1 E[m] ˜

˜ = E[R]

˜ Therefore, 2 cov(w ˜m , R).

1 1 ˜ + cov(m, ˜ R) 2 E[⌧ (w ˜m )] E[⌧ (w ˜m )] ˜ 1 var(w ˜m ) cov(m, ˜ R) = + . 2 E[⌧ (w ˜m )] E[⌧ (w ˜m )] var(w ˜m )

˜ = E[R]

Therefore,

= var(w ˜m )/E[⌧ (w ˜m )].

7.4. Suppose returns and end-of-period endowments are joint normally distributed and there is a representative investor with constant absolute risk aversion ↵. Show that

in the CAPM (6.12)

equals ↵ var(w ˜m ). Thus, the risk premium is higher when market wealth is riskier or when the representative investor is more risk averse. We have ˜ = E[R] where m ˜ = ↵e

↵w ˜m

1 E[m] ˜

1 ˜ , cov(m, ˜ R) E[m] ˜

for some . By Stein’s Lemma, cov( ↵e

↵w ˜m

˜ = , R)

↵2 E[e

↵w ˜m

˜ . ] cov(w ˜m , R)

Thus,

This implies

˜ = E[R]

1 ↵E[e

=

1 ↵E[e

↵w ˜m ] ↵w ˜m ]

˜ + ↵ cov(w ˜m , R) + ↵ var(w ˜m )

˜ cov(w ˜m , R) . var(w ˜m )

= ↵ var(w ˜m ).

˜ and log(˜ 7.5. Assume in (7.11) that log R c1 /c0 ) are joint normally distributed. Specifically, let ˜ = ey˜ and c˜1 /c0 = ez˜ with E[˜ R y ] = µ, var(˜ y) =

2,

2 c,

E[˜ z ] = µc , var(˜ z) =

and corr(˜ y , z˜) = .

(a) Show that µ=

log + ⇢

c

1 2 ⇢ 2

+ ⇢µc

2 c

1 2

2

.

(b) Let r = log Rf denote the continuously compounded risk-free rate. Show that r=

1 2 ⇢ 2

log + ⇢µc

µ=r+⇢

c

1 2

2

.

2 c ,

62

7 Representative Investors

Note that

c

is the covariance of the continuously compounded return y with the continuously

compounded consumption growth rate z, so (7.21b) has the usual form Expected Return = Risk-Free Return + ✓ ⇥ Covariance , 2 /2.

with ✓ = ⇢, except for the extra term

The extra term, which involves the total and hence

idiosyncratic risk of the return, is usually called a Jensen’s inequality term, because it arises from the fact that E[ey˜] = eµ+

2 /2

> eµ .

(a) (7.11) states that ˜ = E[R] ˜ = eµ+ We have E[R]

2 /2

. Also, c1 ⇢

eµ+

2 /2

c0 ⇢ h i E c˜1 ⇢

1 = e⇢µc

2 /2

⇢2

2 c /2

h

⇢2

⇢˜ z

⇣ ⌘ ˜ c˜ ⇢ . i cov R, 1

h i ˜c ⇢ E R˜ 1 h i + eµ+ E c˜1 ⇢

2 c /2

2 /2

⇢˜ z]

⇢µc + 12 ( 2 2⇢ exp ⇢µc + 12 ⇢2

⇢

c+

2 /2

c

+ ⇢2

2 c

.

Thus, eµ

⇢

c+

2 /2

1 = e⇢µc

⇢2

2 c /2

,

which implies µ=

.

⇤

exp µ

= eµ

⇢µc +⇢2

h i h i ˜ E c˜ ⇢ yields E R 1

Also, h i ⇥ ˜c ⇢ E R˜ 1 E ey˜ h i = E [e E c˜1 ⇢

1

E c˜1 ⇢

1 = e⇢µc

=

⇣ ⌘ ˜ c˜ ⇢ . i cov R, 1

1

E c˜1 ⇢ h i = c0 ⇢ e ⇢˜z , so E c˜1 ⇢ = c0 ⇢ e

⇣ ⌘ h i ˜ c˜ ⇢ = E R˜ ˜c ⇢ Substituting cov R, 1 1 eµ+

h

log + ⇢

c

+ ⇢µc

1 2 ⇢ 2

2 c

1 2

2

.

2 c)

2 c /2

. Hence,

7 Representative Investors

(b) From Rf = h i and E c˜1 ⇢ = c0 ⇢ e

⇢µc +⇢2

2 c /2

1 c0 ⇢ h i, = E[m] ˜ E c˜1 ⇢

, we obtain Rf = e⇢µc

⇢2

2 c /2

/ . Hence, r =

Substituting r in the formula for µ obtained in Part (a) yields µ = r + ⇢

log +⇢µc ⇢2 c

63

2 c /2.

2 /2.

7.6. Suppose there is a risk-free asset and a representative investor with power utility, so (7.10) is a stochastic discount factor. Let z˜ = log(c˜1 /c0 ) and assume z˜ is normally distributed with mean µc and variance

2 c.

Let  denote the maximum Sharpe ratio of all portfolios. Use the Hansen-

Jagannathan bound (4.13) to show that p log(1 + 2 )

⇢

.

c

Hint: Apply the result of Exercise 1.15. Note that (7.22) implies risk aversion must be larger if consumption volatility is smaller or the maximum Sharpe ratio is larger. Also, using the approximation log(1 + x) ⇡ x (a first-order Taylor series approximation of log(1 + x) around x = 0), the lower bound on ⇢ in (7.22) is approximately / c . We have

p stdev e ⇢˜z stdev(m) ˜ = = e ⇢2 ⇢˜ z E[m] ˜ E [e ]

2 c

1,

using the result of Exercise 1.15. Thus, the Hansen-Jagannathan bound (4.13) implies p e ⇢2

which implies

e⇢ and ⇢

2 2 c

2 c

1

,

1 + 2 ,

p log(1 + 2 )

.

c

7.7. Suppose there is a risk-free asset in zero net supply and the risky asset returns have a factor structure ˜ i = ai + b0i F˜ + "˜i , R where the "˜i have zero means and are independent of each other and of F˜ . Assume there are no end-of-period endowments and there is a representative investor with CARA utility. Let ↵ denote

64

7 Representative Investors

the risk aversion of the representative investor. Let ⇡ ⇤ denote the vector of market weights. Denote ˜ m . Let initial market wealth by w0 and end-of-period market wealth by w ˜ m = w0 R APT pricing error defined in Section 6.7. Assume "˜i

i

denote the

with probability one, for some constant .

Via the following steps, show that | i| 

↵w0 ⇡i⇤ exp(↵ w0 ⇡i⇤ ) var(˜ "i ) . Rf

(a) Show that i

=

E [exp( ↵w ˜m )˜ "i ] . Rf E [exp( ↵w ˜m )]

(b) Show that i

=

E [exp( ↵w0 ⇡i⇤ "˜i )˜ "i ] . ⇤ Rf E [exp( ↵w0 ⇡i "˜i )]

Hint: Use independence and the fact that end-of-period market wealth is w ˜ m = w0

X

˜ j + w0 ⇡i⇤ R ˜i ⇡j⇤ R

j6=i

= w0

X

˜ j + w0 ⇡i⇤ ai + w0 ⇡ ⇤ b0 F˜ + w0 ⇡ ⇤ "˜i . ⇡j⇤ R i i i

j6=i

(c) Show that E [exp( ↵w0 ⇡i⇤ "˜i )]

1.

Hint: Use Jensen’s inequality. (d) Show that |E [exp( ↵w0 ⇡i⇤ "˜i )˜ "i ]|  ↵w0 ⇡i⇤ exp(↵ w0 ⇡i⇤ ) var(˜ "i ) . Hint: Use an exact first-order Taylor series expansion of the exponential function. (a) There is a stochastic discount factor m ˜ = ⌘↵e

↵w ˜m

for some ⌘. Because E[m] ˜ = 1/Rf ,

⌘↵ =

1 Rf E[e

m ˜ =

e ↵w˜m . Rf E[e ↵w˜m ]

=

E[e ↵w˜m "˜i ] . Rf E[e ↵w˜m ]

↵w ˜m ]

,

so

This implies i

7 Representative Investors

65

(b) Using the fact that w ˜ m = w0

X

˜ j + w0 ⇡ ⇤ ai + w0 ⇡ ⇤ b0 F˜ + w0 ⇡ ⇤ "˜i ⇡j⇤ R i i i i

j6=i

˜ j for j 6= i, we obtain and the independence of "˜i from F˜ and from R E[e

↵w ˜m

E[e

"˜i ] = E[e

↵w ˜m

] = E[e

↵(w0 ↵(w0

P P

j6=i

˜ j +w0 ⇡ ⇤ ai +w0 ⇡ ⇤ b0 F˜ ) ⇡j⇤ R i i i

]E[e

↵w0 ⇡i⇤ "˜i

"˜i ] ,

j6=i

˜ j +w0 ⇡ ⇤ ai +w0 ⇡ ⇤ b0 F˜ ) ⇡j⇤ R i i i

]E[e

↵w0 ⇡i⇤ "˜i

].

Thus, i

(c) The random variable

=

E [exp( ↵w0 ⇡i⇤ "˜i )˜ "i ] . ⇤ Rf E [exp( ↵w0 ⇡i "˜i )]

↵w0 ⇡i⇤ "˜i has mean zero and the exponential function is convex, so

Jensen’s inequality implies E[e

↵w0 ⇡i⇤ "˜i

]

e0 = 1 .

(d) The exact first-order series expansion of ex around x = 0 is ex = 1 + e y x for some y between x and 0. Applying this with x = e

↵w0 ⇡i⇤ "˜i

=1

↵w0 ⇡i⇤ "˜i gives

↵w0 ⇡i⇤ "˜i e

↵w0 ⇡i⇤ ˜

for some ˜ between "˜i and 0. Using this and E[˜ "i ] = 0, we obtain E [exp( ↵w0 ⇡i⇤ "˜i )˜ "i ] = Because of the assumption "˜i

, we have ˜ e

↵w0 ⇡i⇤ ˜

↵w0 ⇡i⇤ E[e

↵w0 ⇡i⇤ ˜ 2 "˜i ] .

and therefore ⇤

 e↵w0 ⇡i

Hence, ⇤

|E [exp( ↵w0 ⇡i⇤ "˜i )˜ "i ]|  ↵w0 ⇡i⇤ e↵w0 ⇡i var(˜ "i ) .

Part II

Dynamic Models

8 Dynamic Securities Markets

8.1. Suppose there is a risk-free asset with constant return Rf each period. Suppose there is a single risky asset with dividends given by 8 > < h Dt Dt+1 = > : D ` t where

h

>

`

with probability 1/2 , with probability 1/2 ,

are constants, and D0 > 0 is given. Suppose the price of the risky asset satisfies

Pt = kDt for a constant k. Suppose the information in the economy consists of the history of dividends, so the information structure can be represented by a tree as in Figure ?? with two branches emanating from each node (corresponding to the outcomes h and `). For each date t > 0 and each path, let ⌫t denote the number of dates s  t such that Ds =

h Ds 1 ,

so Dt = D0

⌫t t ⌫t . h `

Recall that, for 0  n  t, the probability that ⌫t = n is the binomial probability 2

t!

t

n!(t

n)!

.

(a) State a condition implying that there are no arbitrage opportunities for each finite horizon T . (b) Assuming the condition in part (a) holds, show that there is a unique one-period stochastic discount factor from each date t to t + 1, given by 8 >

:z if D /D = t+1 t `

h, `,

for some constants zh and z` . Calculate zh and z` in terms of Rf , k,

h

and

`.

(c) Assuming the condition in part (a) holds, show that there is a unique SDF process M , and show that Mt depends on ⌫t and the parameters Rf , k,

h

and

`.

70

8 Dynamic Securities Markets

(d) Assuming the condition in part (a) holds, show that there is a unique risk neutral probability measure for any given horizon T < 1, and show that the risk neutral probability of any path depends on ⌫t and the parameters Rf , k,

h

and

(e) Consider T < 1 and the random variable 8 >

:0 if D t+1 =

`.

h Dt

for each t < T ,

` Dt

for any t < T .

Calculate the self-financing wealth process that satisfies WT = x.

(f) Suppose there is a representative investor with time additive utility, constant relative risk aversion ⇢ and discount factor . Assume the risk free asset is in zero net supply. Calculate Rf and k in terms of

h,

`,

⇢ and .

(g) Given the formula for k in the previous part, what restriction on the parameters

h,

`,

⇢ and

is needed to obtain k > 0? Show that this restriction is equivalent to "1 # X t 1 ⇢ E Dt < 1. t=1

(a) This is a dynamic version of Exercise 4.1. The return on the risky asset when Dt+1 = Pt+1 + Dt+1 (k + 1)Dt+1 k+1 = = Pt kDt k Likewise, the return when Dt+1 = Rh =

` Dt

h Dt

is

h.

is (k + 1) ` /k. Define

k+1 k

h,

k+1 k

R` =

`.

The necessary and sufficient condition for absence of arbitrage opportunities is R h > Rf > R` . (b) As in Exercise 4.1, the unique one-period conditional state prices are qh =

Rf R` , Rf (Rh R` )

q` =

Rh Rf . Rf (Rh R` )

The unique one-period stochastic discount factor is obtained by dividing the state prices by the probabilities, so zh = 2qh , and z` = 2q` . (c) The SDF process is obtained by compounding the one-period stochastic discount factors. This produces Mt = 2t qh⌫t q`t

⌫t

.

8 Dynamic Securities Markets

71

(d) There are 2T distinct events (paths) that can be distinguished at date T, each of which has probability 2

T.

The risk neutral probability of any path is RfT MT /2T = RfT qh⌫T q`T

⌫T

.

(e) By (8.16), the self-financing wealth process satisfies  MT x W t = Et . Mt If x = 1, then MT = 2T qhT and Mt = 2t qht . The conditional probability that x = 1 is 1/2T Ds /Ds

1

=

for all s  t and is zero otherwise. Therefore, 8 ⇣ T T⌘ 2 qh >  1 < = qhT t if Ds /Ds 1 = t MT x 2T t 2t q h Et = > Mt :0 otherwise . h

h

for all s  t ,

(f) Setting Ct = Dt , (8.35) implies Dt = D0

t/⇢

Mt

Given that Dt = D0

1/⇢

= D0

⌫t t ⌫t , h `

h

t/⇢

1/⇢

2t qh⌫t q`t ⌫t

= D0

✓

2qh

◆

⌫t /⇢ ✓

this holds for all values of ⌫t if and only if =

✓

2qh

◆

1/⇢

,

=

`

,

q` =

✓

2q`

◆

1/⇢

,

so qh =

⇢

2

h

⇢

2

`

.

This implies 1 2 Rf = = qh + q`

1 ⇢ h +

⇢ `

!

.

Also, 1 = q h Rh + q ` R` ✓ ◆ k+1 = 2 k This implies k= 2

⇣

1 ⇢ h

+

1 ⇢ h

⇣

2

+

1 ⇢ h

✓

k+1 k

1 ⇢ `

+

⌘

1 ⇢ `

◆

⌘.

1 ⇢ , `

2q`

◆

(t ⌫t )/⇢

.

t

if

72

8 Dynamic Securities Markets

(g) The condition k > 0 is equivalent to 1 ⇢ h

which can be rewritten as

E

"

2

,

⇢

✓

✓

Dt1 ⇢

=

D01 ⇢

and, by independence,

Thus,


1. Consider an investor with an infinite horizon, utility function u(c) = c, and discount factor

= 1/Rf . Suppose he is constrained

to consume 0  Ct  Wt . (a) Show that the value function for this problem is J(w) = w. (b) Show that the value function solves the Bellman equation. ˆ (c) Show that J(w) = 2w also solves the Bellman equation. ˆ (d) Show that, using the true value function J(w) = w in the Bellman equation, the suboptimal policy Ct = 0 for every t achieves the maximum for every value of w. (a) This is a complete market, and the investor can choose at each date t any consumption sequence (Ct , Ct+1 , . . .) satisfying the budget constraint

86

9 Portfolio Choice by Dynamic Programming 1 X

Rft

u

C u = Wt

u=t

and the constraint 0  Cu  Wu for u

t. The last constraint implies finiteness of the sums in

the following. Vt (w) = max = max

1 X

u

Cu

u=t 1 X t u t

subject to

1 X

t

Cu = w

u=t

Cu

subject to

u=t

=

u t 1 X

u t

Cu = w

u=t

w.

(b) We have max {c + J((w

0cw

c)Rf )} = max {c + (w 0cw

c)Rf )}

=w = J(w) . (c) We have ˆ max {c + J((w

0cw

c)Rf )} = max {c + 2 (w 0cw

c)Rf )}

= 2w ˆ = J(w) . (d) We have max {c + J((w

0cw

c)Rf )} = max {c + (w 0cw

c)Rf }

= max w , 0cw

which is independent of c and hence maximized by any 0  c  w. 9.7. This exercise illustrates the fact that the transversality condition (9.25) holds in bounded and negative dynamic programming. Consider the infinite horizon problem with i.i.d. returns studied in Section 9.6. Denote the investor’s utility function by u(c).

9 Portfolio Choice by Dynamic Programming

(a) Case B: Assume there is a constant K such that

87

K  u(c)  K for each c. Show that the

transversality condition (9.25) holds. (b) Case N: Assume u(c)  0 for each c and J(w) >

1 for each w. Show that the transversality

condition (9.25) holds. Hint: Use (9.28) and the definition of a value function to deduce that the limit in (9.25) is nonnegative. (a) If u is bounded in absolute value by K, then VT +1 (WT⇤ +1 )

= max

1 X

t

u(Ct )

t=T +1

is bounded in absolute value by K

1 X

t

=

E

"

t=T +1

K 1

T +1

!0

as T ! 1. (b) By the definition of a value function, V0 (w)

1 X

u(Ct⇤ )

t=0

#

,

so (9.28) implies lim E[VT +1 (WT⇤ +1 )]

T !1

0.

On the other hand, u  0 implies V  0, so we must actually have lim E[VT +1 (WT⇤ +1 )] = 0 .

T !1

9.8. This exercise illustrates the fact that (9.29) is a sufficient condition for any solution Jˆ of the Bellman equation to be the true value function and a sufficient condition for the argmax in the Bellman equation to be the optimum. Consider the infinite horizon problem with i.i.d. returns studied in Section 9.6. Denote the investor’s utility function by u(c). Let Jˆ be a function that solves the Bellman equation. Assume (9.29) holds. ˆ t )] are finite for each t. Suppose (C ⇤ , ⇡ ⇤ ) For arbitrary decisions (Ct , ⇡t ), assume E[u(Ct )] and E[J(W t t attain the maximum in the Bellman equation. Show that Jˆ is the value function and (Ct⇤ , ⇡t⇤ ) are optimal.

88

9 Portfolio Choice by Dynamic Programming

For arbitrary decisions, we have ˆ t) J(W

ˆ t+1 )] E[u(Ct ) + J(W

for each t. Thus, ˆ 0) J(W

ˆ 1 )] E[u(C0 ) + J(W E[u(c0 ) + u(C1 ) + ··· E

"T 1 X

t

#

u(Ct ) +

t=0

2

T

ˆ 2 )] E[J(W

ˆ T )] E[J(W

for any T . Applying (9.29a) and (9.29b) yields "1 # X t ˆ 0) E J(W u(Ct ) + lim sup E

"

t=0 1 X t=0

t

#

T

T !1

ˆ T )] E[J(W

u(Ct ) .

For the decisions that attain the maximum in the Bellman equation, the first inequality is an equality, and the transversality condition (9.29b) implies that the last inequality is an equality also, so Jˆ is the value function, and the decisions are optimal.

10 Conditional Beta Pricing Models

10.1. Suppose all investors have constant absolute risk aversion, with possibly di↵erent risk aversion coefficients Ah . Assume there are no endowments Yht , the return vectors Rt are i.i.d. over time, and there is a risk-free asset with return Rf each period. Assume all investors have infinite horizons, and assume the value functions are as described in Exercise 9.5. (a) Making the approximation (10.13d), what is ↵t in the approximate CCAPM? (b) Making the approximation (10.28d), what is ↵t in the approximate CAPM? (a) The approximation (10.13d) is H

X 1 1 ⇡ , ↵t Ah h=1

so ↵t is approximately the aggregate absolute risk aversion of the utility functions. (b) Under the given assumptions, the value function of each investor has constant absolute risk aversion (Rf

1)↵h /Rf , so the approximation (10.28d) is H

X Rf Rf 1 ⇡ = ↵t (Rf 1)Ah Rf 1 h=1

Therefore, ↵t is approximately (Rf utility functions.

H X 1 Ah h=1

!

.

1)/Rf times the aggregate absolute risk aversion of the

11 Some Dynamic Equilibrium Models

11.1. Consider a generalization of the model studied in Sects. 11.2–11.5 in which, at each date t, log Dt+1 = log Dt + µt + "t+1 , with µ being a Markov process (observed by the representative investor at each date) and independent of "1 , "2 , . . .. This implies that the distribution of µt+1 , µt+2 , . . . at any date t, conditional on the information at that date, depends only on the value of µt . Adopting the other assumptions in Section 11.2, show that the market price-dividend ratio at date t depends on µt as follows: " ( ) # 1 s 1 X X Pt = s t E exp (1 ⇢) µ` µt , Dt s=t+1

where  = exp

`=t

✓

1 (1 2

⇢)

2 2

◆

.

Note: the model analyzed by Mehra and Prescott (1985) is of this form, with µt following a “finitestate Markov chain.” We have, for s > t, Ds = Dt exp

s 1 X `=t

Hence,

µ` +

s X

`=t+1

"`

!

.

92

11 Some Dynamic Equilibrium Models

Pt = Et Dt =

"

1 X

s=t+1

1 X

s t

s=t+1

=

1 X

s t

s=t+1

=

1 X

s t

"

✓

Ds Dt

◆1

Et exp (1

exp (1 "

# ⇢)

s 1 X `=t

⇢)

s t Et exp (1

s=t+1

⇢

2 2

exp (1 "

t) Et exp (1

(s

⇢)

µ`

!

s 1 X `=t

µ`

!#

s X

⇢)

"`

`=t+1

⇢)

s 1 X `=t

µ`

!#

!#

.

11.2. In the model of Sects. 11.2–11.5, the return on the market portfolio is given by (11.5), which implies log Rmt =

log ⌫ + µ + "t , where the "t are i.i.d. standard normals. Using the parameter

values µ = 0.017,

= 0.00125,

= 0.99 and ⇢ = 10, calculate the standard deviation of the market

return. Note: you will find it is much smaller than the sample standard deviation of 16.54% in the data studied by Mehra and Prescott (1985). This is a very simple version of the excess volatility puzzle. p The parameter values imply ⌫ = 0.8937. Thus, log Rmt = 0.129 + 0.00125 "t . From the calculation in the previous exercise, this implies stdev(Rmt ) =

p e2⇥0.129+2⇥0.0125

e2⇥0.129+0.0125 = 0.040 = 4.0% .

11.3. The model studied in Sections 11.2–11.5 has the properties assumed in Exercise 7.6; hence, (7.22), derived from the Hansen-Jagannathan bound, must hold. In our current notation, (7.22) is p log(1 + 2 ) ⇢ , where  denotes the maximum Sharpe ratio of all portfolios. In particular, the inequality (11.37) must hold for  equal to the market Sharpe ratio. Combining the estimate of the standard deviation of the market return of 16.54% from Mehra and Prescott (1985) with the historical equity premium cited in Section 11.4 and the estimate

= 0.035, derive a numerical lower bound on ⇢ from (11.37).

Note: you will find this bound is more reasonable than the estimate of 47.6 presented in the text. This is because of two o↵setting factors: both the risk premium of the market and the volatility of the market are higher in the data than the model would predict, given reasonable values of and ⇢.

11 Some Dynamic Equilibrium Models

93

An equity premium of 6.18% and a standard deviation of 16.54% imply a Sharpe ratio of 0.374. Using

= 0.035, we have

p log(1 + 0.3742 ) = 16.098 . 0.035

⇢

11.4. This exercise shows that the condition ⌫ < 1 is necessary for the Bellman equation of a representative investor to have a solution, in the model of Sections 11.2–11.5. Consider the infinite-horizon portfolio choice model studied in Section 9.6. Suppose there is a riskfree asset and a single risky asset with return Rt =

1 µ+ e ⌫

"t

for a sequence of independent standard normals "t , where ⌫ is defined in (11.2). Suppose B 1

⇢

2. This process is Gram-Schmidt orthogonalization. Let B1 and B2 be imperfectly correlated Brownian motions with correlation process ⇢. Define Z1 = B1 . Define Z20 = 0 and 1 dZ2 = p (dB2 ⇢ dZ1 ) . 1 ⇢2 Show that Z1 and Z2 are independent Brownian motions. The following shows that Z2 is a Brownian motion: (dZ2 )2 = =

1 ⇢2

1 1 1

= dt .

⇢2

⇥ ⇥

(dB2 )2 dt

2⇢(dB2 )(dZ1 ) + ⇢2 (dZ1 )2

2⇢2 dt + ⇢2 dt

⇤

⇤

12 Brownian Motion and Stochastic Calculus

103

The following shows that Z1 and Z2 are independent: (dZ1 )(dZ2 ) = p =p

1 ⇢2

1 1

⇢2

1

(dZ1 )(dB2 (⇢ dt

⇢ dZ1 )

⇢ dt

= 0.

12.10. This exercise illustrates the implementation of Gram-Schmidt orthogonalization via the Cholesky decomposition discussed in Sections 4.9 and 12.13. Let B1 and B2 be independent Brownian motions. Suppose dYi = ↵i dt +

i1 dB1

+

i2 dB2

for i = 1, 2. (a) What are the elements of the matrix A such that 0 1 ⌘ dY1 ⇣ @ A dY1 dY2 = A dt ? dY2

(b) Let

0 a L=@ b

0 c

1

A.

Calculate a, b and c with a > 0 and c > 0 such that LL0 = A. (c) Define Z = (Z1 Z2 )0 by Zi0 = 0 and 0

L dZ = @

10 1 dB1 A@ A, dB2 22

11

12

21

so dY = ↵ dt + L dZ. Show that Z1 and Z2 are independent Brownian motions. (d) Define correlated Brownian motions ˆi = q dB

1 2 i1

2 i2

+

(

i1 dB1

+

i2 dB2 ) ,

ˆ1 and as in (12.26). Show that Z1 = B dZ2 = p 1

1 ⇢2

ˆ2 (dB

⇢ dZ1 ) ,

ˆ1 and B ˆ2 . as in Exercise 12.9, where ⇢ is the correlation process of B

104

12 Brownian Motion and Stochastic Calculus

(a) We have (dY1 )2 = (

2 11

+

2 12 ) dt ,

(dY1 )(dY2 ) = (

Hence,

0

A=@

(b) We want to solve

Hence,

2 11

0 a @ b

0 c

10 A@

a

11 21

b

0

c

1

2 12

+ +

0

2 21

12 22

11 21

12 22

+

2 22

11

+

12 22

+

2 12 ,

a2 =

2 11

ab =

11 21

+

2 21

2 22 .

b2 + c 2 =

+

2 12

+

+

(dY2 )2 = (

12 22 ) dt ,

11 21

2 11

A=@

+

11 21

1

A.

21 +

12 22

2 21

2 22

+

12 22 ,

This implies a= b= c= =

q

2 11

11

s |

p

2 , + 12 21 + 12 2 11

2 21

+

+

11 22 p 2 11

(

2 22

+

22

2 12

, 11 21 + 12 22 ) 2 2 11 + 12

12 21 | 2 12

(c) We want to compute (dZ1 dZ2 )0 such that 0 10 1 0 a 0 dZ1 @ A@ A=@ b c dZ2

2

.

11 dB1

+

12 dB2

21 dB1 +

22 dB2

Thus,

This implies

a dZ1 =

11 dB1

+

12 dB2 ,

b dZ1 + c dZ2 =

21 dB1

+

22 dB2 .

1

A.

1

A.

2 21

+

2 22 ) dt .

12 Brownian Motion and Stochastic Calculus

1 ( a 1 dZ2 = ( c dZ1 =

11 dB1

+

12 dB2 ) ,

21 dB1

+

22 dB2

105

b dZ1 ) .

One can calculate directly that (dZ1 )2 = (dZ2 )2 = dt and (dZ1 )(dZ2 ) = 0, but it also follows from Exercise 12.9 and Part (d) of this exercise. ˆ1 . The correlation process of B ˆ1 (d) Substituting for a from the previous part shows that Z1 = B ˆ2 is ⇢ given by and B ˆ1 )(dB ˆ2 ) = p ⇢ dt = (dB (

11 21 + 12 22 2 2 2 2 11 + 12 )( 21 + 22 )

dt

Simple algebra shows that

1 ( c

21 dB1

+

22 dB2 )

and b dZ1 = c

p 1

Thus, dZ2 = p 1

1

⇢2

=p 1 ⇢

ˆ2 (dB

⇢2

1 ⇢2

ˆ2 dB

dZ1 .

⇢ dZ1 ) .

12.11. This exercise is to express the conditional variance formula (12.25b) and conditional covariance formula (12.25d) in terms of processes being martingales. A more general fact, which does not require the finite variance assumption, and which can be used as the definition of (dMi )2 and Rt (dMi )(dMj ), is that 0 (dMis )2 is the finite-variation process such that (12.28a) is a local martingale, Rt and 0 (dM1s ) (dM2s ) is the finite-variation process such that (12.28b) is a local martingale. Suppose dMi = ✓ dBi for i = 1, 2, where Bi is a Brownian motion and ✓i satisfies (12.5), so Mi is a finite-variance martingale. (a) Show that (12.25b) is equivalent to Z

Mit2

t

(dMis )2

0

being a martingale. (b) Show that (12.25d) is equivalent M1t M2t being a martingale.

Z

t

(dM1s ) (dM2s ) 0

106

12 Brownian Motion and Stochastic Calculus

(a) Drop the i subscript. For u > s, we have ⇥ ⇤ Ms ) = Es (Mu Ms )2 Es [Mu ⇥ ⇤ = Es (Mu Ms )2 ⇥ ⇤ = Es Mu2 2Ms Mu + Ms2

vars (Mu

= Es [Mu2 ]

2Ms Es [Mu ] + Ms2

= Es [Mu2 ]

Ms2 ,

Ms ] 2

using the martingale property Es [Mu ] = Ms . For Z

Mit2

t

(dMis )2

0

to be a martingale means that, for u > s,  Z u Es Mu2 (dMt )2 = Ms2

Z

0

s

(dMt )2 .

0

By the calculation above, this is equivalent to vars (Mu

Ms ) = E s

Z

u

(dMt )2 ,

s

which is (12.25b). (b) We have, for u > s, covs (M1u

M1s , M2u

M2s ) = Es [(M1u

M1s )(M2u

M2s )]

= Es [(M1u

M1s )(M2u

M2s )]

= Es [M1u M2u

M1u M2s

= Es [M1u M2u ]

M2s Es [M1u ]

= Es [M1u M2u ]

M1s M2s ,

Es [M1u

M1s ]Es [M2u

M1s M2u + M1s M2s ] M1s Es [M2u ] + M1s M2s

using the martingale property Es [Miu ] = Mis . For M1t M2t

Z

t

(dM1s ) (dM2s ) 0

to be a martingale means that, for u > s,  Z u Es M1u M2u (dM1t )(dM2t ) = M1s M2s 0

Z

s

(dM1t )(dM2t ) . 0

M2s ]

12 Brownian Motion and Stochastic Calculus

107

By the calculation above, this is equivalent to covs (M1u

M1s , M2u

M2s ) = Es

Z

u

(dM1t )(dM2t ) , s

which is (12.25d).

12.12. Let B be a Brownian motion. Define Yt = Bt2

t.

(a) Use the fact that a Brownian motion has independent zero-mean increments with variance equal to the length of the time interval to show that Y is a martingale. (b) Apply Itˆ o’s formula to calculate dY and verify condition (12.5) to show that Y is a martingale. Hint: To verify (12.5) use the fact that Z T Z E Bt2 dt = 0

T 0

E[Bt2 ] dt .

(c) Let dM = ✓ dB for a Brownian motion B. Use Itˆ o’s formula to show that Z t 2 Mt (dMs )2 0

is a local martingale. (d) Let dMi = ✓i dBi for i = 1, 2, and Brownian motions B1 and B2 . Use Itˆ o’s formula to show that Z t M1t M2t (dM1s ) (dM2s ) 0

is a local martingale. (a) We want to show that Es [Bt2

t] = Bs2

s. Equivalently, Es [Bt2 ]

Bs2 = t

in Part (a) of the preceding exercise, Es [Bt2 ]

Bs2 = vars (Bt

and, because B is a Brownian motion, vars (Bt (b) For f (t, x) = x2

Bs ) ,

Bs ) = t

s.

t, we have @f = @t

1,

@f = 2x , @x

@2f = 2. @x2

Thus, dYt = df (t, Bt ) =

dt + 2Bt dBt + (dB)2 = 2Bt dBt .

s. By the calculation

108

12 Brownian Motion and Stochastic Calculus

To verify that Y is a martingale on [0, T ], we need to show that E We have E (c) Set Zt = Mt2 and

Z

T

2

(2Bt ) dt = 4 0

Z

Z

T 0

T 0

(2Bt )2 dt < 1 .

E[Bt2 ] dt Z

Yt = Zt

t

=4

Z

T 0

t dt = 2T 2 < 1 .

(dMs )2 .

0

Then, dYt = dZt

(dMt )2 .

As in the preceding part, applying Itˆ o’s formula to Zt = f (Mt ) = Mt2 gives dZt = 2Mt dMt + (dMt )2 . Thus, dYt = 2Mt dMt , which inherits the local martingale property of M . (d) Set Zt = M1t M2t and Yt = Zt

Z

t

(dM1s ) (dM2s ) . 0

Then dYt = dZt

(dM1t ) (dM2t ) .

Applying Itˆ o’s formula to Zt as in Exercise 12.2 gives dZt = M1t dM2t + M2t dM1t + (dM1t ) (dM2t ) . Thus, dYt = M1t dM2t + M2t dM1t , implying Y is a local martingale.

12.13. Let dMi = ✓i dBi for i = 1, 2 and Brownian motions B1 and B2 . Suppose ✓1 and ✓2 satisfy condition (12.5), so M1 and M2 are finite-variance martingales. Consider discrete dates s = t0 < t1 < · · · < tN = u for some s < u. Show that 2 N X covs (M1u M1s , M2u M2s ) = Es 4 (M1tj j=1

M1tj 1 )(M2tj

3

M2tj 1 )5 .

12 Brownian Motion and Stochastic Calculus

109

Hint: This is true of discrete-time finite-variance martingales, and the assumption that the Mi are stochastic integrals is neither necessary nor helpful in this exercise. However, it is interesting to compare this to (12.25d). As calculated in Exercise 12.11, covs (M1u

M1s , M2u

M2s ) = Es [M1u M2u ]

M1s M2s .

Also, the calculation in Exercise 12.11 shows that Etj 1 [(M1tj

M1tj 1 )(M2tj

M2tj 1 )] = Etj 1 [M1tj M2tj ]

Therefore, by iterated expectations, 2 3 N N X X Es 4 (M1tj M1tj 1 )(M2tj M2tj 1 )5 = Es [Etj 1 [(M1tj j=1

M1tj 1 M2tj

1

M1tj 1 )(M2tj

.

M2tj 1 )]]

j=1

=

N X

Es [Etj 1 [M1tj M2tj ]]

Es [M1tj 1 M2tj 1 ]

j=1

=

N X

Es [M1tj M2tj ]

Es [M1tj 1 M2tj 1 ]

j=1

= Es [M1tN M2tN ] = Es [M1u M2u ]

Es [M1t0 M2t0 ] M1s M2s .

13 Continuous-Time Securities Markets and SDF Processes

13.1. Let rd denote the instantaneously risk-free rate in the domestic currency, and let Rd denote the domestic currency price of the domestic money market account: ✓Z t ◆ d d Rt = exp rs ds . 0

As in Section 13.12, let X denote the price of a unit of a foreign currency in units of the domestic currency. Let rf denote the instantaneously risk-free rate in the foreign currency, and let Rf denote the foreign currency price of the foreign money market account: ✓Z t ◆ f f Rt = exp rs ds . 0

Suppose M d is an SDF process for the domestic currency, so M f ⌘ M d X/X0 is an SDF process for the foreign currency. Assume dX = µx dt + X

x dBx

for a Brownian motion Bx . (a) Show that dM f = Mf

rf dt + dZ

for some local martingale Z. (b) Deduce from the previous result and Itˆ o’s formula that ✓ ◆✓ ◆ dX dM d µx dt = (rd rf ) dt . X Md (c) Suppose M d Rd is a martingale under the physical probability and define the risk neutral probability corresponding to M d . Assume M d XRf is also a martingale under the physical probability. Show that

112

13 Continuous-Time Securities Markets and SDF Processes

dX = (rd X

rf ) dt +

⇤ x dBx ,

where Bx⇤ is a Brownian motion under the risk neutral probability. Note: The result of Part (c) is called uncovered interest parity under the risk neutral probability. Suppose for example that r⇤ < r. Then it may appear profitable to borrow in the foreign currency and invest in the domestic currency money market. The result states that, under the risk neutral probability, the cost of the foreign currency is expected to increase so as to exactly o↵set the interest rate di↵erential. (a) Set Y = M f Rf . Then Y is a local martingale, and dY dM f = + rf dt , Y Mf so dM f = Mf

rf dt + dZ ,

where Z is a local martingale defined by dZ = dY /Y . (b) From the previous part, the formula M f = M d X/X0 , and the formula (13.16a) for the SDF process M d , we have dM f Mf ✓ ◆✓ ◆ dM d dX dX dM d = + + X X Md Md

rf dt + dZ =

=

d

r dt

0

dB + µx dt +

x dBx

+

✓

dX X

◆✓

dM d Md

◆

.

For this to be true, the dt terms on each side must match, implying ✓ ◆✓ ◆ dX dM d d f µx dt = (r r ) dt . X Md (c) If M d XRf is a martingale under the physical probability, then XRf /Rd is a martingale under the risk neutral probability. Setting Y = XRf /Rd , we have dY = (rf Y

rd ) dt +

dX . X

This process has zero drift under the risk neutral probability, so the drift of dX/X under the risk neutral probability is (rd

rf ) dt. Because volatilities do not change when we make an

equivalent change of measures, we know that

13 Continuous-Time Securities Markets and SDF Processes

dX = (rd X

113

⇤ x dBx

rf ) dt +

for some Brownian motion Bx⇤ under the risk neutral probability. If fact, defining Bx⇤ by ✓ ◆ 1 dX µx r d + r f ⇤ d f dBx = (r r ) dt = dt + dBx , X x x the facts that Bx⇤ has no drift under the risk neutral probability and (dBx⇤ )2 = (dBx )2 = dt implies Bx⇤ is a Brownian motion under the risk neutral probability, by Levy’s theorem. 13.2. Generalizing from the discrete-time model in Section 8.7, assume an investor’s optimal consumption satisfies the first-order condition e

t u0 (C

u0 (C for a constant discount factor

t)

0)

= Mt

and stochastic discount factor process M . Assume u(c) =

1 1

⇢

c1

⇢

.

(a) Assume (or prove) that the optimal consumption process is an Itˆ o process: dC = ↵ dt + ✓ dZ C for a Brownian motion Z. Use the fact that the drift of dM/M must be r = + ⇢↵

r to show that

1 ⇢(1 + ⇢)✓2 . 2

(b) Interpreting the investor as a representative investor, give an economic explanation for why the equilibrium interest rate r should be higher when (i) the discount rate

is higher, (ii) the

expected growth rate ↵ of consumption is higher, or (iii) the variance ✓2 of consumption growth is smaller. (c) Assume there is a constant risk-free rate and a single risky asset. Assume the dividend-reinvested price of the risky asset is a geometric Brownian motion: dS = µ dt + dB . S Assume the Brownian motion B is the only source of uncertainty in the economy. Show that ✓=

µ r . ⇢

114

13 Continuous-Time Securities Markets and SDF Processes

(d) Assume the risk-free rate is consistent with the statistics reported by Mehra and Prescott (1985); i.e., r = log 1.008. Use the numbers for ↵, ✓, µ and they imply ⇢ > 10 and

calculated in Exercise 12.3. Show that

< 0.

(a) We can prove C is an Itˆ o process by applying Itˆ o’s formula to t ⇢

Ct = C0 e

Mt

1 ⇢

and using Levy’s theorem to construct the Brownian motion Z. However, to derive the formula for r, it is convenient to use Mt = C0⇢ e Setting f (t, c) = C0⇢ e

tc ⇢,

we have @f /@t =

t

Ct

⇢

.

f , @f /@c =

⇢f /c and @ 2 f /@c2 = ⇢(1+⇢)f /c2 .

Hence, dM = M = Equating the drift to

✓ ◆ dC 1 dC 2 dt ⇢ + ⇢(1 + ⇢) C 2 C ✓ ◆ 1 + ⇢↵ ⇢(1 + ⇢)✓2 dt ⇢✓ dZ . 2

r dt yields the result.

(b) A higher means greater impatience, implying the representative investor would want to borrow against future consumption, unless the risk-free rate rises. Likewise, a higher expected growth rate of consumption or less risk in the growth rate of consumption would induce borrowing, unless the risk-free rate rises. (c) The unique SDF process M satisfies dM = M

µ

r dt

r

dB .

Matching coefficients to the formula in the solution to part (a), we obtain µ (d) Given µ = 0.0675,

r

= ⇢✓

,

✓=

µ r . ⇢

= 0.1537, ↵ = 0.0178, ✓ = 0.035, and r = log 1.008, from (µ

we obtain ⇢ = 11.07. Given this value of ⇢, the formula r = + ⇢↵ implies

=

0.107.

1 ⇢(1 + ⇢)✓2 2

r)/ = ⇢✓,

13 Continuous-Time Securities Markets and SDF Processes

115

13.3. Assume there is a representative investor with utility u(c) = log c and suppose that his optimal consumption satisfies the first-order condition (13.48). Assume WT = 0, so the wealth of the investor at any date t is the value of receiving the market dividend from date t to T . Assume (13.34) is a martingale. Show that the market price-dividend ratio Wt /Ct is given by Z

T

(s t)

e

ds .

t

Conclude that if the horizon T is infinite, then Wt /Ct = 1/ . We have, for s

t, Ms e = Mt

and W t = Et

Z

T t

(s t) C

t

Cs

Ms Cs ds = Ct Mt

For T = 1, this implies Wt =

Ct

Z

T

e

(s t)

.

t

.

13.4. Adopt the same assumptions as in the previous problem, but suppose that (13.49) is the representative investor’s utility function. Assume that the market dividend is a geometric Brownian motion: dC = ↵ dt + ✓ dZ C for constants ↵ and ✓ and a Brownian motion Z. Define ⌫=

1 ⇢)↵ + ⇢(1 2

(1

⇢)✓2 .

Show that the market price-dividend ratio Wt /Ct is given by Z

T

e

⌫(s t)

ds .

t

Conclude that if the horizon is infinite and ⌫ > 0, then Wt /Ct = 1/⌫. We have, for s

t, Ms e = Mt

and

(s t) C ⇢ t Cs ⇢

116

13 Continuous-Time Securities Markets and SDF Processes

Wt = Et Ct = Et

Z

T

"Zt

Ms Mt

T

✓ (s

e

t

◆ Cs ds Ct ✓ ◆1 t) Cs Ct

#

⇢

ds .

From dC = ↵ dt + ✓ dZ , C it follows that Cs = Ct exp so

✓

Cs Ct

◆1

✓✓

✓

⇢

◆ 1 2 ✓ (s 2

↵ ✓

= exp (1

t) + ✓(Zs

◆ 1 2 ✓ (s 2

⇢) ↵

t) + (1

Zt )

◆

,

⇢)✓(Zs

Zt )

◆

.

Given date–t information, this is the exponential of a normally distributed random variable with mean

✓

(1

◆ 1 2 ✓ (s 2

⇢) ↵

t)

and variance ⇢)2 ✓2 (s

(1 Therefore, "Z

T

Et

e

(s t)

t

✓

Cs Ct

◆1

⇢

#

ds = = = =

Z Z Z

T

t) .

"✓

Cs Ct

◆1

⇢

#

e

(s t)

e

(s t) ((1 ⇢)↵(s t) (1 ⇢)✓ 2 (s t)/2+(1 ⇢)2 ✓ 2 (s t)/2

Et

t T

e

ds ds

t T

e

⌫(s t)

ds

t

1⇣ 1 ⌫

e

⌫(T t)

⌘

.

For ⌫ > 0 and T = 1, this implies Wt /Ct = 1/⌫. 13.5. Consider an investor with initial wealth W0 > 0 who seeks to maximize E[log WT ]. Assume the investor must choose among portfolio processes ⇡ satisfying (13.13a) and the following stronger version of (13.13b): E Recall that this condition implies

Z

T 0

⇡t0 ⌃t ⇡t dt < 1 .

13 Continuous-Time Securities Markets and SDF Processes

E

Z

T 0

⇡t0

t dBt

117

= 0.

Using the formula (13.12) for Wt show that the optimal portfolio process is ⇡t = ⌃t 1 (µt

rt 1) .

Hint: the objective function obtained by substituting the formula (13.12) for Wt can be maximized in ⇡ separately at each date and in each state of the world. From (13.12), the realized utility is Z T✓ log W0 + rt + ⇡t0 (µt

rt 1)

0

1 0 ⇡ ⌃ t ⇡t 2 t

◆

dt +

Z

T 0

⇡t0

t dBt .

Thus, ignoring the constant log W0 , the expected utility is Z T ✓ ◆ 1 0 0 E rt + ⇡t (µt rt 1) ⇡ ⌃t ⇡t dt . 2 t 0 The optimal portfolio process maximizes ⇡t0 (µt

rt 1)

1 0 ⇡ ⌃t ⇡t 2 t

for each t and in each state of the world, implying ⇡t = ⌃t 1 (µt

rt 1) .

13.6. This exercise establishes the market completeness result asserted in Section 13.10. It uses martingale representation under the physical probability measure. Adopt the assumptions of Section 13.10. Let M = Mp . Define WT = x and W t = Et

Z

T t

Ms MT Cs ds + WT Mt Mt

for t < T . (a) Apply the result of Section 12.7 to deduce that there is a stochastic process t  T,

Z

t

Ms Cs ds + Mt Wt = W0 + 0

Z

t 0

0

dB .

such that, for all

118

13 Continuous-Time Securities Markets and SDF Processes

(b) Take the di↵erential of the formula in Part (a) to show that ✓ ◆0 1 dW = C dt + rW dt + W p + dB M Use this formula to compute (dW )

✓

dM M

(dW )

✓

dM M

◆

.

◆0

dB .

◆

and show that ✓ C dt + rW dt + W

dW =

p+

◆0

1 M

✓ dt + W p

p+

1 M

(c) Define = W⌃

1

(µ

r1) +

1 ( M

1 0

)

.

Show that (W, C, ) satisfies the intertemporal budget constraint (13.33). (a) We have Z

Z T Ms Cs ds + Et Ms Cs ds + MT WT 0 t Z T = Et Ms Cs ds + MT WT ,

t

Ms Cs ds + Mt Wt = 0

Z

t

0

which is a martingale adapted to the vector B. Hence, the martingale representation theorem implies

Z

for some process

t

Ms Cs ds + Mt Wt = W0 + 0

Z

t

0

dB

0

.

(b) By taking the di↵erential of the formula from the previous part, we obtain M C dt + M dW + W dM + (dW )(dM ) =

0

dB .

Dividing by M and rearranging produces dW = =

✓

dM M

◆

1 (dW ) + M ✓ ◆0 1 C dt + rW dt + W p + dB M C dt

dM W M

From this, we can compute the covariation as ✓ ◆ ✓ dM (dW ) = W M

1 p+ M

◆0

0

dB (dW )

p dt ,

✓

dM M

◆

.

13 Continuous-Time Securities Markets and SDF Processes

so

✓

◆0

1 C dt + rW dt + W p + M

dW =

(c) From the definition of 0

and the definition of

(µ

=W 0

0 p p

0 p

1

1 M

0

r1)0 ⌃

= W (µ =W

+

+

1 M

0

1 p dt + W p + M

◆0

dB .

p,

r1)0 ⌃

r1) = W (µ

✓

119

(µ

r1) +

1 M

0

1

(µ

r1)

p,

1

+

1 M

0

1

.

Therefore, the intertemporal budget constraint (13.34) holds.

13.7. This exercise establishes the market completeness result asserted in Section 13.10, using martingale representation under the risk neutral probability. Adopt the assumptions of Section 13.10. Let M denote the unique stochastic discount factor process. Assume M R is a martingale. Consider T < 1, and define the probability QT in terms of ⇠T =

MT RT by (13.38). Define B ⇤ by (13.41). Let x be a random variable that depends only on the path of the vector process B ⇤ up to time T and let C be a process adapted to B ⇤ . Assume Z T Cs x ⇤ E ds + < 1. RT 0 Rs For t  T , define

Wt⇤ = E⇤t

Observe that

which is a QT –martingale.

Z

t 0

Z

T t

Cs x ds + Rs RT

Cs ds + Wt⇤ = E⇤t Rs

Z

T 0

.

Cs x ds + Rs RT

,

(a) Apply the result of Section 12.7 to deduce that there is a stochastic process ⌘ such that, for all t  T,

(b) Set W =

RW ⇤

Z

t 0

Cs ds + Wt⇤ = W0⇤ + Rs

Z

t

⌘ (µ 0

(so, in particular, WT = x) and

of the previous part to deduce that W , C and (13.33).

0

r1) ds +

Z

t

⌘ 0 dB .

0

= R⌘. Apply Itˆ o’s formula and use the result satisfy the intertemporal budget constraint

120

13 Continuous-Time Securities Markets and SDF Processes

(a) Recall that dB ⇤ =

1

dt + dB =

(µ

r1) dt + dB .

By the martingale representation theorem, applied under the risk neutral probability, there exists

such that Z

t 0

Cs ds + Wt⇤ = W0⇤ + Rs =

Setting ⌘ =

1 0

W0⇤

+

Z Z

t

0

dB ⇤

0 t

0

1

(µ

r1) dt +

0

Z

t

0

dB .

0

, we have Z

t 0

Cs ds + Wt⇤ = W0⇤ + Rs

Z

t

0

⌘ (µ

r1) dt +

0

Z

t

⌘ 0 dB .

0

(b) From the previous part, C dt + dW ⇤ = ⌘ 0 (µ R

r1) dt + ⌘ 0 dB .

Setting W = RW ⇤ yields dW ⇤ =

1 dW R

W dR , R2

so 1 dW R

W dR = R2 =

C dt + ⌘ 0 (µ r1) dt + ⌘ 0 dB R C 1 0 1 0 dt + (µ r1) dt + dB . R R R

Multiplying by R and using dR/R = r dt produces dW = W r dt

C dt +

0

(µ

r1) dt +

0

dB .

13.8. This exercise verifies that, as asserted in Section 13.9, condition (13.43) is sufficient for M W to be a martingale. Let M be an SDF process such that M R is a martingale. Define B ⇤ by (13.41). Let W be a positive self-financing wealth process. Define W ⇤ = W/R. (a) Use Itˆ o’s formula, (13.8), (13.16b) and (13.41) to show that dW ⇤ =

1 R

0

dB ⇤ .

13 Continuous-Time Securities Markets and SDF Processes

(b) Explain why the condition E

⇤

Z

T

1 R2

0

0

121

⌃ dt < 1

implies that W ⇤ is a martingale on [0, T ] under the risk neutral probability defined from M , where E⇤ denotes expectation with respect to the risk neutral probability. (c) Deduce from the previous part that (13.43) implies M W is a martingale on [0, T ] under the physical probability. (a) We have dW ⇤ dW = ⇤ W W

dR R

= ⇡ 0 (µ

r1) dt + ⇡ 0 dB

= ⇡0

dt + ⇡ 0 dB

= ⇡ 0 dB ⇤ , using Itˆo’s formula, (13.8), (13.16b) and (13.41) successively. This implies dW ⇤ = W ⇤ ⇡ 0 dB ⇤ W 0 ⇡ dB ⇤ R 1 0 = dB ⇤ , R =

where

= W ⇡. p 0 (b) Define ⌘ = ⌃ , Z0 = 0 and 1 ⌘

dZ =

0

dB ⇤ .

By Levy’s theorem, Z is a Brownian motion under the risk neutral probability. We have dW ⇤ =

⌘ dZ . R

The sufficient condition (12.5) for W ⇤ to be a martingale under the risk neutral probability is that E⇤ This is equivalent to E

⇤

Z

Z T 0

T 0

⌘2 dt < 1 . R2

1 R2

0

⌃ dt < 1 .

122

13 Continuous-Time Securities Markets and SDF Processes

(c) We have E

⇤

Z

T 0

1 R2

0

⌃ dt = = =

Z Z Z

T



E

⇤

E



0 T 0 T

E 0



1 Rt2

0 t ⌃t t

MT R T Rt2 Mt Rt

dt

0 t ⌃t t

0 t ⌃t t

dt

dt ,

using iterated expectations and Et [MT RT ] = Mt Rt for the last equality. Thus, (13.43) is a sufficient condition for W ⇤ = W/R to be a martingale on [0, T ] under the risk neutral probability, hence a sufficient condition for M W to be a martingale under the physical probability.

13.9. This exercise verifies that Novikov’s condition can be expressed as (13.32), as asserted in Section 13.7. For a local martingale Y satisfying dY /Y = ✓0 dB for some stochastic process ✓, Novikov’s condition is that



⇢ Z T 1 E exp ✓0 ✓ dt 2 0

< 1.

Under this condition, Y is a martingale on [0, T ]. Consider Y = M W , where M is an SDF process and W is a self-financing wealth process. (a) Show that dY /Y = ✓0 dB, where ✓ =

0⇡

p

⇣ and ⇣ = 0.

(b) Deduce that Novikov’s condition is equivalent to (13.32). (a) For Y = M W , we have dY dM dW = + + Y M W =

using

=µ

(b) Setting ✓ =

)0 dB

= ( 0⇡

p

p

=

p

dM M

◆✓

dW W

◆

dB + r dt + ⇡ 0 (µ

= ( 0⇡

r1 and 0⇡

0

r dt

✓

r1) dt + ⇡ 0 dB

⇡0

⇣)0 dB ,

+ ⇣.

⇣, we have ✓0 ✓ = ⇡ 0 ⌃⇡ +

0 p p

+ ⇣ 0⇣

2⇡ 0

p

2⇡ 0 ⇣ + 2

0 p⇣

.

dt

13 Continuous-Time Securities Markets and SDF Processes

Substituting

p

=µ

123

r1, ⇣ = 0 and 0 p⇣

r1)0 ⌃

= (µ

1

⇣ = 0,

we obtain ✓0 ✓ = ⇡ 0 ⌃⇡ +

0 p p

+ ⇣ 0⇣

2⇡ 0 (µ

r1) .

13.10. By specializing (13.32), state sufficient conditions for M Si to be a martingale for i = 1, . . . , n. To apply (13.32) to W = Si , take the portfolio process ⇡ to be the i–th basis vector ei . Condition (13.32) in this case is  ⇢ Z T 1 E exp 2 0

0 p p

+ ⇣ 0 ⇣ + e0i ⌃ei

2(µi

r) dt

< 1.

Note that e0i ⌃ei is the squared volatility of the asset return, i.e., the i–th diagonal element of ⌃. 13.11. This exercise verifies that if M W is a martingale, where W is a “consumption-reinvested” wealth process (in particular, self-financing), then the valuation formula (13.350 ) for the consumption process is valid, as asserted in Section 13.8. Suppose W > 0, C and ⇡ satisfy the intertemporal budget constraint (13.33). Define the consumption-reinvested wealth process W † by (13.36). (a) Show that W † satisfies the intertemporal budget constraint (13.37). (b) Show that Wt†

Wt =

Wt†

Z

t 0

Cs Ws†

ds

for each t. Hint: Define Y = W/W † and use Itˆ o’s formula to show that dY = Conclude that Wt Wt† for all t.

=1

C dt . W† Z

t 0

Cs Ws†

ds

124

13 Continuous-Time Securities Markets and SDF Processes

(c) Let M be an SDF process and assume M W † is a martingale. Use this assumption and iterated expectations to show that, for any t < T ,  Z T Z T Z t Cs Cs † † E t MT W T ds = Mt Wt ds + Et Ms Cs ds . † † 0 Ws 0 Ws t (d) Let M be an SDF process and assume M W † is a martingale. Use the results of the previous two parts to show that (13.34) is a martingale. (a) From (13.36) and (13.33), we have dW † dW C = + dt W W W† = r dt + ⇡ 0 (µ

r1) dt + ⇡ 0 dB .

(b) Setting Y = W/W † yields ✓ ◆✓ ◆ ✓ ◆2 dY dW dW † dW dW † dW † = + Y W W W† W† W† C = dt ⇡ 0 ⌃⇡ dt + ⇡ 0 ⌃⇡ dt W C = dt , W so dY =

CY dt = W

C dt . W†

This implies Wt Wt†

= Yt = Y0 =1

Z

t

C ds † W 0 Z t C ds , † 0 W

using W0† = W0 , which follows from (13.36). Multiplying by Wt† yields Z t C Wt = Wt† Wt† ds . † W 0 (c) For t < s < T , iterated expectations and the fact that Es [MT WT† ] = Ms Ws† imply   h i Cs † Cs † Et MT W T † = Et E M W s T T Ws Ws† = Et [Ms Cs ] .

13 Continuous-Time Securities Markets and SDF Processes

125

Therefore, Et



MT WT†

Z

T

Cs Ws†

0

ds = = = =

Z

t

Cs

† 0 Ws

Mt Wt†

ds ⇥

Z

t

Et MT WT†

Cs

† 0 Ws Z t Cs Mt Wt† † 0 Ws Z t Cs Mt Wt† † 0 Ws

(d) Let X t = Et



h

MT WT†

ds + ds +

Z Z

T t

T

+ Et 



MT WT†

Et MT WT†

T

Cs Ws†

Z

T t

Cs Ws†

ds

ds

Et [Ms Cs ] ds t

ds + Et Z

i

Z

T

Ms Cs ds . t

Cs

ds . Ws† This is a martingale. Using Parts (c) and (b) successively, we have Z T Z t Cs † X t = Mt W t ds + Et Ms Cs ds † 0 Ws t Z T † = Mt (Wt Wt ) + Et Ms Cs ds . 0

t

From the assumption that

MW†

is a martingale, it follows that Z T Mt W t E t Ms Cs ds t

is a martingale. The second term in this expression is zero at t = T . Therefore, Z T Mt W t E t Ms Cs ds = Et [MT WT ] . t

This implies Mt W t +

Z

t

Z

t

T

Ms Cs ds + Et Ms Cs ds + Et [MT WT ] 0 t  Z T = E t MT W T + Ms Cs ds .

Ms Cs ds = 0

Z

0

13.12. This exercise provides an alternate proof that requiring M W to be a martingale for selffinancing wealth processes W validates the valuation formula (13.350 ) for consumption processes. It considers reinvesting consumption in the money market account rather than in the portfolio generating the wealth process. Suppose W , C and ⇡ satisfy the intertemporal budget constraint (13.33). Define Z t Cs Wt† = Wt + Rt ds . R s 0

126

13 Continuous-Time Securities Markets and SDF Processes

(a) Show that W † satisfies the intertemporal budget constraint (13.80 ). (b) Let M be an SDF process. Assume M R is a martingale and M W † is a martingale. Deduce that (13.34) is a martingale. (a) We have †

dW = dW + C dt +

✓Z

t 0

= dW + C dt + (W † 0

= rW dt + = rW † dt +

(µ 0

◆ Cs ds dR Rs dR W) R

r1) dt +

(µ

r1) dt +

0

dB + (W † 0

W )r dt

dB .

(b) From the definition of W † , we have Z

t

Ms Cs ds + Mt Wt = 0

Z

t 0

Ms Cs ds + Mt Wt†

Mt R t

Z

t 0

Cs ds . Rs

Given the assumption that M W † is a martingale, it suffices to show that Z

t

Ms Cs ds 0

Mt R t

Z

t 0

Cs ds Rs

is a martingale. By iterated expectations and the assumption that M R is a martingale, we obtain, using the same reasoning as in the previous exercise,   Z T Z t Z T Cs Cs Cs E t MT R T ds = ds ⇥ Et [MT RT ] + Et MT RT ds 0 Rs 0 Rs t Rs Z T Z t Cs = Mt R t ds + Et Ms Cs ds . 0 Rs t Thus, Z T Et Ms Cs ds 0

MT R T

Z

T 0

Z T Z t Cs Cs ds = Et Ms Cs ds Mt R t ds Rs 0 0 Rs Z t Z t Cs = Ms Cs ds Mt Rt ds 0 0 Rs

Et

Z

T

Ms Cs ds t

14 Continuous-Time Portfolio Choice and Beta Pricing

14.1. Assume the continuous-time CAPM holds: ✓ ◆✓ ◆ dSi dWm (µi r) dt = ⇢ Si Wm for each asset i, where Wm denotes the value of the market portfolio, ⇢ = ↵Wm , and ↵ denotes the p aggregate absolute risk aversion. Define i = e0i ⌃ei to be the volatility of asset i, as described in Section 13.2, so we have

dSi = µi dt + Si

i dZi

for a Brownian motion Zi . Likewise, the return on the market portfolio is dWm = µm dt + Wm for some µm ,

m

and Brownian motion Zm . Let

im

m dZm

denote the correlation process of the Brownian

motions Zi and Zm . (a) Using the fact that the market return must also satisfy the continuous-time CAPM relation, show that the continuous-time CAPM can be written as µi (b) Suppose r, µi , µm ,

i,

m

r=

i m im (µm 2 m

r) .

and ⇢i are constant over a time interval

t, so both Si and Wm

are geometric Brownian motions over the time interval. Define the annualized continuously compounded rates of return over the time interval: ri =

log Si t

and

rm =

log Wm . t

128

14 Continuous-Time Portfolio Choice and Beta Pricing

Let r¯i and r¯m denote the expected values of ri and rm . Show that the continuous-time CAPM implies r¯i

r=

cov(ri , rm ) (¯ rm var(rm )

1 r) + [cov(ri , rm ) 2

(a) We have (µm so ⇢ = (µm

r)/

2 m.

r) dt = ⇢

◆2

=⇢

✓

dSi Si

◆✓

(

i m im ) dt .

dWm Wm

t.

2 m dt ,

Therefore (µi

r) dt = =

(b) We have E[

✓

var(ri )]

log Si ] = (µi

2 i /2)

r 2 m

µm

r 2 m

t and E[ 1 2

r¯i = µi

µm

dWm Wm

2 m /2)

log Sm ] = (µm

2 i ,

and

◆

1 2

r¯m = µm

t, so

2 m.

From Part (a), r¯i

r = µi

1 2

r

2 i

=

i m im (µm 2 m

=

i m im 2 m

✓

=

i m im 2 m

(¯ rm

1 2

r)

1 r¯m + 2

2 m

r) +

2 i

r

1 2

◆

1 2

i m im

2 i

1 2

2 i .

Also, var(ri ) = var(rm ) = cov(ri , rm ) =

2 i

t 2 m

t

, ,

i m im

t

.

Making these substitutions yields the result.

14.2. This exercise derives the ICAPM from the portfolio choice formula (14.24). For each investor h = 1, . . . , H, let ⇡h denote the optimal portfolio presented in (14.24). Using the notation of Section 14.7, (14.24) implies

14 Continuous-Time Portfolio Choice and Beta Pricing

1

W h ⇡h = ⌧ h ⌃

(µ

` X

r1)

⌧h ⌘hj ⌃

1

129

⌫ 0 ej .

j=1

(a) Deduce that µ

r1 = ↵W ⌃⇡ +

` X

⌘j ⌫ 0 e j ,

j=1

where ⇡ denotes the market portfolio: ⇡=

H X Wh h=1

W

⇡h .

(b) Explain why (14.42) is the same as (14.29a). (a) We have ⌧h (µ

` X

r1) = Wh ⌃⇡h +

⌧h ⌘hj ⌫ 0 ej .

j=1

Summing over h yields ⌧ (µ

r1) = ⌃

H X

W h ⇡h +

` H X X j=1

h=1

⌧h ⌘hj

h=1

which implies µ

` H X X ⌧h ⌘hj r1 = ↵W ⌃⇡ + ⌧ j=1

h=1

!

!

⌫ 0 ej ,

⌫ 0 ej .

The result follows from the definition

⌘j =

H X ⌧h ⌘hj

.

⌧

h=1

(b) Stacking the equations (14.29a) for i = 1, . . . , n yields (µ

r1) dt = ↵W ( dB)

✓

dW W

◆

+

` X

⌘j ( dB)(dXj )

j=1

= ↵W ( dB)(dB 0 ) 0 ⇡ +

` X

⌘j ( dB)(dB 0 )⌫ 0 ej

j=1

= ↵W ⌃⇡ dt +

` X

⌘j ⌫ 0 ej dt .

j=1

14.3. Consider an investor with log utility and an infinite horizon. Assume the capital market line is constant, so we can write J(w) instead of J(w, x) for the stationary value function defined in Section 14.10.

130

14 Continuous-Time Portfolio Choice and Beta Pricing

(a) Show that log w

J(w) = K + solves the HJB equation (14.31), where K= 1 (µ

Show that c = w and ⇡ = ⌃

log

+

+ 2 /2

r

2

.

r) achieve the maximum in the HJB equation.

(b) Show that the transversality condition h lim E e

t

T !1

i J(WT⇤ ) = 0

holds, where W ⇤ denotes the wealth process generated by the consumption and portfolio processes in part (a). (a) Substituting J = K + log w/ , Jw = 1/( w) and Jww = 1/( w2 ), the HJB equation (14.31) is ⇢ 1h ci 1 0 0 = max log c K log w + r + ⇡ 0 (µ r1) ⇡ ⌃⇡ . c,⇡ w 2 The maximum is achieved at c = w and ⇡ = ⌃

1 (µ

r1). Substituting these into the HJB

equation, it reduces to the formula given for K. (b) We have dW ⇤ = W⇤

✓

0

r + ⇡ (µ

= (r + 2 Hence, ⇤

d log W =

✓

r)

◆

◆

dt + ⇡ 0 dB

r)0 ⌃

) dt + (µ

1 + 2 2

r

C⇤ W⇤

1

r)0 ⌃

dt + (µ

dB .

1

dB .

This implies that h

E e

t

i

log WT = e

T

log W0 + e

T

✓

r

◆ 1 2 +  T !0 2

as T ! 1. 14.4. Consider an investor with power utility and an infinite horizon. Assume the capital market line is constant, so we can write J(w) instead of J(w, x) for the stationary value function defined in Section 14.10.

14 Continuous-Time Portfolio Choice and Beta Pricing

131

(a) Define (1 ⇢)r ⇢

⇠=

(1

⇢)2 . 2⇢2

Assume (14.32) holds, so ⇠ > 0. Show that ⇢

J(w) = ⇠

✓

1 1

⇢

w

1 ⇢

◆ 1 (µ

solves the HJB equation (14.31). Show that c = ⇠w and ⇡ = (1/⇢)⌃

r1) achieve the

maximum in the HJB equation. (b) Show that, under the assumption ⇠ > 0, the transversality condition h lim E e

t

T !1

i J(WT⇤ ) = 0

holds, where W ⇤ denotes the wealth process generated by the consumption and portfolio processes in part (a). (a) Substituting J = ⇠

⇢ w 1 ⇢ /(1

⇢), wJw = ⇠

⇢ w1 ⇢

and w2 Jww =

⇢⇠

⇢ w1 ⇢ ,

w1

⇢

the HJB equation

(14.31) is 0 = max c,⇡

⇢

1 1

⇢

c1

⇢

1

⇢

⇢

⇠

w1

⇢

h + r + ⇡ 0 (µ

ci ⇠ w

r1)

1 (µ

The maximum is achieved at c = ⇠w and ⇡ = (1/⇢)⌃

⇢

1 ⇢⇠ 2

⇢

w1

⇢ 0

⇡ ⌃⇡

r1). Substituting these into the

HJB equation, it reduces to the formula given for ⇠. (b) We have ◆ C⇤ r + ⇡ (µ r) dt + ⇡ 0 dB W⇤ ✓ ◆ 2 1 = r+ ⇠ dt + (µ r)0 ⌃ 1 dB . ⇢ ⇢

dW ⇤ = W⇤

Hence, WT⇤

= W0 exp

✓✓

✓

r

0

2 ⇠+ ⇢

2 2⇢2

◆

1 T+ ⇢

Z

T

(µ

0

r) ⌃

1

dBt

(µ

0

0

◆

.

This implies (WT⇤ )1 ⇢ Thus,

=

W01 ⇢ exp

✓

(1

✓

⇢) r

2 ⇠+ ⇢

2 2⇢2

◆

T+

.

1

⇢ ⇢

Z

T 0

r) ⌃

1

dBt

◆

.

132

14 Continuous-Time Portfolio Choice and Beta Pricing

h E e

T

(WT⇤ )1

⇢

i

= W01 =

⇢

exp

W01 ⇢ exp

✓⇢ ✓⇢

+ (1

✓ ⇢) r

+ (1

⇢)(r

⇠+ ⇠) +

2 ⇢ (1

2 2⇢2 ⇢)2 2⇢

◆

(1 ⇢)2 2 + 2⇢2 ◆ T .

T

◆

The transversality condition holds if and only if + (1

⇢)(r

⇢)2 < 0. 2⇢

(1

⇠) +

Substituting the formula for ⇠ into this and rearranging shows that it is equivalent to (1 ⇢)r ⇢

(1

⇢)2 > 0. 2⇢2

14.5. Consider an investor who seeks to maximize E[log WT ]. Assume Z T Z T E |rt | dt < 1 and E 2t dt < 1 , 0

0

where  denotes the maximum Sharpe ratio. Assume portfolio processes are constrained to satisfy Z T E ⇡t0 ⌃t ⇡t dt < 1 . 0

Maximizing at each date and in each state of the world as in Exercise 13.5, show that V (t, w, x) = log w + f (t, x), where f (t, x) = E

Z

T t

✓

From (13.12), the realized utility is Z T✓ log Wt + rs + ⇡s0 (µs

1 rs + 2s 2

E implies

Z

T 0

◆

ds +

Z

T t

⇡s0

s dBs .

⇡s0 ⌃s ⇡s ds < 1 ,

Z

t 0

⇡s0

s dBs

is a martingale. Thus, the expected utility is Z T ✓ log Wt + Et rs + ⇡s0 (µs t

ds Xt = x .

1 0 ⇡ ⌃s ⇡s 2 s

rs 1)

t

The assumption

◆

rs 1)

1 0 ⇡ ⌃s ⇡s 2 s

◆

ds .

14 Continuous-Time Portfolio Choice and Beta Pricing

133

This is maximized by maximizing ⇡s0 (µs

1 0 ⇡ ⌃ s ⇡s 2 s

rs 1)

for each s, implying ⇡s = ⌃s 1 (µs

rs 1) .

Making this substitution, the expected utility is Z T ✓ ◆ Z T ✓ ◆ 1 1 2 0 1 log Wt + Et r + (µ r1) ⌃ (µ r1) ds = log Wt + Et r+  ds . 2 2 t t Hence, V (t, w, x) = log w + f (t, x), where Z f (t, x) = E

T t

✓

1 r + 2 2

◆

ds Xt = x .

14.6. This exercise completes the proof in Section 14.3 that the tangency portfolio is optimal when the capital market line is constant. Assume the capital market line is constant and the horizon is finite. Define Z by (14.9), W ⇤ by (14.12),

by (14.13), and

by (14.14).

(a) Take the di↵erential of (14.13) to show that dW = rW dt

C dt

(dW )

✓

dMp Mp

◆

✓ + W+

(b) Use the result of the previous part and the formula dMp /Mp = ✓ ◆ dMp (dW ) , Mp and use the definition (14.14) of

to show that W , C and

Mp

◆

 dZ .

r dt

 dZ to calculate

satisfy the intertemporal budget

constraint (13.33). (a) (14.13) implies Mp C dt + W dMp + Mp dW + (dW )(dMp ) = dZ . Dividing by Mp and rearranging yields dW = rW dt + W  dZ

(dW )

✓

dMp Mp

◆

C dt +

Mp

dZ .

134

14 Continuous-Time Portfolio Choice and Beta Pricing

(b) From the previous part, we have (dW )

✓

dMp Mp

Therefore, dW = rW dt

◆

✓

=

✓

C dt + W +

W+

Mp

◆

Mp

◆

2 dt .

✓

2

 dt + W +

Mp

◆

 dZ .

From (14.14), we obtain 0

(µ

r1) = = 0

dB = = =

✓ ✓ ✓ ✓ ✓

W+ W+ W+ W+ W+

Mp Mp Mp Mp Mp

◆ ◆ ◆

(µ

1

r1)0 ⌃

1

(µ

r1)

2 , (µ

◆ ◆

r1)0 ⌃

dB

0 p dB

 dZ .

Therefore, the intertemporal budget constraint (13.33) holds. 14.7. Consider a power-utility investor with a finite horizon (case (d) in Section 14.10). Assume the capital market line is constant and the investor is constrained to always have nonnegative wealth. Let M = Mp . Calculate the optimal portfolio as follows. (a) Using (14.10), show that, for s > t, h Et Ms1

1/⇢

for a constant ↵.

i

1 1/⇢ ↵(s t)

= Mt

e

,

(b) Define Ct and WT from the first-order conditions and set Z T Ms MT W t = Et Cs ds + WT Mt t Mt

.

Show that Wt = g(t)Mt

1/⇢

for some deterministic function g (which you could calculate). (c) By applying Itˆ o’s formula to W in Part (b), show that the optimal portfolio is 1 ⌃ ⇢

1

(µ

r1) .

14 Continuous-Time Portfolio Choice and Beta Pricing

(a) For s

135

t, we have Ms = Mt exp

✓

t)

1 2  (s 2

1)/⇢

1 2  (s 2

r(s

t)

(Zs

Zt )

◆

.

Therefore, Ms1 1/⇢

=

1 1/⇢ Mt exp

✓

r(s

t)(⇢

t)(⇢

1)/⇢

(Zs

Zt )(⇢

1)/⇢

The exponential is of a normally distributed variable, so h Et Ms1

1/⇢

i

1 1/⇢ ↵(s t)

= Mt

e

,

where ↵=

r(⇢

1 2  (⇢ 2

1)/⇢

1 1)/⇢ + 2 (⇢ 2

1)2 /⇢2 .

(b) The first order conditions are t

⇢

= Mt

)

WT ⇢ = MT

)

e

Ct

⇣ Ct = e

t

Mt

⌘

1/⇢

WT = ( MT / )

1/⇢

, .

Thus, 1 Wt = Et Mt

Z

T

Ms Cs ds + MT WT t

Z T ⇣ ⌘ 1/⇢ 1 Et es Ms1 Mt t Z T⇣ ⌘ 1/⇢ h 1 = es Et Ms1 Mt t Z T⇣ ⌘ 1/⇢ 1/⇢ = Mt es e↵(s =

1/⇢

1/⇢

t)

1/⇢

ds + ( / ) i

ds +

1 1/⇢

MT

1 ( / ) Mt

ds + ( / )

1/⇢

t

1/⇢

Mt

h 1 E t MT

1/⇢ ↵(T t)

e

1/⇢

i

.

Hence, W t = Mt where g(t) = (c) From Itˆo’s formula,

Z

T t

⇣

dW dg dM 1/⇢ = + W g M 1/⇢ g0 (1/⇢)M = dt + g

e

s

⌘

1/⇢

e↵(s

1 1/⇢ dM

1/⇢

t)

g(t) ,

ds + ( / )

1/⇢ ↵(T t)

e

+ (1/2)(1/⇢)(1 + 1/⇢)M M 1/⇢

.

2 1/⇢ (dM )2

.

◆

.

136

14 Continuous-Time Portfolio Choice and Beta Pricing

The stochastic part of this comes from (1/⇢)M M

1 1/⇢ dM 1/⇢

1 dM r 1 = dt + ⇢ M ⇢ ⇢

=

0 p dB .

Hence the stochastic part of the portfolio return must be 1 ⇢

0 p dB ,

=

1 (µ ⇢

implying ⇡0 = Postmultiplying by

0⌃ 1

1 ⇢

0 p

r1)0 ⌃

1

.

shows that this has the unique solution ⇡0 =

1 (µ ⇢

r1)0 ⌃

1

.

14.8. This exercise derives a linear partial di↵erential equation (PDE) for the optimal wealth of a power utility investor in complete markets. This specific example is due to Wachter (2002). In general this approach leads to a PDE for a function of M and the state variables that influence M , but only the state variable appears here due to homotheticity (power utility). Suppose the risk-free rate r is constant. Suppose there is a single risky asset. Suppose the asset does not pay dividends prior to T and its price S satisfies dSt = µt dt + St

t dBt

for a Brownian motion B. Define t

Assume

=

µt

r

.

t

is an Ornstein-Uhlenbeck process d

t

= (✓

t ) dt

+ dB

for constants , ✓ and , where B is the same Brownian motion that appears in dS/S. Define Ct and WT by the first-order conditions (14.6), and set Z T Ms MT W t = Et Cs ds + WT Mt t Mt for t < T .

14 Continuous-Time Portfolio Choice and Beta Pricing

137

(a) Show that Wt =

1/⇢

1/⇢

Mt

Explain why 1/⇢

is some function f (t,

Et

"Z

Et

"Z

T

T

e

Ms Mt

◆1

t

s/⇢

e

s/⇢

✓

t

✓

Ms Mt

◆1

1/⇢

✓

MT Mt

◆1

1/⇢

ds +

1/⇢ 1/⇢

ds +

✓

MT Mt

◆1

1/⇢

1/⇢

#

.

#

t ).

(b) Using the formula Wt = Mt

1/⇢

f (t,

t ),

show that

dW = something dt + W

✓

f + f ⇢

◆

dB .

(c) Using the fact that the drift of the martingale Z t Ms Cs ds + Mt Wt , 0

must be zero, derive a PDE that must be satisfied by f and its partial derivatives ft , f and f . (d) Define the portfolio weight on the risky asset by ✓ ◆ 1 f ⇡= + . f ⇢ Show that (C, W, ⇡) satisfies the intertemporal budget constraint. Hint: Follow the steps in Rt Exercise 14.6, using the fact that the drift of 0 Ms Cs ds + Mt Wt is zero, i.e., f satisfies the PDE.

(a) The first order conditions are t

e

⇢

= Mt

)

WT ⇢ = MT

)

Ct

⇣ Ct = e

t

Mt

⌘

1/⇢

WT = ( MT / )

,

1/⇢

.

Thus, 1 Wt = Et Mt = =

Z

T

Ms Cs ds + MT WT Z T 1 1/⇢ 1 Et e s/⇢ Ms1 1/⇢ ds + 1/⇢ MT Mt t "Z ✓ ◆ T Ms 1 1/⇢ 1/⇢ 1/⇢ Mt Et e s/⇢ ds + Mt t t

1/⇢

1/⇢

✓

MT Mt

◆1

1/⇢

#

.

138

14 Continuous-Time Portfolio Choice and Beta Pricing

We have dM = M so, for s > t, Ms = exp Mt

✓

r(s

r dt

1 2

t)

Z

dB , s

t

2 u du

Conditional on date–t information, the distribution of Z s Z 2 and u du t

depend only on (b) Writing M

1/⇢

Z

s u dBu t

◆

.

s u dBu t

t.

= eX , where X = dM M

1/⇢ 1/⇢

log M/⇢ and using deX /eX = dX + (dX)2 /2, we have ✓ ◆ 2 1 1 2 = r dt dt dB + 2 dt ⇢ 2 2⇢ 2 2⇢r + (1 + ⇢) = dt + dB . 2⇢2 ⇢

Also, 1 df = ft dt + f d + f (d )2 2 1 2 = ft dt + f [(✓ ) dt + dB] + f dt 2  1 2 = ft + (✓ )f + f dt + f dB . 2 Hence, dW dM = W M

1/⇢ 1/⇢

df + + f

dM M

1/⇢ 1/⇢

!✓

df f

ft + (✓ 2⇢r + (1 + ⇢) 2 = dt + 2 2⇢ Rt (c) The di↵erential of 0 Ms Cs ds + Mt Wt is

◆

)f + f

M C dt + M dW + W dM + (dW )(dM ) = M W = MW

e

t

M

M

1/⇢

1/⇢ f

1 2 f 2

dt +

✓

f + ⇢ f

✓

C dW dM + + + W W M

+

dW W

r dt

dB

✓

◆

dB +

f dt . ⇢f

◆◆ dM M ◆ ! f + dt . ⇢ f

dW W ✓

◆✓

Using the previous formula for dW/W and equating the drift to zero yields e

1/⇢

t

f

2⇢r + (1 + ⇢) + 2⇢2

2

+

ft + (✓

)f + f

1 2 f 2

f + ⇢f

r

✓

f + ⇢ f

◆

= 0.

14 Continuous-Time Portfolio Choice and Beta Pricing

139

Multiplying by 2⇢2 f and simplifying yields ⇣ 2⇢2 e

t

⌘

1/⇢

+ (1

2

⇢)(2⇢r +

)]f + 2⇢2 ft + 2⇢[⇢(✓

) + (1

⇢)

]f + ⇢2

2

f

= 0.

Rt (d) Given that f satisfies the PDE from Part (c), the di↵erential of 0 Ms Cs ds + Mt Wt is ✓ ✓ ◆✓ ◆◆ ✓ ◆ C dW dM dW dM f MW + + + = MW + dB dB W W M W M ⇢ f = M W (⇡

) dB .

Thus, ✓ ◆✓ ◆ C dM dW dM dt W M W M ✓ ◆✓ ◆ C dW dM dt + r dt + ⇡ dB W W M C dt + r dt + ⇡ dt + ⇡ dB W C dt + r dt + ⇡(µ r) dt + ⇡ dB . W

dW = (⇡ W = = =

) dB

14.9. This exercise demonstrates the equivalence between the intertemporal and static budget constraints in the presence of non-portfolio income when the investor can borrow against the income, as asserted in Section 14.2. Let M be an SDF process and Y a non-portfolio income process. Assume Z T E Mt |Yt | dt < 1 0

for each finite T . The intertemporal budget constraint is dW = rW dt +

0

(µ

r1) dt + Y dt

C dt +

0

dB .

(a) Suppose that (C, W, ) satisfies the intertemporal budget constraint (14.43), C

0, and the

nonnegativity constraint (14.7) holds. (i) Suppose the horizon is finite. Show that (C, W ) satisfies the static budget constraint Z T Z T W0 + E Mt Yt dt E Mt Ct dt + MT WT 0

by showing that

0

Z

t

Ms (Cs 0

Ys ) ds + Mt Wt

140

14 Continuous-Time Portfolio Choice and Beta Pricing

is a supermartingale. Hint: Show that it is a local martingale and at least as large as the martingale

Xt , where X t = Et

Z

T

Ms Ys ds . 0

This implies the supermartingale property (Appendix A.13.) (ii) Suppose the horizon is infinite and limT !1 E[MT WT ]

0. Assume Y

0. Show that the

static budget constraint W0 + E

Z

1

E

Mt Yt dt

0

Z

1

Mt Ct dt

0

holds. (b) Suppose the horizon is finite, markets are complete, C

0, and (C, W ) satisfies the static

budget constraint (14.44) as an equality. Show that there exists

such that (C, W, ) satisfies

the intertemporal budget constraint (14.43). (a) (i) The di↵erential of

Z

is M (C

t

Ms (Cs

Ys ) ds + Mt Wt

0

Y ) dt + M dW + W dM + (dM )(dW )

= rM W dt + M 0 (µ

r1) dt + M

0

dB

rM W dt

MW =M

0

dB 0

dB

M

0

MW

dt 0

dB .

Thus, it is a local martingale. By the nonnegative wealth constraint (14.7) and the nonnegativity of C,

This implies

Hence,

Z Z

t

Ms Cs ds + Mt Wt + Et 0

Z

T

Ms Ys ds

t

Ms (Cs

Et

Ys ) ds + Mt Wt

0

Z

0.

t

Z

T

Ms Ys ds . 0

t

Ms (Cs

Ys ) ds + Mt Wt

0

is a supermartingale, which implies Z W0 E

T

Ms (Cs

Ys ) ds + MT WT

0

Rearranging produces the static budget constraint.

.

14 Continuous-Time Portfolio Choice and Beta Pricing

141

(ii) Taking the limit of the finite-horizon static budget constraint as T ! 1, using the nonnegativity of C, Y and M and the Monotone Convergence Theorem yields Z 1 Z 1 Z W0 + E Mt Yt dt E Mt Ct dt + lim E[MT WT ] E 0

T !1

0

1

(b) The proof is the same as in Exercise 13.6, replacing C in that exercise by C defining W t = Et it follows that

Z

Z

T t

Ms (Cs Mt

Ys ) ds +

MT WT Mt

,

t

Ms (Cs

Ys ) ds + Mt Wt

0

is a martingale. Therefore, Z

t

Ms (Cs

Ys ) ds + Mt Wt = W0 +

0

for some stochastic process

Z

t

0 s dBs

0

. Setting = W⌃

1

(µ

r1) +

1 ( M

1 0

)

,

it follows that (C, W, ) satisfies the intertemporal budget constraint.

Mt Ct dt .

0

Y . Specifically,

Part III

Derivative Securities

15 Option Pricing

15.1. The fundamental PDE (15.23) can be written as ⇥ + rS

+

1 2

2 2

S

= rV .

For the Black-Scholes call option formula, calculate the theta and gamma and verify the fundamental PDE. The delta, gamma and theta can be calculated as =

@ ⇥ SN (d1 ) @S

e

rT

K N(d2 )

⇤

@d1 @d2 e rT K n(d2 ) @S @S ✓ ◆ @d1 @d2 = N(d1 ) + S n(d1 ) @S @S = N(d1 ) + S n(d1 )

= N(d1 ) , =

@ @S

@d1 @S 1 = n(d1 ) p , S T ⇤ @ ⇥ ⇥= SN (d1 ) e rT K N(d2 ) @T @d1 @d2 = S n(d1 ) + e rT K n(d2 ) re rT K N(d2 ) @T @T ✓ ◆ @d2 @d1 = S n(d1 ) re rT K N(d2 ) @T @T = n(d1 )

=

S n(d1 ) p 2 T

Thus, we want to confirm that

re

rT

K N(d2 ) .

146

15 Option Pricing

S n(d1 ) p 2 T

re

rT

1 K N(d2 ) + rS N(d1 ) + 2

2 2

S



n(d1 )

1 p

S

⇥ = r SN (d1 )

T

e

rT

It suffices to note that the two terms on the left-hand side involving the reciprocal of

⇤ K N(d2 ) .

p

T cancel.

15.2. Use put-call parity to show that a European put and a European call with the same strike and time to maturity have the same gamma. Letting C denote the call price and P the put price, put-call parity at date 0 is P = C + e

rT K

S.

Therefore, @P @C = @S @S

1,

and @2P @2C = . @S 2 @S 2 15.3. Consider an asset with a constant dividend yield q. Assume the price S of the asset satisfies dS = (µ S

q) dt + dB ,

where B is a Brownian motion under the physical measure, and µ and

are constants. Consider a

European call and a European put with strike K on the asset. Assume the risk-free rate is constant, and adopt the assumptions of Section 15.3. (a) Let A denote the event ST > K. Show that E[ST 1A ] = e(µ d⇤1 =

log(S0 /K) + µ q + p T

q)T S

1 2 2

T

⇤ 0 N(d1 ),

where

.

Hint: This can be computed directly under the physical measure or by changing measures using e(q

µ)T S

T /S0

as the Radon-Nikodym derivative.

(b) Show that E[K1A ] = K N(d⇤2 ), where d⇤2 = d⇤1

p

T.

(c) It follows from the previous parts that the expected return of the European call under the physical measure, if held to maturity, is e(µ e

q)T S

qT S

⇤ 0 N(d1 )

0 N(d1 )

K N(d⇤2 ) , e rT K N(d2 )

where d1 and d2 are defined in (15.29). Assuming T = 1, µ = 0.12, r = 0.04, q = 0.02, and = 0.20, show that the expected rate of return of a European call that is 20% out of the money (S0 /K = 0.8) is 118%.

15 Option Pricing

147

(d) Show that the expected return of the European put under the physical measure, if held to maturity, is K N( d⇤2 ) e(µ q)T S0 N( d⇤1 ) . e rT K N( d2 ) e qT S0 N( d1 ) Assuming T = 1, µ = 0.12, r = 0.04, q = 0.02, and

= 0.20, show that the expected rate of

return of a European put that is 20% out of the money (S0 /K = 1.2) is

54%.

(a) Using the physical measure, recall that ✓

log ST = log S0 + µ

1 2

q

2

◆

T+

p

Tx ˜,

p where x ˜ ⌘ BT / T is a standard normal under the physical measure. The event A is the event d⇤2 . Thus,

that log ST > log K, which is equivalent to x ˜> 1 E[ST 1A ] = p 2⇡ =e

Z

1 d⇤2

S0 e(µ

q

Z 1 S0 p 2⇡ Z 1 q)T S0 p 2⇡

1

(µ q)T

= e(µ

d⇤2 1 d⇤1

q)T

S0 [1

= e(µ

q)T

S0 N(d⇤1 ) ,

p

e

(x

e

y 2 /2

p p

Tx

e

T )2 /2

x2 /2

dx

dx

dy

N( d⇤1 )]

= e(µ

changing variables as y = x

2 /2)T +

T to obtain the third equality. To compute E[ST 1A ] via the

change of measure, note that (µ q)T

E[ST 1A ] = e

S0 E

"

e(q

µ)T S

S0

T

#

1A = e(µ

q)T

S0 E⇤ [1A ] = e(µ

q)T

S0 prob⇤ (A) ,

where the asterisks denote the measure defined by the Radon-Nikodym derivative e(q Set ⇠t = e(q

µ)t S

t /S0 .

Then d⇠ = ⇠

dB ,

so ⇠ is a martingale under the physical measure. Girsanov’s theorem implies that ✓ ◆ d⇠ ⇤ dB = dB (dB) = dB dt ⇠ defines a Brownian motion under prob⇤ . Thus,

µ)T S

T /S0 .

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15 Option Pricing

dS = (µ S

2

q+

) dt + dB ⇤ ,

and A is the event that ✓

log S0 + µ This is equivalent to

1 q+ 2

2

◆

T + BT⇤ > log K .

B⇤ pT < d⇤1 . T

Therefore, prob⇤ (A) = N(d⇤1 ), and E[ST 1A ] = e(µ

q)T S

⇤ 0 N(d1 ).

(b) We have E[K1A ] = K prob(A). As shown in the solution to Part (a), A is the event that x ˜>

d⇤2 ,

where x ˜ is a standard normal under the physical measure. Thus, prob(A) = 1 N( d⇤2 ) = N(d⇤2 ), and E[K1A ] = K N(d⇤2 ). (c) The expected payo↵ of the call under the physical measure is 0.0305K, and its Black-Scholes price is 0.0140K. Thus, the expected rate of return is 0.0305/0.0140

1 = 118%.

(d) Denote the complement of A by Ac . The expected payo↵ of the put under the physical measure is E[K1Ac ]

E[ST 1Ac ] = K prob(Ac ) = K[1

e(µ

N(d⇤2 )]

= K N( d⇤2 )

e(µ

q)T

e(µ

S0 prob⇤ (A)

q)T

q)T

S0 [1

N(d⇤1 )]

S0 N( d⇤1 ) .

Thus, the expected return is as claimed. For the parameters given, the expected payo↵ of the put is 0.0080K, and its Black-Scholes price is 0.0173K, so the expected rate of return is 0.0080/0.0173

1=

54%.

15.4. Adopt the Black-Scholes assumptions. Consider an American put and exercise boundary f with associated exercise time 8 >

:T otherwise . Show that there is a trading strategy with value max(0, K

S⌧ ) at date ⌧ and value (15.26) at

date 0. Hint: Consider replicating the date–T payo↵ X = er(T

⌧ ) max(0, K

S⌧ ).

15 Option Pricing

149

From the market completeness result, there exists a self-financing wealth process W such that r(T t) ER [X] t

WT = X and Wt = e W0 = e

rT

h ER er(T

for each t 2 [0, T ]. Taking t = 0 yields ⌧)

i ⇥ S⌧ ) = E R e

max(0, K

Taking t = ⌧ yields W⌧ = e

r(T ⌧ ) R E⌧

h

er(T

⌧)

r⌧

max(0, K

i S⌧ ) = E R ⌧ [max(0, K

max(0, K

⇤ S⌧ ) .

S⌧ )] = max(0, K

S⌧ ) .

Note that this calculation for t = ⌧ depends on the stopping theorem.

15.5. Consider a perpetual American call on an asset with price S given by (15.1) and with a constant dividend yield q > 0. Assume the risk free rate r is constant. Let V (St ) denote the value of the call (exercised optimally). Assume V is twice continuously di↵erentiable. (a) Using the fact that M V is a local martingale, derive an ordinary di↵erential equation (ODE) that V must satisfy in the continuation region. (b) Show that AS satisfies the ODE for constants A and equation 1 2 Show that the positive root

2 2

✓

+ r

q

1 2

2

if and only if

◆

r = 0.

of this equation satisfies

(c) The general solution of the ODE is A1 S

1

is a root of the quadratic

> 1.

+ A2 S 2 , where the

i

are the roots of the quadratic

equation (15.32). Use the fact that limS#0 f (S) = 0 to show that f (S) = AS for some constant A, where

is the positive root of the quadratic equation.

(d) Use the value matching and smooth pasting conditions to show that the optimal exercise point is S ⇤ = K/(

1), where

is the positive root of the quadratic equation, and to derive A.

(e) Show that, if S0  S ⇤ , the value of the call is ✓

K 1

◆✓

1

◆ ✓

S0 K

◆

.

(a) We have 1 dV = V 0 (S) dS + V 00 (S) (dS)2 2 1 2 2 00 = SV 0 (S)[µ dt + dB] + S V (S) dt . 2

150

15 Option Pricing

Also, dM = M

r dt

µ+q

r

db +

d" . "

Therefore ✓ ◆✓ ◆ d (M V ) dM dV dM dV = + + MV M V M V µ+q r d" SV 0 1 = r dt dB + + [µ dt + dB] + " V 2

2 2V

S

00

V

dt

(µ + q

r)

SV 0 dt . V

For this to be a local martingale, we must have rV (S) + (r

q)SV 0 (S) +

1 2

S V 00 (S) = 0 .

2 2

(b) If V (S) = AS , then SV 0 (S) = AS , and S 2 V 00 (S) = ( rAS + (r This is satisfied by (c) If

1 ( 2

q)AS +

and A 6= 0 if and only if

1)

1)AS . Hence, the ODE is 2

AS = 0 .

is a root of the quadratic equation (15.32).

< 0, we have limS#0 S = 1, so the coefficient Ai corresponding to the negative root of the

quadratic equation must be zero. (d) The value matching condition is A(S ⇤ ) = S ⇤ K. The smooth pasting condition is A(S ⇤ )

1

=

1. Multiplying the latter equation by S ⇤ and substituting from the former gives (S ⇤ K) = S ⇤ . Thus, S ⇤ = K/(

1). This implies A = (S ⇤

K)(S ⇤ )

=

K 1

✓

K 1

◆

.

(e) Substituting the values of A and S ⇤ from the previous part, we have ✓ ◆ ✓ ◆ ✓ ◆ K K 1 S0 K V (S0 ) = AS0 = S0 = . 1 1 1 K 15.6. Consider a perpetual American put under the assumptions of Exercise 15.5. Consider the exercise time ⌧ = min{t | St  S ⇤ } for a constant S ⇤ . Let V (St ) denote the value of the put (exercised optimally). Assume V is twice continuously di↵erentiable and limS!1 V (S) = 0. Show that the optimal exercise point is S⇤ =

K 1

,

where

is the negative root of the quadratic equation (15.32). Show that the value of the put is,

for S0

S⇤,

15 Option Pricing

✓

K 1

◆

1

✓

K S0

◆

151

.

The same calculations as in the previous exercise show that V (S) = A1 S + A2 S , for constants Ai , where

and

are the roots of the quadratic equation (15.32). The fact that

limS!1 V (S) = 0 implies that the coefficient Ai corresponding to the positive root must be zero. Thus, V (S) = AS , where give A(S ⇤ ) = K

is the negative root. The value matching and smooth pasting conditions

S ⇤ and A(S ⇤ )

from the former gives (K

1

S⇤) =

A = (K

1. Multiplying the latter equation by S ⇤ and substituting

=

S ⇤ . Thus, S ⇤ = S ⇤ )(S ⇤ )

=

K/(1 ). This implies ✓ ◆ K . 1

K 1

Thus, V (S0 ) = AS0 = (K

⇤

⇤

S )(S )

=

K 1

✓

1

◆

✓

K S0

◆

.

15.7. Consider perpetual American calls and puts under the assumptions of Exercise 15.5. Let ER denote expectation under the infinite-horizon risk neutral probability. Let ⌧ = min{t | St = S ⇤ } for a constant S ⇤ . For any stochastic process X,

ER [X⌧ 1{⌧ S⇤ : if S ⇤  S0 , S0 where

is the positive root and

the negative root of the quadratic equation (15.32). See Borodin

and Salminen (2000, p. 622). Use these facts to derive the values of perpetual American calls and puts.

152

15 Option Pricing

The value of a call option exercised at S ⇤ (S

⇤

S0 is

R

r⌧

K)E [e

] = (S

⇤

K)

✓

S0 S⇤

◆

.

Maximizing this in S ⇤ gives K . 1

S⇤ =

Hence the value of a call, when S0  S ⇤ , is ✓ ◆✓ ◆ K ( 1)S0 K = 1 K

K 1

✓

The value of a put option exercised at S ⇤  S0 is (K

S ⇤ )ER [e

r⌧

S⇤)

] = (K

✓

1

S⇤ S0

◆✓ ✓

◆

S0 K

◆

.

.

Maximizing this in S ⇤ gives S⇤ = Hence, the value of a put, when S0

S ⇤ , is ✓ K 1 1

K 1 ◆

.

✓

K S0

◆

.

15.8. Consider a second risky non-dividend-paying asset with price Z. Assume dZ = µz dt + Z

z

dBz ,

where Bz is a Brownian motion under the physical measure. Let ⇢ denote the correlation process of Bz and B. (a) Define ⇠t = Mt St = Et [MT ST ]. Show that d⇠ dM = r dt + dB + . ⇠ M (b) Prove that (dBz )

✓

d⇠ ⇠

◆

=

✓

⇢

µz

r z

◆

dt .

(c) Use Girsanov’s theorem to show that dZ = (r + ⇢ Z where Bz⇤ is a Brownian motion under probS .

z ) dt

+

z

dBz⇤ ,

15 Option Pricing

153

(a) We have ✓ ◆✓ ◆ d⇠ dM dS dM dS = + + ⇠ M S M S dM = + µ dt + dB (µ r) dt M dM = r dt + dB + . M (b) We have (dBz )

✓

d⇠ ⇠

◆

= (dBz )(dB) + (dBz ) = ⇢ dt

µz

r

✓

dM M

◆

µz

r

dt ,

z

using the fact that

✓

dZ Z

◆✓

dM M

◆

=

(µz

(c) Define Bz⇤ to be zero at date 0, and define ✓ ◆ d⇠ ⇤ dBz = dBz (dBz ) = dBz ⇠

r) dt .

✓

⇢

z

◆

dt .

By Girsanov’s theorem, Bz⇤ is a Brownian motion under probs . We have dZ = µz dt + Z

dBz  ✓ ⇤ = µz dt + z dBz + ⇢ = (r + ⇢

z

z ) dt

+

z

µz

r z

◆

dt

dBz⇤ .

15.9. This exercise verifies the assertion (15.6) regarding the dynamics of an SDF process. Suppose the information in the economy is given by independent Brownian motions B1 , . . . , Bn . Consider a non-dividend-paying asset with price satisfying n

X dS = µ dt + S

i dBi

i=1

for stochastic processes µ and (a) Define a stochastic process

i.

and Brownian motion B s such that dS = µ dt + dB s . S

154

15 Option Pricing

(b) Consider an SDF process M . We have dM = M for some stochastic processes

i.

n X

r dt

i dBi

i=1

and Brownian motion B m such

Define a stochastic process

that dM = M

dB m .

r dt

(c) Let ⇢ denote the correlation process of B s and B m . Show that ⇢ = (µ (d) Show that there is a local martingale Z such that dM = M

r dt

µ

r

dB s + dZ ,

and (dB s )(dZ) = 0. (a) Define B0s = 0,

v u n uX =t

2 i ,

i=1

dB s =

n 1X

i dBi .

i=1

By Levy’s theorem, B s is a Brownian motion, and we have dS = µ dt + dB s . S (b) Define B0m = 0,

v u n uX =t

2, i

i=1

dB m =

n 1X

i dBi .

i=1

By Levy’s theorem, B m is a Brownian motion, and we have dM = M (c) We have (µ

so ⇢ = (µ

r)/ .

dB m .

r dt

r) dt =

✓

dS S

◆✓

dM M

=

(dB s )(dB m )

=

⇢ dt ,

◆

r)/ .

15 Option Pricing

155

(d) We want dM = M

µ

r dt

r

dB s + dZ ,

which is equivalent to dM µ r + r dt + dB s M µ r = dB m + dB s .

dZ =

This defines a local martingale Z, and (dB s )(dZ) =

(dB s )(dB m ) +

µ

r

dt =

⇢ dt +

µ

r

dt = 0 .

15.10. A compound option is an option on an option. Suppose there is a constant risk-free rate, and the underlying asset price has a constant volatility. Consider a European call option with strike K 0 maturing at T 0 . Assume the underlying asset does not pay dividends during [0, T 0 ]. Consider a European option maturing at T < T 0 to purchase the first call option at price K. The purpose of this exercise is to value the “call on a call.” The value at T of the underlying call is given by the Black-Scholes formula with T 0

T being the time to maturity. Denote this value by V (T, ST ). The

value at T of the call on a call is max(0, V (T, ST )

K) .

Let S ⇤ denote the value of the underlying asset price such that V (T, S ⇤ ) = K. The call on a call is in the money at its maturity T if ST > S ⇤ , and it is out of the money otherwise. Let A denote the set of states of the world such that ST > S ⇤ , and let 1A denote the random variable that equals 1 when ST > S ⇤ and 0 otherwise. The value at its maturity T of the call on a call is V (T, ST )1A

K1A .

(a) What is the value at date 0 of receiving the payo↵ K1A at date T ? (b) To value receiving V (T, ST )1A at date T , let C denote the set of states of the world such that ST 0 > K 0 , and let 1C denote the random variable that equals 1 when ST 0 > K 0 and 0 otherwise. Recall that V (T, ST ) is the value at T of receiving ST 0 1C

K 0 1C at date T 0 . Hence,

the value at date 0 of receiving V (T, ST )1A at date T must be the value at date 0 of receiving (ST 0 1C

K 0 1C )1A at date T 0 . Via the following steps, show that this date–0 value is M(d1 , d01 ,

p T /T 0 )

e

rT 0

K 0 M(d2 , d02 ,

p T /T 0 ) ,

156

15 Option Pricing

where M(a, b, ⇢) denotes the probability that ⇠1 < a and ⇠2 < b, where ⇠1 and ⇠2 are standard normals with correlation ⇢. (i) Show that the value at date 0 of receiving V (T, ST )1A at date T is S0 probS (D)

e

rT 0

K 0 probR (D) ,

where D = A \ C.

(ii) Show that probS (D) is the probability that B⇤ p T < d1 T

B⇤ 0 pT < d01 , T0

and

where log(S0 /S ⇤ ) + r + 12 p T 0 log(S0 /K ) + r + 12 p d01 = T0

2

d1 =

2

T

,

T0

,

and where B ⇤ denotes a Brownian motion under probS . Note that the random variables in p (15.35) are standard normals under probS with a correlation equal to T /T 0 .

(iii) Show that probR (D) is the probability that B⇤ p T < d2 T

B⇤ 0 pT < d02 , T0

and

where d2 = d1

p

T,

d02 = d01

p

T0 ,

and where B ⇤ now denotes a Brownian motion under probR . (a) This is a digital option. Its value is e

rT K

probR (A). The calculation of probR (A) is the same

calculation made in Section 15.6, replacing K with S ⇤ . Therefore, the value is e (b) (i) The random variable (ST 0 1C

rT K

N(d2 ).

K 0 1C )1A equals

ST 0 1C 1A

K 0 1C 1A = ST 0 1D

K 0 1D .

The random variable ST 0 1D is the payo↵ of a share digital paying on the event D. Following the same calculation as in Section 15.6, the value at date 0 of this share digital is S0 probS (D). The random variable K 0 1D is the payo↵ of a digital paying on the event D. Following the same calculation as in Section 15.6, the value at date 0 of this digital is e

rT 0 K 0 probR (D).

15 Option Pricing

(ii) We have d log S =

✓

1 r+ 2

2

◆

157

dt + dB ⇤

where B ⇤ is a Brownian motion under probS . The event D occurs if and only if log ST > log S ⇤ and log ST 0 > log K. This is equivalent to ✓ ◆ 1 2 T + BT⇤ > log S ⇤ , log S0 + r + 2 ✓ ◆ 1 2 log S0 + r + T 0 + BT⇤ 0 > log K . 2 Rearranging shows that D is the event (15.35). (iii) We have d log S =

✓

r

1 2

2

◆

dt + dB ⇤

where B ⇤ is a Brownian motion under probR . The event D occurs if and only if log ST > log S ⇤ and log ST 0 > log K. This is equivalent to ✓ ◆ 1 2 log S0 + r T + BT⇤ > log S ⇤ , 2 ✓ ◆ 1 2 log S0 + r T 0 + BT⇤ 0 > log K . 2 Rearranging shows that D is the event (15.36).

15.11. Calculate the value of a call on a put assuming a constant risk-free rate and a constant volatility for the underlying asset price. Assume the underlying asset does not pay dividends during [0, T 0 ], where T 0 is the time to maturity of the put. Assume K 0 such that the Black-Scholes value of the put at time T equals K when ST = S ⇤ . The call on the put will be in the money at T if ST < S ⇤ . Let A denote the event ST < S ⇤ . The put will be in the money at T 0 if ST 0 < K. Let C denote this event, and set D = A \ C. The value at date 0 of the call on the put is the value at date 0 of receiving the cash flows T and K 0 1D

K1A at date

ST 0 1D at date T 0 . Following the same calculations as in the previous exercise, the

value at date 0 of the call on the put is

158

15 Option Pricing

e

rT

K probR (A) + e

We have d log S =

✓

rT 0

r

K 0 probR (D)

1 2

2

◆

S0 probS (D) .

dt + dB ⇤

where B ⇤ is a Brownian motion under probR . The event A is the event ✓ ◆ 1 2 log S0 + r T + BT⇤ < log S ⇤ . 2 This is equivalent to

B⇤ pT < T

d2 .

Thus, probR (A) = N( d2 ). The event D is the event ✓ ◆ 1 2 log S0 + r T + BT⇤ < log S ⇤ , 2 ✓ ◆ 1 2 log S0 + r T 0 + BT⇤ 0 < log K . 2 This is equivalent to B⇤ pT < T ⇤ B 0 pT < T0

d2 , d02 .

Let M(a, b, ⇢) denote the probability that two standard normals with correlation ⇢ are less than a p and b respectively. Then probR (D) = M( d2 , d02 , T /T 0 ). We also have ✓ ◆ 1 2 d log S = r + dt + dB ⇤ 2 where B ⇤ is a Brownian motion under probS . The same calculation shows that probS (D) = p M( d1 , d01 , T /T 0 ). Thus, the value at date 0 of the call on the put is e

rT

K N(d2 ) + e

rT 0

K 0 M( d2 , d02 ,

p T /T 0 )

S0 M( d1 , d01 ,

p T /T 0 ) .

16 Forwards, Futures, and More Option Pricing

16.1. Under the assumption that S1 and S2 have volatilities d(S1 /S2 ) = something dt + S1 /S2

q

2 2

i

+

2 2

+

✓

and correlation ⇢, show that 2⇢

1 2 dB

for a Brownian motion B. Hint: Use Levy’s theorem. Set Y = S1 /S2 . We have dY dS1 = Y S1 = µ1 dt + = (µ1

✓

dS2 S2

dS1 S1

1 dB1

µ2

◆✓

dS2 S2

µ2 dt

1 2

+

2 2 ) dt

=

q

2 1

2⇢

B0 = 0, and dB =

1

(

dS2 S2

2 dB2

⇢

Define

◆

+

1 2

1 dB1

⇢

1 dB1

+

◆2

1 2 dt

+

2 2 dt

2 dB2 .

2 2,

2 dB2 )

.

Then B is a Brownian motion by Levy’s theorem, and dY = (µ1 Y

µ2

⇢

1 2

+

2 2 ) dt

+ dB .

16.2. This exercise implements Merton’s formula when the discount bond price is given by the Vasicek model (Section 17.1).  is the rate of mean reversion of the short rate process, and (absolute) volatility of the short rate process.

is the

160

16 Forwards, Futures, and More Option Pricing

Suppose the price of a non-dividend-paying stock has a constant volatility . Assume the volatility at date t of a discount bond maturing at T > t is ⇣ 1 e  for constants  > 0 and

(T t)

⌘

> 0. Assume the discount bond and stock have a constant correlation ⇢.

(a) Using the result of the previous exercise, write the volatility of S/P (T ) as a function (t). (b) Define avg

s

=

1 T

Show that 2 avg

=

2

1 + 2 



2

2⇢

2

(2

avg

>

T

(t)2 dt . 0

2⇢

(c) Use l’Hˆopital’s rule to show that for small T , (d) Show that

Z

avg

)

✓

1

e T

T

◆

+

2

✓

1

e 2T 2T

◆

.

⇡ .

for large T if ⇢ is sufficiently small.

(a) Taking S1 = S and S2 = P in the previous exercise, it follows that the volatility of S/P is r 2 2⇢ 2 2+ (t) = 1 e (T t) 1 e (T t) . 2   (b) We have Z 1 T (t)2 dt = T 0

2

=

2 S

+

=

2

1 + 2 

Z

2

+

2 T 2

2 T



T

0 T 2

⇣

1

2e

2 1 

(T t)

T

e

2⇢

(2

2

Z T⇣ ⌘ 2⇢ 1 e (T t) dt T 0  1 2⇢ 1 2T + 1 e T 1 e T 2 T  ✓ ◆ ✓ ◆ 1 e T e 2T 2 1 2⇢ ) + T 2T

+e

2(T t)

⌘

dt

(c) By l’Hˆ opital’s rule, lim

1

e T

T !0

T

= lim

T !0

Therefore, 1 lim T !0 T (d) We have

This is larger than

2

1 lim T !1 T

Z

T

1

Z

T

e 2T = 1. 2T

(t)2 dt =

2

.

0

(t)2 dt =

2

2

+

0

2⇢ 2

if and only if

2⇢ > 0

,

⇢
K) for a constant K. Show that W must satisfy the PDE ⇥ Wt + rSWS + ⇤ ✓⇤

⇤ V

⇢(µ

r)e

V

⇤

1 1 WV + e2V S 2 WSS + 2 2

2

WV V + ⇢eV SWSV = 0 .

(a) As shown in the previous chapter, dM = M where

r dt

µ

r

dB1 +

is a local martingale uncorrelated with B1 . Define ✓ ◆ d dt = (dB2 ) ,

d

, by

and set d" d = "

+ dB2 .

Then, we have dM = M

r dt

µ

r

dB1

dB2 +

d" , "

and ✓

◆ ✓ ◆ d" d (dB1 ) = (dB1 ) + (dB2 )(dB1 ) = 0 , " ✓ ◆ ✓ ◆ d" d (dB2 ) = (dB2 ) + (dB2 )2 " =

dt + dt = 0 .

(b) By Girsanov’s theorem, dB1⇤ = dB1 +

µ

r

dt

and

dB2⇤ = dB2 + dt

define Brownian motions under the risk neutral probability. Substituting yields

16 Forwards, Futures, and More Option Pricing

dS = r dt + dB1⇤ , S dV = (✓ = (✓⇤

V ) dt V ) dt

⇢(µ

r)

⇢(µ

dt

r)

p

⇢2 dt + [⇢ dB1⇤ +

1

dt + [⇢ dB1⇤ +

where

p 1 

⇤

✓ =✓ (c) From Itˆo’s formula, the drift of dW is  ⇢(µ Wt + rSWS + (✓⇤ V )

r)

WV +

p 1

⇢2

1 2

⇢2 dB2⇤ ] ,

p 1

167

⇢2 dB2⇤ ]

.

2 2

S WSS +

1 2

2

WV V +

⇢SWSV .

Setting this to zero yields the PDE.

16.8. This problem is adapted from Hull and White (1987). Assume dSt = µt dt + St d

t

t dB1t ,

= ( t ) dt + ( t ) dB2t ,

for some functions (·) and (·), where B1 and B2 are independent Brownian motions under the physical probability measure and µ may be a stochastic process. Assume

t

in (16.26) equals

( t ) for some function (·). Assume there is a constant risk-free rate and the asset does not pay dividends. (a) Show that dSt = r dt + St d for some function

⇤ (·),

t

=

⇤

⇤ t dB1t ,

⇤ ( t ) dt + ( t ) dB2t ,

where B1⇤ and B2⇤ are independent Brownian motions under the risk

neutral probability. (b) Use iterated expectations to show that the date–0 value of a call option equals ⇥ ER S0 N(d1 )

e

rT

⇤ K N(d2 ) ,

168

16 Forwards, Futures, and More Option Pricing

where log(S0 /K) + r + p d1 = avg T p d2 = d1 avg T ,

1 2 2 avg

T

,

and the risk-neutral expectation in (16.27) is taken over the random “average” volatility s Z 1 T 2 = dt avg T 0 t on which d1 and d2 depend. This average volatility is explained in a nonrandom context in Section 15.12. (c) Implement the Black-Scholes formula numerically. Plot the value of an at-the-money (S0 = K) call option as a function of the volatility . Observe that the option value is approximately an affine (linear plus constant) function of . (d) Explain why the value of an at-the-money call option on an asset with random volatility is approximately given by the Black-Scholes formula with 2s 3 Z T 1 2 dt5 ER 4 T 0 t input as the volatility.

(e) From Part (c), one should observe that the Black-Scholes value is not exactly linear in the volatility. Neither is it uniformly concave nor uniformly convex; instead, it has di↵erent shapes in di↵erent regions. Explain why if it were concave (convex) over the relevant region, then the Black-Scholes formula with

2s

ER 4

1 T

Z

T 0

3

2 5 t dt

input as the volatility would overstate (understate) the value of the option. (a) Setting dB1⇤ = dB1 +

µ

r

dt

and

dB2⇤ = dB2 + dt

yields dS = r dt + dB1⇤ , S d = ( ) dt + ( )[dB2⇤ =

⇤

( ) dt + ( ) dB2⇤ ,

( ) dt]

16 Forwards, Futures, and More Option Pricing

169

where ⇤

(b) Conditioning on

and variance S0 N(d1 )

e

avg ,

2T . rT K

( )= ( )

( ) ( ).

log ST is normally distributed under probR with mean ✓ ◆ 1 2 log S0 + r T 2 avg

We know that, given this distribution, the mean of e

rT

max(0, ST

K) is

N(d2 ). By iterated expectations,

ER [e

rT

max(0, ST

⇥ K)] = ER ER [e

rT

K) |

max(0, ST

= ER [S0 N(d1 )

e

rT

K N(d2 )] .

avg ]

⇤

(d) If f is an affine function and x ˜ is a random variable, then E[f (˜ x] = f (E[˜ x]). Because the function avg

7! S0 N(d1 (

avg ))

e

rT

K N(d2 (

avg ))

is approximately affine, we have ER [S0 N(d1 (

avg ))

e

rT

K N(d2 (

avg ))]

⇡ S0 N(d1 (ER [

avg ]))

e

rT

K N(d2 (ER [

avg ])) .

(e) If f is a concave function, then Jensen’s inequality states that E[f (˜ x]  f (E[˜ x]). Thus, if the function avg

7! S0 N(d1 (

avg ))

e

rT

K N(d2 (

avg ))

were concave, we would have ER [S0 N(d1 (

avg ))

e

rT

K N(d2 (

avg ))]

 S0 N(d1 (ER [

avg ]))

e

rT

K N(d2 (ER [

avg ])) .

16.9. This exercise generalizes the price of risk specification in the Heston model. Under the condition in Part (d), it is a member of the extended affine family defined by Cheridito, Filipovi´c, and Kimmel (2007), and M R is a martingale. Part (a) transforms the vector process (log S, V ) into a “standard form.” See Section 16.9 for discussion, including the significance of the condition in Part (d). The usefulness of generalizing the price of risk process is that it permits more flexible dynamics under the physical measure (and hence more flexible expected returns) while preserving the pricing formulas (that depend on dynamics under the risk neutral probability). In the Heston model (16.17), define Y1 = V /

2

and Y2 = log S

⇢V / .

170

16 Forwards, Futures, and More Option Pricing

(a) Derive the constants ai , bij and such that 0 1 0 1 0 10 1 0p 10 1 dY1 a1 b11 b12 Y1 Y1 0 dB1 @ A = @ A dt + @ A @ A dt + @ A@ A. p dY2 a2 b21 b22 Y2 0 Y1 dB2

(b) Consider a price of risk specification

for constants 21

ij .

1t

=

10

2t

=

20

+ p

11 Y1t

Y1t

+ p

21 Y1t

Y1t

, ,

The specification in Section 16.8 is the special case

as functions of

10

and

11

10

= 0. Derive

10

and

11

is such that M R is a martingale. Derive

constants a⇤i and b⇤ij in terms of 10 and 11 such that 0 1 0 1 0 10 1 0p 10 1 dY1 a⇤1 b⇤11 b⇤12 Y1 Y1 0 dB1⇤ @ A = @ A dt + @ A @ A dt + @ A@ A, p dY2 a⇤2 b⇤21 b⇤22 Y2 0 Y1 dB2⇤

where the Bi⇤ are independent Brownian motions under the risk-neutral probability. 2

1/2. Under what condition on

10

is a⇤1

1/2?

(a) We have dY1 = = =

1 2

 2

✓ 2

dV

Thus,

and

=

2 (1

p

(✓

V ) dt +

dt

Y1 dt +

⇢ dY2 = d log S dV ✓ ◆ 1 = µ V dt 2 ✓ ◆ ⇢✓ = µ dt

V

p

dB1 Y1 dB1 ,

⇢

(✓

V ) dt +

p

V [⇢ dB1 +

2

2

Y1 dt + ⇢ Y1 dt +

0 1 0 1 a1 ✓/ 2 @ A=@ A, a2 µ ⇢✓/ ⇢2 ).

and

from the fact that (16.19) must hold for all V .

(c) Assume that M defined in terms of

(d) Assume ✓/

20

p 1

0 1 0 b11 b12 @ A=@ b21 b22

p 1

⇢2

⇢2 dB2 ]

p ⇢ V dB1

p Y1 dB2 . 

2 /2

+ ⇢

1 0 A, 0

16 Forwards, Futures, and More Option Pricing

(b) Given 1

=

10 Y1

1/2

+

1/2 , 11 Y1

and

2

the condition (16.19) is equivalent to ✓ ◆ ✓ 1/2 20 ⇢ 10 + Y1 + ⇢ 1/2

Multiplying by Y1

11

21

+

1/2

20 = p Y1 1 ⇢2

◆

1/2 Y1

=

✓

µ

+ p

r

◆

Y1

1/2

21

⇢2

1

1/2

Y1

,

.

and noting that this must hold for all values of Y1 , we obtain µ

r

⇢

10

+

20

=

⇢

11

+

21

= 0.

,

Thus, 20

=µ

21

=

r ⇢

⇢

10 ,

11 .

(c) By Girsanov’s theorem, dB1⇤ = dB1 +

1 dt ,

dB2⇤ = dB2 +

2 dt

are Brownian motions under the risk-neutral probability. Making these substitutions yields 0p 10 1 0p 10 1 0 1 Y1 0 dB1 Y1 0 dB1⇤ + Y 11 1 @ A@ A=@ A@ A @ 10 A dt . p p ⇤ 0 Y1 dB2 0 Y1 dB2 + Y 20 21 1 Thus,

and

(d) a⇤1

0 @

0 1 0 a⇤ a @ 1A = @ 1 a⇤2 a2 b⇤11 b⇤12 b⇤21

b⇤22

1

0

A=@

1/2 if and only if ✓/

2

b11 b21 1/2

10 20

11 21

1

A=@

b12 b22 10 .

0

1

✓/ r 0

A=@

2

10

⇢✓/ + ⇢  2 /2

10

1

A,

11

+ ⇢ + ⇢

11

1 0 A. 0

171

172

16 Forwards, Futures, and More Option Pricing

16.10. This exercise values an American call on an asset paying a known discrete dividend at a known date. Part (c) is similar to the valuation of a compound option in Exercise 15.10. Consider an American call option with strike K on an asset that pays a single known discrete dividend

at a known date T < u, where u is the date the option expires. Assume the asset price

S drops by

when it goes ex-dividend at date T (i.e., ST = limt"T St

) and otherwise is an Itˆ o

process. Assume there is a constant risk-free rate. (a) Show that if

< 1

e

r(u T )

K, then the call should not be exercised early.

For the remainder of this exercise, assume following process Zt =

8 > < St > :S

e

> 1

e

r(T t)

r(u T )

K. Assume the volatility of the

if t < T , if T  t  u ,

t

is constant over [0, u]. Note that Z is the value of the following non-dividend-paying portfolio: borrow e

r(T t)

at any date t < T to partially finance the purchase of the asset and use its

dividend to repay the debt. Let V (t, St ) denote the value of a European call on the asset with strike K maturing at u. Let S ⇤ denote the value of the stock price just before T such that the holder of the American option would be indi↵erent about exercising just before the stock goes ex-dividend. This value is given by S ⇤ equivalently, ST > S ⇤

K = V (T, S ⇤

). Exercise is optimal just before T if limt"T St > S ⇤ ;

. Let A denote the event ST > S ⇤

of the world such that ST  S ⇤ exercises optimally are (ST +

and let C denote the set of states

and Su > K. The cash flows to a holder of the option who K)1A at (or, rather, “just before”) date T and (Su

date u. (b) Show that the value at date 0 of receiving (ST + e

rT

log(S0

e

S0

N(d1 )

K)1A at date T is

e

rT

(K

) N(d2 ) ,

where d1 =

d2 = d1

p

rT

)

log(S ⇤ p T

)+ r+

1 2 2

T.

(c) Show that the value at date 0 of receiving (Su

K)1C at date u is

T

,

K)1C at

16 Forwards, Futures, and More Option Pricing

S0

e

rT

p T /u)

M( d1 , d01 ,

ru

e

173

p T /u) ,

K M( d2 , d02 ,

where M(a, b, ⇢) denotes the probability that ⇠1 < a and ⇠2 < b when ⇠1 and ⇠2 are standard normal random variables with correlation ⇢, and where d01 =

log(S0

e

p

d02 = d01

rT

)

log K + r + p u

1 2 2

u

,

u.

(a) Exercise is optimal just before T if and only if ST +

K exceeds the value of the call at T .

The value of the call at T is bounded below by ST

e

r(u T ) K.

Thus, a necessary condition

for exercise to be optimal is that ST +

K > ST

e

r(u T )

,

K

> (1

e

r(u T )

)K .

(b) The cash flow ST 1A = ZT 1A is a share digital paying in the event A = {ZT > S ⇤

}. Its value

at date 0 is Z0 probZ (A) = (S0 where log d1 = The cash flow (

⇣

Z0

S⇤

⌘

+ r+ p T

1 2 2

T =

log(S0

e

rT

) N(d1 ) ,

e

rT

)

log(S ⇤ p T

)+ r+

1 2 2

K)1A is a share digital paying in the event A = {ZT > S ⇤

T

.

}. Its value at

date 0 is e

rT

K) probR (A) = e

(

rT

(

K) N(d2 ) .

(c) The cash flow Su 1C = Zu 1C is a share digital paying in the event C = {ZT  S ⇤ Its value at date 0 is Z0 probZ (C) = (S0

e

rT

) probZ (C) .

To calculate probZ (C), recall that d log Z =

✓

1 r+ 2

2

◆

dt + dB ⇤ ,

where B ⇤ is a Brownian motion under probZ . Thus, the event C is equivalent to ✓ ◆ 1 2 log Z0 + r + T + BT⇤  log(S ⇤ ), 2 ✓ ◆ 1 2 log Z0 + r + u + Bu⇤ > log K . 2

, Zu > K}.

174

16 Forwards, Futures, and More Option Pricing

This is equivalent to log(S ⇤ B⇤ pT  T log Z0 B⇤ pu < u

)

log Z0 p T log K + r + p u

r+ 1 2 2

1 2 2

u

T

=

d1 ,

= d01 .

(d) The cash flow K1C = Zu 1C is a digital paying in the event C = {ZT  S ⇤ value at date 0 is e

ru K

, Zu > K}. Its

probR (C). To calculate probR (C), recall that ✓ ◆ 1 2 d log Z = r dt + dB ⇤ , 2

where B ⇤ is a Brownian motion under ✓ log Z0 + r ✓ log Z0 + r

probR . Thus, the event C is equivalent to ◆ 1 2 T + BT⇤  log(S ⇤ ), 2 ◆ 1 2 u + Bu⇤ > log K . 2

This is equivalent to log(S ⇤ B⇤ pT  T log Z0 Bu⇤ p < u

)

log Z0 p T log K + r p u

r 1 2 2

1 2 2

u

T

= d02 .

=

d2 ,

17 Term Structure Models

17.1. In the Vasicek model, set f (t, r) = exp( ↵(T

t)

(T

t)r).

(a) Show that f satisfies the fundamental PDE and the boundary condition f (T, r) = 1 if and only if 0

↵0

✓ +

+  = 1,

1 2

2 2

= 0,

and ↵(0) = (0) = 0. (b) Verify that the formula (17.6) for yields with

(⌧ ) = ⌧ b(⌧ ) and ↵(⌧ ) = ⌧ a(⌧ ) satisfies these

conditions. (a) The fundamental PDE for the Vasicek model is ft + (✓ Taking f (t, r) = exp( ↵(T

t)

(T

r)fr +

1 2

2

frr = rf .

t)r) implies ft = (↵0 + fr = frr =

0

r)f ,

f, 2

f.

Substituting these into the fundamental PDE yields ↵0 +

0

r

(✓

r) +

For this to hold for all values of r, we must have

1 2

2 2

= r.

176

17 Term Structure Models

↵0

1 2

✓ +

0

2 2

= 0,

+  = 1.

Also, f (T, r) = 1 implies ↵(0) + (0)r = 0. For this to hold for all values of r, we must have ↵(0) = (0) = 0. (b) From (17.6c), we have (⌧ ) = 0 0

1

⌧

e 

,

(⌧ ) = e

⌧

,

(⌧ ) +  (⌧ ) = e

⌧

+1

e

⌧

= 1,

(0) = 0 . Also, (17.6c) implies ↵(⌧ ) =

✓

2

✓

↵0 (⌧ ) = ✓ ↵0 (⌧ )

✓ (⌧ ) +

1 2

2 2

22 2

22

◆

+

2

(⌧ ) = ✓

22

+

✓ 1

⌧+ ✓ 2 ✓ e

✓

2

2

2 ⌧

2

✓2

3 ◆ ✓ e ⌧ ◆ ✓ e ⌧

◆

1 2

22

1

e

2⌧

⌧

+e

22 2e

43 2⌧

2

2

⌧

e

22

2

+

e

1

e

2⌧

,

,

2⌧

= 0, ↵(0) = 0 .

17.2. Consider a single-factor affine model as defined in (17.15). (a) Show that dr =

dt

r dt +

for constants , , ↵ and . (b) Assume

p

↵ + r dB ⇤

> 0 in (17.35). Show that r is a translation of a square-root process — i.e., there

exists ⌘ and Y such that rt = ⌘ + Y t , dY = ˆ dt ˆ and ˆ . for constants  ˆ , ✓,

p  ˆ Y dt + ˆ Y dB ⇤ .

17 Term Structure Models

(c) The condition ˆ > 0 is necessary and sufficient for Yt square root to exist. Assuming

177

0 for all t in (17.36b) and hence for the

> 0, what are the corresponding conditions on the coefficients

in (17.35) that guarantee ↵ + rt

0 for all t?

(a) Assume r=

0

+

1X

,

dX = ⌘ dt + kX dt + with

1

6= 0. Then dr = =

1 ⌘ dt

1 ⌘ dt +

= ( 1⌘ = where

=

(b) Define ⌘ =

1⌘

+

dt

k 0,  =

1 kX 1k

✓

1

p

a + bX dB ⇤ s ◆ ✓ r 0 dt + 1 a + b 1

2 1

b

0 1,

and

b

0 1

0 1

◆

dB ⇤

+ b 1 r, dB ⇤

= b 1.

⌘. Then

dY = dr =

dt

=

dt

= ˆ dt where ˆ =

r

a + bX dB ⇤ ,

q k 0 ) dt + kr dt + a 12 p r dt + ↵ + r dB ⇤ ,

k, ↵ = a

↵/ and Y = r

dt +

p

⌘,  ˆ = , and ˆ =

p

p ↵ + r dB ⇤ , p (⌘ + Y ) dt + ↵ + (⌘ + Y ) dB ⇤ , p  ˆ Y dt + ˆ Y dB ⇤ , r dt +

.

(c) We have ↵ + r = ↵ + (⌘ + Y ) = Y . Thus, Y to

0 , ↵+ r

0. Because ˆ =

⌘ =

+ ↵/ , the condition ˆ > 0 is equivalent

+ ↵ > 0.

17.3. Consider a two-factor affine model with Gaussian factors — i.e., (Xt ) in (17.15) is a constant matrix. Show that the two factors can be taken to be the short rate and its drift in the sense that ⇤ drt = Yt dt + dZ1t , ⇤ ⇤ dYt = (a + brt + cYt ) dt + ⇣1 dZ1t + ⇣2 dZ2t

178

17 Term Structure Models

, a, b, c, ⇣1 and ⇣2 and independent Brownian motions Zi⇤ under the risk neutral

for constants probability. Assume

r= and

0

1 X1

+

2 X2 ,

0 1 0 1 0 10 1 0 X1 ✓1 k11 k12 X1 A @ A dt + @ d @ A = @ A dt + @ X2 ✓2 k21 k22 X2

Define

Y =

⇣

1

2

⌘

Then

0 1 ⇣ ✓ @ 1A + ✓2

dr = Y dt +

⇣

for constants ⇣

1

1

2

⌘

and 0 @

2.

1

2

0

⌘

@

+

⌘

11 21

10 1 ⇤ dB 12 A @ 1A . dB2⇤ 22

0 10 1 k11 k12 X @ A @ 1A . k21 k22 X2

11 21

12 22

⇤ 2 dB2

10 A@

dB1⇤ dB2⇤

1 A

Also,

10 1 ⇣ ✓ A @ 1 A dt + k22 ✓2

k11 k12 k21

2

1

⇤ 1 dB1

= Y dt +

dY =

+

Furthermore,

0

1

r P2

where

@ Y

If L is singular, then

L = @P Y

2 X i=1

for some , which implies

2

⌘

i ✓i

=

@

10 10 1 k11 k12 X A@ A @ 1 A dt k22 k21 k22 X2 0 10 ⇣ ⌘ k11 k12 A @ 11 + 1 2 @ k21 k22 21

k11 k12 k21

0

i=1

0

0

i ✓i

1

0

A = L@

1 2 i=1

⇣

X1 X2

1

A,

2 i ki1

1

2

P2

⌘

i=1

0 @

X1 X2

i ki2

1

1

A.

A = (r

0)

10 1 ⇤ dB 12 A @ 1A . dB2⇤ 22

17 Term Structure Models

dY =

dr = Y dt +

If L is nonsingular, then we have 0 @

Hence,

dY =

⇣

1

2

⌘

0 @

k11 k21

⇣

X2 X2

1

0

A=L

10 1 ⇣ k12 ✓ A @ 1 A dt + k22 ✓2

1

1@

2

⌘

1

2

@

0 @

k21

In either case,

12

21

0 i ✓i

i=1

k11

10 1 dB1⇤ A@ A. ⇤ dB2 22

11

r P2

Y

0

⌘

179

1

A.

10 1 0 k12 k k A @ 11 12 A L 1 @ k22 k21 k22 Y 0 10 ⇣ ⌘ k11 k12 A@ + 1 2 @ k21 k22

r P2

0

21

A dt

i ✓i 10 1 dB1⇤ 12 A@ A. ⇤ dB2 22 i=1

11

1

dY = (a + br + cY ) dt + ⌘1 dB1⇤ + ⌘2 dB2⇤ for constants a, b, c, ⌘1 and ⌘2 . p 2 2 Define = 1 + 2 and

dZ1⇤ =

1

(

⇤ 1 dB1

+

⇤ 2 dB2 ) .

Then Z1⇤ is a Brownian motion under the risk neutral probability, and dr = Y dt + dZ1⇤ . Let ⇠ be a vector of unit length that is orthogonal to (

1

2)

0,

e.g., ⇠ = ( 2 /

1/

)0 . Define

dZ2⇤ = ⇠1 dB1⇤ + ⇠2 dB2⇤ . Then, under the risk neutral probability, Z2⇤ is a Brownian motion that is independent of Z1⇤ . Defining

we have

0 1 0 ⇣ / @ 1A = @ 1 ⇣2 2/

⇠1 ⇠2

1 A

10

1 ⌘1 @ A, ⌘2

180

17 Term Structure Models

0 1 dB1⇤ A dY = (a + br + cY ) dt + ⌘1 ⌘2 @ ⇤ dB2 0 10 1 ⇣ ⌘ dB1⇤ 1/ 2/ A@ A = (a + br + cY ) dt + ⇣1 ⇣2 @ ⇠1 ⇠2 dB2⇤ 0 1 ⇣ ⌘ dZ ⇤ 1A = (a + br + cY ) dt + ⇣1 ⇣2 @ . dZ2⇤ ⇣

⌘

17.4. This exercise shows that the factors in a two-factor CIR model can be taken to be the short rate and its volatility. This idea is developed by Longsta↵ and Schwartz (1992). Suppose r is the sum of two independent square-root processes X1 and X2 as in Section 17.3. Define Yt =

2 1 X1t

Assume

1

+

2 2 X2t .

6=

2.

Note that the instantaneous variance of r = X1 + X2 is ⇣ p ⌘2 p ⇤ ⇤ = Yt dt . 1 X1t dB1t + 2 X2t dB2t

Show that 0

for a constant vector

@

dr dY

1

A=

0 1 r dt + K @ A dt + ⌃S(r, Y ) dB ⇤ , Y

, and constant matrices K and ⌃, where S(r, Y ) is a diagonal matrix the

squared elements of which are affine functions of (r, Y ). It is without loss of generality to take

1

> 0 and

2

motion if necessary). It is also with loss of generality to take reordered if necessary). From r = X1 + X2 and Y = X1 = We have

0 1 0 X1 1 d@ A = @ X2 0 0 1 =@ 0

Y 2 1

2 2r 2 2

,

and

Bi⇤ can be used as the Brownian

> 0 (because

2 1 X1

+

X2 =

1

>

2 2 X2 , 2 1r 2 1

2

(because X1 and X2 can be

we obtain Y 2 2

.

10 1 0 10 1 0 p 10 1 ✓1 1 0 X1 0 dB1⇤ 1 X1 A @ A dt @ A @ A dt + @ p A @ ⇤A 2 ✓2 0 2 X2 0 dB2 2 X2 10 1 0 10 1 2 0 ✓ 1 r 1 A @ 1 A dt + @ 2 A @ A dt 2 2 2 1 2 2 ✓2 Y 1 1 0 p 10 1 ⇤ 2r Y 0 dB 1 2 @ 1 A @ 1A . +p 2 p 2 2 0 Y dB2⇤ 1 2 2 1r 0

17 Term Structure Models

181

Thus, 0 1 0 10 1 r 1 1 dX1 A@ A d@ A = @ 2 2 Y dX 2 1 2 0 10 10 1 1 1 1 0 ✓ 1 A@ A @ 1 A dt + =@ 2 2 2 1 0 2 ✓2 1 2 0 10 p 1 1 Y 1 @ A@ 1 +p 2 2 2 2 0 1 2 1 2

0

10 1 r @ A@ A @ A dt 2 2 2 2 2 Y 1 2 1 1 10 1 2 0 dB1⇤ 2r A @ A. p ⇤ 2r Y dB 2 2 1 1 1

10

2 2

1

17.5. This exercise develops the completely affine version of the Vasicek model. Assume the short rate is an Ornstein-Uhlenbeck process under the physical measure; i.e., dr =  ˆ (✓ˆ

r) dt + dB ,

for constants  ˆ , ✓ˆ and , where B is a Brownian motion under the physical measure. Assume there is an SDF process M with dM = M where

r dt

dB +

d" , "

is a constant and " is a local martingale uncorrelated with B.

(a) Show that the short rate is an Ornstein-Uhlenbeck process under the risk neutral probability corresponding to M (i.e., the Vasicek model holds). (b) Show that the risk premium of a discount bond depends only on its time to maturity and is independent of the short rate. (a) Girsanov’s theorem states that B ⇤ is a Brownian motion under the risk neutral probability, where dB ⇤ = dB + dt. We have

where  =  ˆ and ✓ = ✓ˆ

dr =  ˆ (✓ˆ

r) dt + (dB ⇤

= (✓

r) dt + dB ⇤ ,

/.

(b) The price at t of a discount bond maturing at T is e

↵(⌧ )

(⌧ )rt

,

dt)

182

17 Term Structure Models

where ⌧ = T

t, ↵(⌧ ) = ⌧ a(⌧ ), (⌧ ) = ⌧ b(⌧ ), and a(·) and b(·) are the functions given in (17.6).

Setting Xt =

↵(T

t)

(T

t)rt , Itˆ o’s formula implies that the return of a discount bond is

1 dX + (dX)2 = ↵0 (⌧ ) dt + 2

0

(⌧ )r dt

(⌧ ) dr +

1 2

= ↵0 (⌧ ) dt +

0

(⌧ )r dt

(⌧ )(✓ˆ

r) dt

2

(⌧ )(dr)2 (⌧ ) dB +

Therefore, the risk premium of a discount bond is ✓ ◆✓ ◆ dM dP = ( dB)( (⌧ ) dB) = M P

1 2

2

(⌧ )

2

dt .

(⌧ ) dt .

An alternative calculation of the risk premium uses the the fundamental PDE (Exercise 17.1). The fundamental PDE is ↵0 (⌧ )

✓ (⌧ ) + 0

1 2

2 2

(⌧ ) = 0 ,

(⌧ ) +  (⌧ ) = 1 .

The first equation is equivalent to ✓ˆ (⌧ ) +

↵0

(⌧ ) +

1 2

2 2

2

2

= 0.

Substituting these, the return of a discount bond is ↵0 (⌧ ) dt +

0

(⌧ )r dt

(⌧ )(✓ˆ

Thus, the risk premium is

r) dt

(⌧ ) dB +

1 2

(⌧ )

dt = r dt

(⌧ ) dt

(⌧ ) dB .

(⌧ ).

17.6. This exercise develops the completely affine version of the multi-factor CIR model. Assume r = X1 + X2 where the Xi are independent square-root processes under the physical measure; i.e., dXi =  ˆ i (✓ˆi for constants  ˆ i , ✓ˆi and

i,

Xi ) dt +

i

p Xi dBi ,

where the Bi are independent Brownian motions under the physical

measure. Assume there is an SDF process M with dM = M where

1

and

2

r dt

1

p

X1 dB1

2

p d" X2 dB2 + , "

are constants and " is a local martingale uncorrelated with B.

17 Term Structure Models

183

(a) Show that the Xi are independent square-root processes under the risk neutral probability corresponding to M . (b) Show that the risk premium of a discount bond is a linear function of the factors X1 and X2 , with coefficients depending on the time to maturity. (a) By Girsanov’s theorem, setting dBi⇤ = dBi +

i

p

X i dt produces independent Brownian motions

Bi⇤ under the risk neutral probability. We have dXi =  ˆ i (✓ˆi = i (✓i where i =  ˆi +

i i

p Xi (dBi⇤ p Xi ) dt + i Xi dBi⇤ , Xi ) dt +

and ✓=  ˆ i ✓ˆi /i .

i

i

p

X i dt)

(b) The price at t of a discount bond maturing at T is Pt (T ) = eYt , where Yt =

2 X

[↵i (⌧ ) +

i (⌧ )Xit ]

i=1

with ⌧ = T

t and ↵i (·) and

dP 1 = dY + (dY )2 P 2  2 X = ↵i0 (⌧ ) dt + =

i=1 2  X

↵i0 (⌧ ) dt +

i (·)

being defined in (17.14). By Itˆ o’s formula,

0 i (⌧ )Xi dt

i (⌧ ) dXi

0 i (⌧ )Xi dt

i ( ✓ i i (⌧ )ˆ

+

ˆ

1 2

2 2 i (⌧ )(dXi )

Xi ) dt

i (⌧ ) i

p

X i dBi +

i=1

1 2

2 2 i (⌧ ) i Xi dt

.

Therefore, the risk premium of a discount bond is ✓

dM M

◆✓

dP P

◆

=

2 X

i i i (⌧ )Xi dt .

i=1

17.7. Assume there is an SDF process with dM = M

r dt

⇥

S(Xt ) + S(Xt )

1

⇤Xt

⇤0

dB +

d" , "

where B is a vector of independent Brownian motions under the physical measure, " is a local martingale uncorrelated with B, S(X) is a diagonal matrix the squared elements of which are

184

17 Term Structure Models

affine functions of X, S(X)

1

denotes the inverse of S(X),

is a constant vector, and ⇤ is a

constant matrix. Assume M R is a martingale, so there is a risk neutral probability corresponding to M . [Warning: This assumption is not valid in general. See the end-of-chapter notes.] (a) Assume r =

0+

0X

and dX = ( + KX) dt + S(X) dB ⇤ , where B ⇤ is a vector of independent

Brownian motions under the risk neutral probability, vectors, and K and

0

is a constant,

and

are constant

are constant matrices. Show that ˆ dX = ( ˆ + KX) dt + S(X) dB

ˆ for a constant vector ˆ and constant matrix K. (b) Using the fact that bond prices are exponential-affine, calculate ✓ ◆✓ ◆ dP dM P M to show that the risk premium of a discount bond is affine in X. (c) Consider the Vasicek model with the price of risk specification (17.38). Show that, in contrast to the completely affine model considered in Exercise 17.5, the risk premium of a discount bond can depend on the short rate. (a) By Girsanov’s theorem,

Thus,

⇥ dB ⇤ = dB + S(X) + S(X)

1

⇤ ⇤X dt .

dX = ( + KX) dt + S(X) dB ⇤ ⇥ = ( + KX) dt + S(X) dB + S(X) + S(X)

1

⇤ ⇤X dt

= [ + KX + S(X)S(X) + ⇤X] dt + S(X) dB .

The vector S(X)S(X) is an affine function of X, say A + CX for a vector A and matrix C. Set ˆ =

ˆ = K + C + ⇤. + A and K

(b) Let k denote the dimension of X. The price at t of a discount bond maturing at T is Pt = eYt , where Yt =

k X i=1

↵i (⌧ )

⌧ X i=1

i (⌧ )Xi ,

17 Term Structure Models

for some functions ↵i (·) and dP 1 = dY + (dY )2 P 2 k X = [↵i0 (⌧ ) dt +

i (·)

and where ⌧ = T

185

t. By Itˆ o’s formula,

k

k

1 XX i (⌧ ) dXi ] + i (⌧ ) j (⌧ )(dXi )(dXj ) . 2 i=1 i=1 j=1 P The stochastic part of dP/P comes from i dXi and hence is 0 i (⌧ )Xi dt

(⌧ )> S(X) dB ,

where (⌧ )> denotes the row vector ( 1 (⌧ ) · · · k (⌧ )). Thus, ✓ ◆✓ ◆ ⇥ dP dM = (⌧ )> S(X) (dB)(dB)0 S(X) + S(X) P M =

(⌧ )> S(X)S(X) dt

1

⇤X

(⌧ )> ⇤X dt ,

⇤

which is an affine function of X. (c) In the Vasicek model, we can take X = r and S(X) = 1. Hence, ✓ ◆✓ ◆ dP dM = (⌧ )> S(X)S(X) dt (⌧ )> ⇤X dt P M =

(⌧ )

dt

(⌧ ) ⇤r dt .

17.8. This example is from Du↵ee (2002). The model in (b) is an essentially affine model. Assume

for constants , ✓,

0 @

dr dY

, ,

1

0

A=@

(✓

r)

(

Y)

1

0

A dt + @

10 10 1 1 0 dB1⇤ A@ p A@ A ⌘ 0 Y dB2⇤ 0

0

and ⌘, where the Bi⇤ are independent Brownian motions under a risk

neutral probability. (a) Given the completely affine price of risk specification (17.21), where X = (r Y )0 and 0 1 1 0 S(X) = @ p A , 0 Yt show that

dr = (✓ˆ

r) dt + dB1 ,

ˆ where B1 is a Brownian motion under the physical measure. Show that the for some constant ✓, risk premium of a discount bond depends only on its time to maturity and does not depend on r or Y .

186

17 Term Structure Models

(b) Consider the price of risk specification (17.38), 0 1 @ 0

1

replacing S(X) 1 0 A. 0

by

Show that the risk premium of a discount bond can depend on r and Y .

(a) By Girsanov’s theorem, B ⇤ is related to a vector B of independent Brownian motions under the physical measure as dB ⇤ = dB + S(X) dt 0 10 1 0 = dB + @ p A @ 0 Y 0 1 = dB + @

Thus,

dr = (✓

where ✓ˆ = ✓+

1 /.

1 2

1

A dt

1 p A dt . 2 Y

r) dt + dB1⇤

= (✓

r) dt + (dB1 +

= (✓ˆ

r) dt + dB1 ,

1 dt)

Note that r satisfies the Vasicek model (under the risk neutral probability).

Thus, the price at t of a discount bond maturing at T is Pt = eZt , where Zt = for functions ↵(·) and (·), where ⌧ = T

↵(⌧ )

(⌧ )rt ,

t. It follows that

dP 1 = dZ + (dZ)2 P 2 = ↵0 (⌧ ) dt + 

= ↵0 (⌧ ) + Hence,

0

0

(⌧ )r dt

(⌧ )r

(⌧ )[(✓ˆ (⌧ )(✓ˆ

r) +

r) dt + dB1 ] + 1 2

2 2

(⌧ ) dt +

1 2 ⇣

2 2

(⌧ ) dt

(⌧ )

⌘

0

0 @

dB1 dB2

1

A.

17 Term Structure Models

✓

dM M

◆✓

dP P

◆

=

=

0

⇣

0

S(X)0 (dB)(dB)0 @

=

1

0 10 ⌘ 1 0 @ p A@ 2 0 Yt

1

(⌧ ) 0

187

1 A

(⌧ ) 0

(⌧ ) dt .

1

A dt

(b) By Girsanov’s theorem, B ⇤ is related to a vector B of independent Brownian motions under the physical measure as 0 1 1 0 A ⇤X dt dB ⇤ = dB + S(X) dt + @ 0 0 0 10 1 0 10 10 1 1 0 1 0 ⇤ ⇤ r 1 A @ 11 12 A @ A dt = dB + @ p A @ A dt + @ 0 Y 0 0 ⇤21 ⇤22 Y 2 0 1 0 1 ⇤11 r + ⇤21 Y 1 A dt . = dB + @ p A dt + @ Y 0 2

Thus,

dr = (✓

where  ˆ=

r) dt + dB1⇤

= (✓

r) dt + (dB1 +

= ˆ (✓ˆ

r) dt + ⇤21 Y dt + dB1 ,

⇤11 and ✓ˆ = (✓ +

1 dt

+ ⇤11 r + ⇤21 Y dt)

. 1 )/ˆ

Note that r satisfies the Vasicek model (under the risk neutral probability). Thus, the price at t of a discount bond maturing at T is Pt = eZt , where Zt = for functions ↵(·) and (·), where ⌧ = T

↵(⌧ )

(⌧ )rt ,

t. It follows that

dP 1 = dZ + (dZ)2 P 2 = ↵0 (⌧ ) dt + 

= ↵0 (⌧ ) +

0

0

(⌧ )r dt

(⌧ )r

(⌧ )[ˆ (✓ˆ (⌧ )ˆ (✓ˆ

r) dt + ⇤21 Y dt + dB1 ] +

r) + ⇤21 Y dt +

1 2

2 2

(⌧ ) dt +

1 2 ⇣

2 2

(⌧ ) dt

(⌧ )

0 1 dB1 A. 0 @ dB2 ⌘

188

17 Term Structure Models

Hence, ✓

dM M

◆✓

dP P

◆

2

0 1 30 0 1 1 0 (⌧ ) A ⇤X 5 (dB)(dB)0 @ A = 4S(X) + @ 0 0 0 2 0 1 0 10 13 0 ⇣ ⌘ 1 0 ⇣ ⌘ ⇤11 ⇤21 1 0 A@ A5 @ =4 1 2 @ p A+ r Y @ 0 Yt ⇤12 ⇤22 0 0 0 1 ⇣ (⌧ ) p ⌘ @ A dt = 1 + ⇤11 r + ⇤12 Y 2 Y 0 =

(⌧ )[

1

(⌧ ) 0

1

A dt

+ ⇤11 r + ⇤12 Y ] dt .

17.9. This exercise verifies that the two-factor version of the model of Constantinides (1992) is a quadratic model. Assume Mt = exp(X1t + (X2t

a)2 ) is an SDF process, where dX1 = µ dt + dB1t , dX2 =

with µ, ,  and

X2t dt + dB2t ,

being constants and with B1 and B2 being independent Brownian motions under

the physical measure. (a) Derive dM/M , and deduce that r is a quadratic function of X1 and X2 and the market prices of risk are affine functions of X1 and X2 . (b) Given the prices of risk calculated in the previous part, find Brownian motions B1⇤ and B2⇤ under the risk neutral probability and show that the dX satisfy (17.24b). (a) We have M = eY , where Y = X1 + (X2 dY = dX1 + 2(X2

a)2 . Itˆ o’s formula implies

a) dX2 + (dX2 )2

= µ dt + dB1 + 2(X2 = [µ + and

2

2(X2

a)[ X2 dt + dB2 ] +

a)X2 ] dt + dB1 + 2 (X2

2

dt

a) dB2 ,

17 Term Structure Models

189

dM 1 = dY + (dY )2 M 2 = [µ +  = µ+

2 2

2(X2 +

1 2

2

a)X2 ] dt + dB1 + 2 (X2 2(X2

2

a)X2 + 2

a) dB2 +

1 2

2

dt + 2

2

a)2 dt + dB1 + 2 (X2

(X2

(X2

a)2 dt

a) dB2 .

Because the drift of dM/M is minus the short rate, we have r=

1 2

2

µ

2

+ 2(X2

a)X2

2

2

(X2

a)2 .

Also, dM = M

r dt

1 dB1

2 dB2 ,

where 1

=

2

=

, 2 (X2

a) .

(b) By Girsanov’s theorem, B ⇤ is a vector of independent Brownian motions under the risk neutral probability associated to M , where dB1⇤ = dB1

dt ,

dB2⇤ = dB2

2 (X2

a) dt .

We have dX1 = µ dt + (dB1⇤ + dt) = (µ + ) dt + dB1⇤ , dX2 = =

X2 dt + [dB2⇤ + 2 (X2 2a

2

dt + (2

2

a) dt]

)X2 dt + dB2⇤ .

This is a special case of (17.24b).

17.10. This exercise develops the Vasicek model with time-dependent parameters studied by Hull and White (1990). Consider the Vasicek model with time-dependent parameters:

190

17 Term Structure Models

rt dt + (t) dBt⇤ ,

drt = (t) ✓(t)

where B ⇤ is a Brownian motion under a risk neutral probability. Define ✓ Z t ◆ ✓ Z t ◆ Z t rˆt = exp (s) ds r0 + exp (s) ds (u) dBu⇤ , 0 0 u ✓ Z t ◆ Z t g(t) = exp (s) ds (u)✓(u) du . 0

u

(a) Show that rˆ defined in (17.41a) satisfies dˆ rt =

(t)ˆ rt dt + (t) dBt⇤ .

(b) Define rt = rˆt + g(t). Show that r satisfies (17.40). (c) Given any functions (·) and (·), explain how to choose ✓(·) to fit the current yield curve. (a) By Leibniz’s rule, ✓ Z t ◆ dˆ r = (t) exp (s) ds r0 dt + (t) dBt⇤

(t)

0

=

(t)ˆ rt dt +

Z

t

exp 0

✓ Z

t

(s) ds u

◆

(u) dBu⇤ dt

(t) dBt⇤ .

(b) By Leibniz’s rule again, g 0 (t) = (t)✓(t) = (t)[✓(t)

(t)

Z

t

exp 0

✓ Z

t u

◆ (s) ds (u)✓(u) du

g(t)] .

Thus, drt = dˆ rt + g 0 (t) dt =

(t)ˆ rt dt + (t) dBt⇤ + (t)[✓(t)

g(t)] dt

rt ] dt + (t) dBt⇤ .

= (t)[✓(t)

(c) The price at t of a discount bond maturing at T is  ✓ Z T ◆ ✓ Z T ◆  ✓ Z R ER exp r du = exp g(u) du E exp u t t t

t

Note that the conditional expectation ER t



exp

✓ Z

T

rˆu du t

◆

T

rˆu du t

◆

.

17 Term Structure Models

191

does not involve the function ✓(·). Thus, one can match the market price Qt (T ) by setting ✓ Z T ◆ Q (T ) h ⇣t R ⌘i . exp g(u) du = T t ER exp r ˆ du u t t Equivalently, taking logs, Z T g(u) du = log Qt (T ) t

log ER t



exp

✓ Z

T

rˆu du t

◆

.

We want to hold for each T > t. Di↵erentiating in T gives  ✓ Z T ◆ d d R g(T ) = log Qt (T ) + log Et exp rˆu du . dT dT t Taking another derivative in T and using the previous formula for g 0 yields  ✓ Z T ◆ d2 d2 R (T )[✓(T ) g(T )] = log Qt (T ) + log Et exp rˆu du . dT 2 dT 2 t Thus, to fit the yield curve at t, one should choose ✓(T ) for T > t by  ✓ Z T ◆ 1 d2 1 d2 R ✓(T ) = g(T ) log Qt (T ) + log Et exp rˆu du . (T ) dT 2 (T ) dT 2 t 17.11. This exercise asks for the Hull-White model to be written in the Heath-Jarrow-Morton form. Assume the short rate is rt = rˆt + g(t), where dˆ r= for constants  and

ˆ r dt + dB ⇤ ,

and g(·) is chosen to fit the current yield curve.

(a) Calculate the forward rates fs (u) using the Vasicek bond pricing formula. (b) Calculate ↵s (u) and

s (u)

such that, as s changes, dfs (u) = ↵s (u) ds +

(c) Prove that ↵s (u) =

s (u)

Z

⇤ s (u) dBs .

u s (t) dt . s

(a) The price Ps (u) at s of a discount bond maturing at u is  ✓ Z u ◆ ✓ Z u ◆  ✓ Z u ◆ R R Es exp rt dt = exp g(t) dt Es exp rˆt dt s s ✓ Zs u ◆ = exp g(t) dt eYs , s

192

17 Term Structure Models

where Ys = with ⌧ = u

s, ↵(⌧ ) = ⌧ a(⌧ ),

↵(⌧ )

(⌧ )ˆ rs

(⌧ ) = ⌧ b(⌧ ) and a(·) and b(·) being defined in (17.6), taking

✓ = 0. Thus, 2

↵(⌧ ) = (⌧ ) =

2

2

⌧+ 2

1 1 

1

3 ⌧

e

2

⌧

e

43

1

e

2⌧

,

e

2⌧

+e

⌧

.

It follows that the forward rate fs (u) is d log Ps (u) = g(u) + ↵0 (⌧ ) + du

0

2

= g(u)

(⌧ )ˆ rs

2

22

+



e 2

2

⌧

22

rˆs .

(b) Fixing u, the forward rate evolves for s < u as 2

dfs (u) = =



⌧

e 2





 2

e

⌧

e

⌧

2

=

2

ds  e

e

e

2⌧

2⌧

2⌧

2

and the volatility is

+ e

⌧



⇣

e

s (u)

(u s)

= e

⌧

⌧

rˆs ds + e

rˆs ds

⌧

ds + e

Thus, the drift is ↵s (u) =

ds + e

⌧

e

dˆ rs

rˆs ds + e

⌧

dBs⇤

dBs⇤ .

2(u s)

e

(u s)

⌘

,

.

(c) We have s (u)

Z

u s (t) dt = e

(u s)

s 2

=



e

Z

(u s)

= ↵s (u) .

u

e

(t s)

dt

s

⇣

1

e

(u s)

⌘

17.12. This exercise derives an option pricing formula for discount bonds in the Vasicek/HullWhite model.

17 Term Structure Models

193

Assume the short rate is rt = rˆt + g(t), where ˆ r dt + dB ⇤ ,

dˆ r=

and g(·) is chosen to fit the current yield curve. (a) Consider a forward contract maturing at T on a discount bond maturing at u > T . Let Ft denote the forward price for t  T . What is the volatility of dFt /Ft ? (b) What is the average volatility between 0 and T of dFt /Ft in the sense of (16.15)? (c) Consider a call option maturing at T on a discount bond maturing at u > T . Derive a formula for the value of the call option at date 0. (a) The forward price is Ft = Pt (u)/Pt (T ). The volatility of dF/F is the volatility of d log F , so it is the volatility of d log Pt (u)

d log Pt (T ). From the previous exercise, the price Pt (x) at t of

a discount bond maturing at x is exp

✓ Z

x

◆

g(s) ds eYt (x) ,

t

where Yt (x) =

↵(x

t)

(x

t)ˆ rt

with 2

↵(⌧ ) = (⌧ ) =

2

2

⌧+ 2

1 1 

e

3 ⌧

1

e

2

⌧

43

1

e

2⌧

,

.

The volatility of d log Pt (u) d log Pt (T ) is the volatility of dYt (u) dYt (T ), which is the volatility of [ (u

t)

(T

t)]dˆ rt .

Hence, the volatility is 

⇣

e

(T t)

e

(u t)

⌘

=



e

T

e

u

(b) The average volatility in the sense of (16.15) is avg

= =

 

e e

T

T

e e

u

u

s r

1 T

Z

T

e2t dt 0

e2T 2T

1

et .

194

17 Term Structure Models

(c) Because the risk-free rate is stochastic, we need to use Merton’s (equivalently, Black’s) formula. The value at date 0 is P0 (u) N(d1 )

e

yT

K N(d2 ) ,

where log(P0 (u)/K) + y + p d1 = avg T p d2 = d1 avg T , and y =

log P0 (T )/T .

1 2 2 avg

T

,

Part IV

Topics

18 Heterogeneous Priors

18.1. Suppose each investor h has CARA utility with absolute risk aversion ↵h . Assume the information in the economy is generated by w ˜m . Assume investor h believes w ˜m is normally distributed with mean µh and variance

2,

where

is the same for all investors.

(a) Show that the Radon-Nikodym derivative of investor h’s probability Ph with respect to the average probability P is z˜h =

1 H

exp PH

⇣

(w ˜ m µ h )2 2 2

j=1 exp

⇣

⌘

(w ˜ m µj )2 2 2

⌘.

(b) Show that the sharing rule (18.4) is equivalent to 0 " ✓ # ◆ H J 2 2 X X µj µh ⌧j ⌧j (µh ⌧h h ↵h @ w ˜ h = ⌧h log + + w ˜ + ⌧ m h 2 ⌧ 2 ⌧ ⌧ j ↵j j=1

2

j=1

1

µj ) A

w ˜m .

(c) Show that if investors also disagree about the variance of w ˜m , then the sharing rule (18.4) is quadratic in w ˜m .

(a) We have z˜h = gh (w ˜m ) for some function gh . Consider a random variable x ˜ = f (w ˜m ) for some function f . By the normal distribution assumption and the definition of a Radon-Nikodym derivative, ✓ ◆ Z 1 1 (w µh )2 p f (w) exp dw = Eh [˜ x] 2 2 2⇡ 1 = E[˜ xz˜h ] Z H 1 X 1 p = H 2⇡ j=1 Because this is true for each function f , we must have

1 1

f (w)gh (w) exp

✓

(w µj )2 2 2

◆

dw .

198

18 Heterogeneous Priors

exp

✓

(w µh )2 2 2

◆

✓ ◆ H (w µj )2 1 X = gh (w) exp , H 2 2 j=1

implying gh (w) =

1 H

(b) Notice that log

✓

z˜h z˜j

◆

= log

✓

gh (w ˜m ) gj (w ˜m )

◆

=

exp PH

⇣

(w µh )2 2 2

j=1 exp

⇣

(w µj )2 2 2

µj ) 2 ( w ˜m 2 2

(w ˜m

⌘

µh ) 2

⌘. µ2j

=

µ2h + 2(µh 2 2

µj ) w ˜m

.

Also, the ⌧j /⌧ sum to one, so log(

˜h ) h ↵h z

H X ⌧j j=1

⌧

log( j ↵j z˜j ) =

H X ⌧j j=1

=

⌧

H  X ⌧j

⌧

j=1

=

[log(

H X ⌧j j=1

⌧

log

log

˜h ) h ↵h z ✓

✓

h ↵h j ↵j h ↵h j ↵j

log( j ↵j z˜j )]

◆

◆

⌧j + log ⌧

+

H X ⌧j j=1

✓

z˜h z˜j

◆

µ2j

⌧

µ2h + 2(µh 2 2

µj ) w ˜m

!

.

Substituting this into (18.4) produces the result. (c) Following the reasoning in Part (a) produces 1 h

z˜h = gh (w ˜m ) = 1 H

Hence, log

✓

z˜h z˜j

◆

j

= log

PH

exp

j=1

+

h

1 j

⇣

(w ˜ m µh )2 2 2 h

exp

✓

(w ˜ m µj ) 2 2 j2

⌘

(w ˜ m µ j )2 2 j2

◆.

(w ˜ m µh ) 2 . 2 h2

Now, one can follow the reasoning in Part (b), producing a sharing rule that includes terms 2 / involving the w ˜m

2 j.

18.2. Assume all investors have constant relative risk aversion ⇢ and the same discount factor . Solve the social planning problem in a finite-horizon discrete-time model to show that the social planner’s utility is E

"

T X t=0

t

C1 ⇢ Zt t 1 ⇢

#

18 Heterogeneous Priors

199

for some stochastic process Z. Show that Z is a supermartingale relative to the average beliefs if ⇢ > 1. Hint: For the last statement, use a conditional version of the Minkowski inequality. The Minkowski inequality states that for random variables x ˜h and any ⇢ > 1, E

"

H X

x ˜h

h=1

!⇢ #1/⇢



H X h=1

⇥ ⇢ ⇤1/⇢ Et x ˜h .

The social planning problem is max

H X T X

h

h=1 t=0

t

"

1 ⇢ Cht E Zht 1 ⇢

#

subject to

(8 t)

H X

Cht = Ct .

h=1

This is equivalent to maximizing H X h=1

1 ⇢ Cht h Zht 1 ⇢

H X

subject to

Cht = Ct

h=1

for each t and in each state of the world. Following the calculations in Exercise 3.9 yields ( h Zht )1/⇢ Cht = PH Ct . 1/⇢ j=1 ( j Zjt )

Thus,

1 ⇢ ht Zht Cht

= ⇣P H

and the social planner’s utility is 2 PH T X ( h Zht )1/⇢ t 6 E 4 ⇣P h=1 ⌘1 H 1/⇢ t=0 j=1 ( j Zjt ) where

Zt = h Zh,t+1 )

1/⇢ .

h Zht )

1/⇢

j=1 ( j Zjt )

Ct1 ⇢ ⇢

H X h=1

Set x ˜h = (

(

1

(

⇢

!

h Zht )

1/⇢

⌘1

3

1 ⇢ , ⇢ Ct

" T 7 X t E Zt 5=

1/⇢

t=0

!⇢

.

If ⇢ > 1, then the Minkowski inequality yields

Ct1 ⇢ 1 ⇢

!#

,

200

18 Heterogeneous Priors

Et [Zt+1 ]1/⇢ = Et  =

"

H X h=1 H X

H X

x ˜h

h=1

!⇢ #1/⇢

⇥ ⇢ ⇤1/⇢ Et x ˜h Et [

h Zh,t+1 ]

1/⇢

h=1

=

H X

(

h Zht )

1/⇢

h=1

1/⇢

= Zt

.

Thus, Et [Zt+1 ]  Zt when ⇢ > 1. 18.3. Consider an infinite-horizon version of the model in Section 18.5 in which both investors agree the dividend process is a two-state Markov chain, with states D = 0 and D = 1. Suppose the investors’ beliefs Ph satisfy, for all t

0,

P1 (Dt+1 = 0|Dt = 0) = 1/2 ,

P1 (Dt+1 = 1|Dt = 0) = 1/2 ,

P1 (Dt+1 = 0|Dt = 1) = 2/3 ,

P1 (Dt+1 = 1|Dt = 1) = 1/3 ,

P2 (Dt+1 = 0|Dt = 0) = 2/3 ,

P2 (Dt+1 = 1|Dt = 0) = 1/3 ,

P2 (Dt+1 = 0|Dt = 1) = 1/4 ,

P2 (Dt+1 = 1|Dt = 1) = 3/4 .

Assume the discount factor of each investor is = 3/4. For s = 0 and s = 1, set "1 # X t Vh (s) = Eh Dt | D0 = s . t=1

For each h, use the pair of equations Vh (s)

= Ph (Dt+1 = 0 | Dt = s)Vh (0) + Ph (Dt+1 = 1 | Dt = s)[1 + Vh (1)]

to calculate Vh (0) and Vh (1). Show that investor 2 has the highest fundamental value in both states [V2 (0) > V1 (0) and V2 (1) > V1 (1)] but investor 1 is the most optimistic in state D = 0 about investor 2’s future valuation, in the sense that P1 (Dt+1 = 0 | Dt = 0)V2 (0) + P1 (Dt+1 = 1 | Dt = 0)[1 + V2 (1)] > P2 (Dt+1 = 0 | Dt = 0)V2 (0) + P2 (Dt+1 = 1 | Dt = 0)[1 + V2 (1)] .

18 Heterogeneous Priors

201

Consider investor 1. Set x = V1 (0) and y = V1 (1). We want to solve 4 x= 3 4 y= 3

1 x+ 2 2 x+ 3

1 (1 + y) , 2 1 (1 + y) . 3

The solution is V1 (0) = x = 4/3 = 1.33 and V1 (1) = y = 11/9 = 1.22. Now for investor 2, set x = V2 (0) and y = V2 (1). We want to solve 4 x= 3 4 y= 3

2 x+ 3 1 x+ 4

1 (1 + y) , 3 3 (1 + y) . 4

The solution is V2 (0) = x = 16/11 = 1.45 and V2 (1) = y = 21/11 = 1.91 Hence, investor 2 has the highest fundamental value in both states. The last claim is equivalent to   1 16 1 21 2 16 1 21 · + · 1+ > · + · 1+ . 2 11 2 11 3 11 3 11 Multiplying by 11, this is equivalent to 48 64 > , 2 3 which is true.

19 Asymmetric Information

19.1. In the economy of Section 19.4, assume the uninformed investors are risk neutral. Find a fully revealing equilibrium, partially revealing equilibria in which the price reveals s˜ + b˜ z for any b, and a completely unrevealing equilibrium (an equilibrium in which the price is constant rather than depending on s˜ and/or z˜). The equilibrium condition is that p(˜ s, z˜) =

E[˜ x | p(˜ s, z˜)] . Rf

If p(˜ s, z˜) = µ(˜ s)/Rf , then E[˜ x | p(˜ s, z˜)] E[˜ x | µ(˜ s)] µ(˜ s) = = = p(˜ s, z˜) , Rf Rf Rf so there is a fully revealing equilibrium. On the other hand, if p(˜ s, z˜) = x ¯/Rf , then E[˜ x | p(˜ s, z˜)] E[˜ x] = = p(˜ s, z˜) , Rf Rf so there is a completely unrevealing equilibrium. For the partially revealing equilibria, suppose p(˜ s, z˜) = a0 + a1 (˜ s + b˜ z ) with a1 6= 0. Then

 E[˜ x | p(˜ s, z˜)] E[˜ x | s˜ + b˜ z] 1 cov(˜ x, s˜) = = x ¯+ (˜ s Rf Rf Rf var(˜ s) + b2 var(˜ z)

This equals p(˜ s, z˜) if and only if  1 cov(˜ x, s˜) a0 = x ¯ (¯ s + b¯ z) , Rf var(˜ s) + b2 var(˜ z)  1 cov(˜ x, s˜) a1 = , Rf var(˜ s) + b2 var(˜ z)  b cov(˜ x, s˜) a1 b = ) . Rf var(˜ s) + b2 var(˜ z)

s¯ + b˜ z

b¯ z) .

204

19 Asymmetric Information

For any b, we can define a0 and a1 by the first two equations and the third equation will hold. Thus, there is an equilibrium revealing s˜ + b˜ z for any b. This includes b = 0, which is the fully revealing equilibrium. 19.2. Consider the model of Section 19.5, but assume there is a continuum of investors indexed by h 2 [0, 1] with possibly di↵ering risk aversion coefficients ↵h and possibly di↵ering error variances var(˜ "h ). Suppose, for some b, that each investor observes x ˜ + b˜ y in addition to his private signal s˜h . The market-clearing condition is Z

1

✓h (˜ x + b˜ y , s˜h ) dh = y˜ , 0

where ✓h is the number of shares demanded by investor h. Let on x ˜ + b˜ y and s˜h . Set

h

= 1/

2 h.

Define Z 1 ⌧= ⌧h dh

1 = ⌧

and

0

Z

2 h

denote the variance of x ˜ conditional

1

⌧h

h dh ,

0

where ⌧h is the risk tolerance of investor h. (a) Show that the equilibrium price is a discounted weighted average of the conditional expectations of x ˜ minus a risk premium term, where the weight on investor h is ⌧h

h /(⌧

).

(b) Define =

var(˜ x) var(˜ x) + b2 var(˜ y)

and

h =

(1

(1 ) var(˜ x) . ) var(˜ x) + var(˜ "h )

Show that ⌧h

h h

=

1 . ↵h var(˜ "h )

(c) Assume the strong law of large numbers holds in the sense that Z 1 ⌧h h h "˜h dh = 0 . 0

Define 1 = ⌧

Z

1

⌧h

h h dh .

0

Show that the equilibrium price equals a0 + a1 (˜ x + b˜ y ) if and only if a0 =

(1

)[(1

)¯ x

b¯ y]

, Rf (1 ) +  a1 = , Rf Z 1 1 b= 1 dh . "h ) 0 ↵h var(˜

19 Asymmetric Information

(a) The number of shares of the risky asset demanded by investor h is ⌧h

h

E[˜ x|x ˜ + b˜ y , s˜h ]

Rf p(˜ x, y˜) ,

so the market-clearing condition is Z 1 ⌧h h E[˜ x|x ˜ + b˜ y , s˜h ]

Rf p(˜ x, y˜) dh = y˜ ,

0

which we can rearrange as Rf p(˜ x, y˜)

Z

1

⌧h

h dh =

0

Z

1

⌧h 0

x|x ˜ h E[˜

+ b˜ y , s˜h ] dh

y˜ .

This is equivalent to p(˜ x, y˜) =

Z

1 0

⌧h ⌧

h

✓

E[˜ x|x ˜ + b˜ y , s˜h ] Rf

◆

y˜ . ⌧ Rf

dh

(b) From (19.4c), the conditional variance is 2 h

= (1

h )(1

) var(˜ x) .

This implies 1 1 h = (1  h h h

) var(˜ x) =

(1

var(˜ "h ) (1 ) var(˜ x)

) var(˜ x) = var(˜ "h ) .

Thus, ⌧h

h h

=

h h

↵h

=

1 . ↵h var(˜ "h )

(c) From (19.4a), E[˜ x|x ˜ + b˜ y , s˜h ] = E[˜ x|x ˜ + b˜ y ] + h s˜h = E[˜ x|x ˜ + b˜ y]

E[˜ s|x ˜ + b˜ y]

h E[˜ x|x ˜ + b˜ y ] + h s˜h

Thus, p(˜ x, y˜) =

Z

1 0

⌧h ⌧

1 = ⌧ Rf

h

Z

✓

E[˜ x|x ˜ + b˜ y , s˜h ] Rf

1

⌧h 0

x|x ˜ h E[˜

◆

dh

+ b˜ y ] dh

y˜ ⌧ Rf Z 1 1 ⌧h ⌧ Rf 0

Z 1 1 y˜ + ⌧h h h s˜h dh ⌧ Rf 0 ⌧ Rf E[˜ x|x ˜ + b˜ y ] E[˜ x|x ˜ + b˜ y ] ˜ x = + Rf Rf Rf

y˜ . ⌧ Rf

x|x ˜ h h E[˜

+ b˜ y ] dh

205

206

19 Asymmetric Information

From (19.2a), E[˜ x|x ˜ + b˜ y] = x ¯ + (˜ x

x ¯ + b˜ y

b¯ y) .

Therefore, p(˜ x, y˜) =

(1

)[¯ x + (˜ x x ¯ + b˜ y Rf

b¯ y )]

˜ x Rf

+

y˜ . ⌧ Rf

This is equal to a0 + a1 (˜ x + b˜ y ) if and only if )¯ x b¯ y] , Rf (1 ) +  a1 = , Rf (1 ) b 1 a1 b = . Rf ⌧ Rf a0 =

(1

)[(1

The last two equations imply b = Rf so b=

1 ⌧ 

=

1

Z

1 , ⌧ Rf

1

⌧h

h h dh =

0

1

Z

1 0

1 dh . ↵h var(˜ "h )

19.3. In the single-period Kyle model, assume the informed investor has CARA utility. There is a linear equilibrium. Derive an expression for

as a root of a fifth-order polynomial, assuming

the informed investor observes s˜ = x ˜ + "˜, where "˜ is normally distributed with zero mean and is independent of x ˜. Let v˜ = E[˜ x | s˜], and set 2

= var(˜ x | s˜) =

✓

1

var(˜ x) var(˜ x) + var(˜ ")

◆

var(˜ x) .

A strategy for the informed trader is affine in the signal s˜ if and only if it is affine in v˜, so we are looking for an equilibrium in which the informed trader plays ✓(v) = ↵ + v for some ↵ and

.

With y˜ = ↵ + v˜ + z˜, the normal-normal updating rule implies E[˜ x | y˜] = E[˜ v | y˜] = + y˜, where ✓ ◆ 2 var(˜ v) = 1 v¯ , (19.1a) 2 var(˜ v ) + var(˜ z) var(˜ v) = 2 . (19.1b) var(˜ v ) + var(˜ z) The informed trader maximizes the certainty equivalent

19 Asymmetric Information

E[✓(˜ x

↵ var(✓(˜ x 2

(✓ + z˜)) | s˜]

(✓ + z˜)) | s˜) = ✓(˜ v

↵ 2 ✓ [ 2

✓)

2

+

2

207

var(˜ z )] .

The optimum is attained at ✓=

v˜ 2 +↵ +↵ 2

2 var(˜ z)

2

2 var(˜ z)

.

Thus, ↵=

2 +↵

+↵ 1 = 2 +↵ 2+↵

2 var(˜ z)

Substituting (19.1d) into (19.1b) yields "✓ # ◆2 1 var(˜ v ) + var(˜ z) 2 + ↵ 2 + ↵ 2 var(˜ z)

=

,

(19.1c)

.

(19.1d)

var(˜ v) , 2 2 + ↵ + ↵ 2 var(˜ z)

which can be rearranged as ⇥

var(˜ v) + Thus,

2 +↵

2

+↵

2

var(˜ z)

is a root of a fifth-order polynomial.

⇤2

⇥ var(˜ z) = 2 + ↵

2

+↵

2

⇤ var(˜ z ) var(˜ v) .

19.4. In the continuous-time Kyle model, assume log v˜ is normally distributed instead of v˜ being normally distributed. Denote the mean of log v˜ by µ and the variance of log v˜ by

2.

Set

=

v/ z.

Show that the strategies 1

P0 = eµ+ 2

2 v

dPt = Pt dYt d✓t =

(log v˜

µ)/ 1 t

Yt

dt

form an equilibrium by showing the following: (a) Define Wt = Yt / value (log v˜

z.

µ)/

Show that, conditional on v˜, W is a Brownian bridge on [0, 1] with terminal

v.

Use this fact to show that P satisfies P1 = v˜ and is a martingale relative

to the market makers’ information. (b) For v > 0 and p > 0, define J(t, p) =

p

v + v(log v

log p)

+

1 2

v z (1

t)v .

In the definition of the class of allowed strategies in Section 19.7, modify condition (ii) to (ii0 ) hR i 1 E 0 Pt2 dt < 1. Prove the verification theorem.

208

19 Asymmetric Information

(a) We have 1

dW =

(d✓ +

z

z

dB) =

(log v˜

µ)/ 1 t

W

v

dt + dB .

Hence, W is a Brownian bridge on [0, 1] with terminal value (log v˜ dP = P

dY =

v

µ)/

v.

Moreover,

dW ,

so P is a geometric Brownian motion relative to the market makers’ information. We also have P1 = P0 e

v W1

1 2

2 v

1

= eµ+ 2

2 v

elog v˜

µ

1 2

2 v

= elog v˜ = v˜ ,

so Pt is the conditional expectation of v˜ given the market makers’ information, for each t. Furthermore, P being a geometric Brownian motion implies that ✓ satisfies condition (ii0 ). (b) We have v zv

Jt = Jp = Jpp = (dP )2 = Hence,

Z

2 1

,

v , p

v , p2 2 2 v P dt .

1

1 Jt dt + Jp dP + Jpp (dP )2 2 0 ◆ Z 1✓ 1 v 1v v zv = J(0, P0 ) + dP + 2 P 2 0 Z 1 P v dP = J(0, P0 ) + P 0 Z 1 = J(0, P0 ) + (P v)(d✓ + z dB) .

J(1, P1 ) = J(0, P0 ) +

2 v

0

Rearranging gives

Z

1

(v

P ) d✓ = J(0, P0 )

0

J(1, P1 ) +

Z

1

(P

v)

z

dB .

0

Taking expectations throughout, using the assumption (ii0 ), yields Z 1 E (˜ v P ) d✓ = J(0, P0 ) J(1, P1 )  J(0, P0 ) , 0

the inequality following from J(1, p)

0, which can be deduced from the fact that J(1, p) is

convex in p and hence has a unique minimum at p = v. Thus, any strategy that implies P1 = v˜ is optimal.

20 Alternative Preferences in Single-Period Models

20.1. Consider the following pairs of gambles:

C:

A : 100% chance of $3,000

versus

8 >

:75% chance of $0

B:

8 >

:20% chance of $0 8 >

:80% chance of $0

.

(a) Show that an expected utility maximizer who prefers A to B must also prefer C to D. (b) Show that the preferences A C = ↵A + (1

B and D

↵)Q and D = ↵B + (1

C violate the independence axiom by showing that ↵)Q for some 0 < ↵ < 1 and some gamble Q.

(c) Plot the gambles A, B, C, and D in the probability simplex of Figure ??, taking p1 to be the probability of $0 and p3 to be the probability of $4,000. Show that the line connecting A with B and the line connecting C with D are parallel. Note: the preferences A

B and D

C are common. This example is also due to Allais (1953)

and is a special case of the common ratio e↵ect. See, e.g., Starmer (2000). (a) Preference for C over D is equivalent to 0.25u(x2 ) > 0.2u(x3 )

,

u(x2 ) > 0.8u(x3 )

,

A

B.

(b) Let Q be the gamble that pays 0 for sure, and set ↵ = 0.25. (c) In the probability simplex, A = (0, 0), B = (0.2, 0.8), C = (0.75, 0) and D = (0.8, 0.2). The slope of the line connecting A and B is 0.8/0.2 = 4, and the slope of the line connecting C and D is 0.2/0.05 = 4.

210

20 Alternative Preferences in Single-Period Models

20.2. Consider the following pairs of gambles: 8 8 > >

:10% chance of $0 :55% chance of $0 8 8 > >

:99.8% chance of $0 :99.9% chance of $0

.

Show that an expected utility maximizer who prefers A to B must also prefer C to D. Note: the preferences A

B and D

C are common. This example is due to Kahneman and Tversky (1979).

We have A

B

,

0.002u(x2 ) + 0.998u(x1 ) > 0.001u(x3 ) + 0.999u(x1 )

,

0.002u(x2 ) > 0.001u(x3 ) + 0.001u(x1 )

,

u(x2 ) > 0.5u(x3 ) + 0.5u(x1 ) ,

and C

D

,

0.9u(x2 ) + 0.1u(x1 ) > 0.45u(x3 ) + 0.55u(x1 )

,

0.9u(x2 ) > 0.45u(x3 ) + 0.45u(x1 )

,

u(x2 ) > 0.5u(x3 ) + 0.5u(x1 ) .

20.3. Consider weighted utility. Let "˜ have zero mean and unit variance. For a constant , denote the certainty equivalent of w + "˜ by w

⇡( ). Assume ⇡(·) is twice continuously di↵erentiable.

By di↵erentiating v(w

⇡( ))E[ (w + "˜)] = E[ (w + "˜)v(w + "˜)] ,

assuming di↵erentiation and expectation can be interchanged, show successively that ⇡ 0 (0) = 0 and ⇡ 00 (0) =

v 00 (w) v 0 (w)

2 0 (w) . (w)

Note: This implies that for CRRA weighted utility and small , ⇡( )/w ⇡ (⇢

2 ) var( "˜/w)/2.

We have v 0 (w

⇡( ))E[ (w + "˜)]⇡ 0 ( ) + v(w

⇡( ))E[ 0 (w + "˜)˜ "]

= E[ (w + "˜)v 0 (w + "˜)˜ "] + E[ 0 (w + "˜)v(w + "˜)˜ "] .

20 Alternative Preferences in Single-Period Models

At

211

= 0, using E[˜ "] = 0, we obtain ⇡ 0 (0) = 0. Taking second derivatives and substituting ⇡(0) = 0,

⇡ 0 (0) = 0, E[˜ "] = 0, and E[˜ "2 ] = 1 yields v 0 (w) (w)⇡ 00 (0) + v(w)

00

00

(w) = v(w)

(w) + 2 0 (w)v 0 (w) + (w)v 00 (w) ,

which implies v 00 (w) v 0 (w)

⇡ 00 (0) =

2 0 (w) . (w)

20.4. Consider CRRA weighted utility. (a) Show that g in (20.11) is strictly monotone in y > 0 — so the preferences are monotone with regard to stochastic dominance — if and only if

 0 and ⇢ 

+ 1 with at least one of these

being a strict inequality.  0 and

(b) Show that g in (20.11) is strictly monotone and concave if and only if with either

< 0 or ⇢
0, and min E [W2 | s˜] = W1 + ✓1 (µ = W1 + ✓1 [µ +

P1 ) + s a (˜

s a ✓1 (˜ µ)

µ)

P1 ] .

21 Alternative Preferences in Dynamic Models

We can make this arbitrarily large by taking ✓1 ! µ+ If ✓1 > 0, then ✓1 (˜ s

s a (˜

µ)

221

1 unless P1

0.

µ) < 0, and min E [W2 | s˜] = W1 + ✓1 (µ

P1 ) +

= W1 + ✓1 [µ +

s b (˜

s b ✓1 (˜ µ)

µ)

P1 ] .

We can make this arbitrarily large by taking ✓1 ! 1 unless µ+

s b (˜

µ)

P1  0 .

Thus, the existence of an optimum implies µ+

µ)  P1  µ +

s b (˜

s a (˜

µ) .

If the first inequality is an equality, then any long position is optimal. If the second is an equality, then any short position is optimal. If both are strict inequalities, then ✓1 = 0 is the unique optimum. Thus, ✓1 = 1 is optimal if and only if P1 = µ + (b) If ✓1 < 0, then ✓1 (˜ s

s b (˜

µ) .

µ) < 0, and min E [W2 | s˜] = W1 + ✓1 (µ

P1 ) +

= W1 + ✓1 [µ + We can make this arbitrarily large by taking ✓1 ! µ+ If ✓1 > 0, then ✓1 (˜ s

s b (˜

µ)

s b (˜

s b ✓1 (˜ µ)

µ)

P1 ] .

1 unless P1

0.

µ) > 0, and min E [W2 | s˜] = W1 + ✓1 (µ = W1 + ✓1 [µ +

P1 ) + s a (˜

We can make this arbitrarily large by taking ✓1 ! 1 unless

s a ✓1 (˜ µ)

µ)

P1 ] .

222

21 Alternative Preferences in Dynamic Models

µ+

s a (˜

P1  0 .

µ)

Thus, the existence of an optimum implies µ+

s a (˜

µ)  P1  µ +

s b (˜

µ) .

If the first inequality is an equality, then any long position is optimal. If the second is an equality, then any short position is optimal. If both are strict inequalities, then ✓1 = 0 is the unique optimum. Thus, ✓1 = 1 is optimal if and only if P1 = µ +

s a (˜

µ) .

(c) We have E [W1 ] = W0

✓0 P0 + ✓0 E [P1 (˜ s)] .

If ✓0 < 0, then E [W1 ] is minimized by maximizing E [P1 (˜ s)], which is equivalent to minimizing the variance of s˜; i.e., taking

=

b.

Thus, for ✓0 < 0,

⇣ min E [W1 ] = W0 + ✓0 E b [P1 (˜ s)] We can make this arbitrarily large by taking ✓0 ! E b [P1 (˜ s)]

⌘ P0 .

1 unless

P0

0.

If ✓0 > 0, then E [W1 ] is minimized by minimizing E [P1 (˜ s)], which is equivalent to maximizing the variance of s˜; i.e., taking

=

a.

Thus, for ✓0 > 0,

⇣ min E [W1 ] = W0 + ✓0 E a [P1 (˜ s)]

⌘ P0 .

We can make this arbitrarily large by taking ✓0 ! 1 unless E a [P1 (˜ s)]

P0  0 .

Hence, the existence of an optimum implies E a [P1 (˜ s)]  P0  E b [P1 (˜ s)] . If the first inequality is an equality, then any long position is optimal. If the second is an equality, then any short position is optimal. If both are strict inequalities, then ✓1 = 0 is the unique optimum. Thus, ✓1 = 1 is optimal if and only if

21 Alternative Preferences in Dynamic Models

P0 = E a [P1 (˜ s)] . It remains to calculate this. We have P1 (˜ s) = µ +

s b (˜

µ)1{˜sµ}

=µ+

s b (˜

µ)

s a )(˜

µ)1{˜s>µ} .

(

b

Thus, E a [P1 (˜ s)] = µ =µ

( (

a )E

b

a)

b

Z

a

⇥

1 0

(˜ s

µ)1{˜s>µ} x

p 2⇡

2/

To evaluate the integral, use the fact that ✓ ◆ x x2 p exp dx = 2 2 2⇡ 2

⇤

exp

2

✓

x2 2 2/

Thus,

1

n0 (x) dx = n(1)

2.

p

n(0) =

0

Z

which yields the result.

n(0) =

1 0

x p 2⇡

2

exp

✓

x2 2 2

◆

dx .

n0 (x)

where n denotes the normal density with mean zero and variance Z

◆

dx = p

2⇡

We have

1 2⇡

2

.

,

(d) For any positive number n, consider ✓0 = n and ✓1 (s) = n for all s. Then, for each , ⇣ E [W2 ] = W0 + n E [˜ x] as n ! 1.

⌘ P0 = W0 + n(µ

P0 ) ! 1

223

22 Production Models

22.1. Assume there is costless adjustment of capital, i.e. Kt+1 = Kt + It for any It . (a) Show that a capital stock process is optimal if and only if Kt solves  Mt max Et 1 [⇡(K, Xt ) + K] K K Mt 1 for each t. (b) Show that the first-order condition for the maximization problem in the previous part is a special case of (22.4a). (a) Substituting It = Kt+1

Kt in (22.2), the firm’s value is "1 # X E Mt [⇡(Xt , Kt ) Kt+1 + Kt ] . t=0

This equals M0 [⇡(X0 , K0 ) + K0 ] + E

"

1 X

Mt ⇡(Xt , Kt )

Mt

1 Kt

#

+ M t Kt ] .

t=1

In this infinite sum, Kt appears in only in the single term E[Mt ⇡(Xt , Kt ) Because Kt is chosen at time t

1 Kt

+ M t Kt ] .

1, maximizing this expectation is equivalent to maximizing the

conditional expectation given time t expectation is Mt

Mt

1 Et 1



1 information in each state of the world. The conditional

Mt [⇡(Xt , Kt ) + Kt ] Mt 1

Mt

1 Kt ,

226

22 Production Models

and maximizing this is equivalent to maximizing  Mt Et 1 [⇡(Xt , Kt ) + Kt ] Mt 1

Kt .

(b) The first-order condition is Et

1



Mt [⇡K (Xt , Kt ) + ] = 1 , Mt 1

equals Rt defined in (22.4b0 ).

and ⇡K (Xt , Kt ) +

22.2. Assume a firm combines labor L with capital K to produce output Qt = Kt↵ Lt , where ↵ and Lt

are constants with ↵ +

 1. The firm’s operating cash flow is Pt Qt

Wt Lt , optimized over

0, where P and W are regarded as exogenous stochastic processes. Show that operating cash

flow equals Xt Kt for some exogenous stochastic process X and constant = 1 if ↵ +

 1, and show that

= 1.

The first-order condition for maximizing P K ↵ L

W L is P K ↵ L

1

= W , which implies

W L = P K ↵L , so operating cash flow is )P K ↵ L .

(1 Furthermore, the first-order condition implies L=

✓

W P K↵

◆1/(

1)

=

✓

P K↵ W

◆1/(1

)

.

Therefore, operating cash flow is (1

)P K

↵

✓

P K↵ W

This is of the form claimed with ↵=1

◆

/(1

= ↵/(1

)

= (1

)P

). We have

✓

P W

◆

/(1

)

K ↵/(1

 1 because ↵  1

)

. , and

= 1 if

.

22.3. Assume a firm can produce any output Qt up to a maximum capacity Zt Kt , where Z is a positive stochastic process. Assume the output price is Pt = Yt Qt process Y and constant

1/

for a positive stochastic

> 1. Assume operating cash flow equals revenue Pt Qt .

22 Production Models

227

(a) Show that it is optimal to produce at full capacity Qt = Zt Kt , and operating cash flow is 1 1/

X t Kt

, where X = Y Z 1

1/

.

(b) Assume investment is irreversible and there is no depreciation. Assume the risk-free rate r is constant, the market price of risk

is constant, and X is a geometric Brownian motion with drift

µ and volatility . Assume the correlation of X and M is a constant ⇢, and assume µ

⇢ < r.

Show that the value (22.22) of the asset paying marginal cash flow is ✓ ◆ 1 1 k 1/ Xt , r µ+ ⇢ and the dividend-price ratio is r

µ+

⇢.

(c) Show that the return on the asset, including the dividend, is (r +

⇢) dt + dB ,

where B is the Brownian motion driving X. (d) Use the result of Exercise 15.5 to show that the optimal exercise time for the option is min{t | St (k)

/(

1)}, where

Show that this implies Pt Zt  ⇤

is the positive root of the quadratic equation (22.27).

⇤

for all t and the firm invests only when Pt Zt = ✓ ◆✓ ◆ 1 = (r µ + ⇢) . 1

⇤,

where

Note that Pt Zt is the output price per unit of capital. (e) The value of each growth option is given by (15.33), where the strike price is 1 and ✓ ◆ 1 1 S0 = k 1/ X0 . r µ+ ⇢ Integrate the values of the growth options over k 2 [K0 , 1) to compute the value of the firm. What condition on

and

is needed for the value of the firm to be finite?

(a) The firm chooses Q  ZK to maximize P Q = Y Q1 Q = ZK. Hence, P Q = Y Z 1

1/

K1

1/

= XK 1

1/

(b) Marginal operating cash flow is ⇡K (X, K) = (1 marginal cash flow is ✓ ◆ Z 1 k 1/ Et

1 t

Mu Xu du = k Mt

M X is a geometric Brownian motion with drift

1/

. Because 1

.

1/ )XK

1/

✓ (r

1/ > 0, the optimum is

1 µ+

◆

1/

. The value of the asset paying

Xt

Z

1 t

Et



Mu X u Mt X t

⇢). Therefore,

du .

228

22 Production Models

Et



Mu X u =e Mt X t

(r µ+

⇢)(u t)

.

It follows that the value of the asset paying marginal cash flow is ✓ ◆✓ ◆ 1 Xt k 1/ . r µ + ⇢) The dividend on the asset is the marginal cash flow, which is ✓ ◆ 1 Xt k 1/ . Therefore, the dividend yield is r

µ+

⇢.

(c) From the previous part, dS(k) dX = = µ dt + dB . S(k) X Therefore the return on the asset is (r

µ+

⇢) dt + µ dt + dB = (r +

⇢) dt + dB .

(d) Observing that the strike of the option is 1, Exercise 15.5 shows that the optimal exercise time is /(

1), where

is the positive root of the quadratic equation (15.32). In that equation,

q denotes the dividend yield, which here is r

µ+

the same as (22.27). The condition St (k) = /( k

1/

Xt = (r

µ+

⇢. Substituting this value of q, (15.32) is

1) is equivalent to ✓ ◆✓ ◆ ⇢) , 1 1

and Kt

1/

X t = Kt

1/

1 1/

Yt

Zt = Qt

1/

Y t Z t = Pt Z t .

Thus, the firm invests when Pt Zt = (r

µ+

Via this investment, it ensures St (Kt )  /( Pt Zt  (r for all t.

µ+

⇢)

✓

1

◆✓

1

◆

.

1) for all t, which is equivalent to ✓ ◆✓ ◆ ⇢) . 1 1

22 Production Models

229

(e) From (15.33), the values of the growth options are ✓

1 1

◆✓

1

◆ 

1 µ+

r

✓

⇢ =

The integral of k

/

1

✓

◆

1 1

1/

k

X0

◆ ✓

1

◆✓

over [K0 , 1) is finite if and only if Z

1

/

k

1

)/

K0

K0

and the aggregate value of growth options is ✓ ◆✓ ◆ ✓ ◆✓ 1 1 1

1

◆✓

X0 µ+

r

> . Assuming

(

dk =

◆✓

k

/

.

> ,

,

X0 µ+

r

⇢

◆

⇢

◆

(

K0

)/

.

The value of assets in place is E

Z

1

Mt ⇡(Xt , K0 ) dt =

0

1 1/ K0 1 1/

= K0 =

r

E

Z

X0

X0 µ+

1 0

Z

Mt Xt dt

1

e

(r µ+

⇢)t

dt

0

⇢

1 1/

K0

.

22.4. Under the assumptions of Section 22.8, marginal q (22.23) is a function of Xt . Denote this function by f . The purpose of this exercise is to derive the formula (22.26) for f . (a) Show that

Z

t

Mu Xu du + Mt f (Xt ) 0

is a martingale. (b) Using Itˆ o’s formula and the result of the previous part, show that f satisfies the di↵erential equation x + (µ

⇢)xf 0 (x) +

1 2

2 2 00

x f (x) = rf (x)

for 0 < x < x⇤ . (c) Show that the function r satisfies the di↵erential equation.

x µ+

⇢

230

22 Production Models

(d) Show that the homogeneous equation ⇢)xf 0 (x) +

(µ

is satisfied by Ax for constants A and

1 2

2 2 00

x f (x) = rf (x)

if and only if

is a root of the quadratic equation

(22.27). (e) The general solution of the di↵erential equation in Part (b) is

r for constants Ai , where the

i

x µ+

⇢

+ A1 x

1

+ A2 x

2

are the roots of the quadratic equation. From the definition of f ,

it follows that limx#0 f (x) = 0. Use this fact to show that f (x) = where

x µ+

r

⇢

+ Ax ,

is the positive root of the quadratic equation and A is a constant.

b part. Thus, (f) Because the process in Part (a) is a martingale, its di↵erential cannot have a dK

b t in df (Xt ) calculated from Itˆ the coefficient of dK o’s formula must be zero when Xt = x⇤ . Use this fact to compute the constant A and derive (22.26).

(a) We have Z

t

Z 1 Mu Mu X u + Mt E t Xu du Mt 0 u Z 1 = Et Mu Xu du .

Mu Xu du + Mt f (Xt ) = 0

Z

t

0

(b) The di↵erential is 

dM M X dt + M df + f dM + (df )(dM ) = M X dt + df + f + (df ) M Also, 1 df = f 0 (X) dX + f 00 (X) (dX)2 2 ✓ ◆ dX 1 2 00 dX 2 0 = Xf (X) + X f (X) , X 2 X and dX dW = X W

b 1 dK . b K

✓

dM M

◆

.

22 Production Models

Hence,

✓

and (df )

✓

dM M

◆

0

= Xf (X)

✓

dX X

◆2

=

dM M

◆

dX X

◆✓

✓

◆2

dW W

2

=

0

= Xf (X)

✓

Therefore, the coefficient of dt in the di↵erential is  1 2 2 00 M X + µXf 0 (X) + X f (X) 2

dt ,

dW W

◆✓

dM M

◆

⇢Xf 0 (X) dt .

=

⇢Xf 0 (X) .

rf

Equating this to zero yields the di↵erential equation. (c) For this function f , x + (µ

⇢)xf 0 (x) +

1 2

2 2 00

x f (x) = x +

(µ ⇢)x = r µ+ ⇢ r

rx µ+

⇢

= rf (x) .

(d) Given f (x) = Ax , we have (µ

⇢)xf 0 (x) +

1 2

2 2 00

x f (x)

rf (x) = (µ

⇢)Ax +

1 ( 2

1)

2

Ax

rAx .

This is equal to zero for all x 6= 0 if and only if (µ

⇢) +

1 ( 2

1)

2

r = 0,

which is the quadratic equation (22.27). (e) If

2

< 0, then f (x) ! 1 as x ! 0, unless A2 = 0.

ˆ in the calculation in Part (b) comes from M df and is equal to (f) The term involving dK 1

M Xf 0 (X)

b dK b K

This is equal to zero when Xt = x⇤ if and only if f 0 (x⇤ ) = 0. We have f 0 (x) =

r

Setting this equal to zero at x = x⇤ yields ✓ 1 A= r

1 µ+

+ Ax

⇢

1 µ+

⇢

◆

1

(x⇤ )1

.

.

Hence, f (x) = =

✓ ✓

r r

1 µ+ x µ+

⇢ ⇢

◆ ◆

x 1

1

(x⇤ )1

1⇣x⌘ x⇤

x 1

.

231