Econ 136 Financial Economics Course Reader University of California, Berkeley [Summer 2020 ed.]

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ACKNOWLEDGEMENTS Permission to reproduce the se copyrighted materials has been obtained from the publishers for : Raymond J. Hawkins Econ 136: Financial Economics University of California , Berkel ey Sllllllller2020 Any further rep roduction is a violatio n of the amended Copyright Act of 1976. Civil and criminal penalties may be imposed for copyrig ht infringement .

From Theory of financial risk and derivative pricing: from . Bou chaud , J.P., & Potters, M .

x), which is th e count ercumulative distribut ion function of the city size. The equation of moti on of G, is

G,+1 (x) =

(8)

L)

- fo"'c ,G )r x11•)• i.,r x) x - x) on the vertical axis. Th is figure shows that In P(lrl> x)

= -{, In x + constant

(4 1)

yields a good fit for lrl between 2 and 80 standard deviations. Ord inary least squares (OLS) estimation yields - {,= - 3.1 ± 0.l( i.e., Equation 40 ). It is nor necessary for th is graph to be a straig ht line or for the slope to be - J {e.g.• in a Gaussian world, it would be a concave parabola). Gopikr ishnan ct aL {1999) refer to Equation 40 as the inverse cublc law of returns, The panicular value C, 3 is consistent with a finite varian ce and means that stock marker returns arc not Levy distr ibuted {a Uvy distribution is eit her Gaussian, or has infinite variance, C,< 2). 27 Plerou et al. (1999) examined firms of different sizes. Small firms have higher volatility than large firms, as verified by Figure 4a. Mo reover, Figure 4a also shows similar slopes for the graphs for four quan ilcs of firm size. Figure 4b no rmaliz.csthe distrib ution of each size quamile by irs stan dard deviation, so that the normaliz.ed distr ibutions all have a stan dard deviation of 1. Th e plots collapse on rhc samc curve, and all have expone nts close ro C, 3. Plcrou ct al. (200S) found that the bid-ask spread also follows the cu bic law.

115cc T1\eb (2007) foe a wide-ranging cnay on those rare cvcn11.

"Lo nger-horiion return distributions a.reshaped by two opposite forcn. One {om: is du-1a Finitesum of ind x) • • forx2:0. u ~foct. cJchc l1tcn1urcatiruta 1/C nthcr than (, hmcr the 11 - 2 •nd 11 - J facron htrc , nlflc r th.n the usu.al" · I hor brm unable to find•• catltcr rcftrmcc for dtc»c upraatOM, IO I dtn'rCIII1!iianfor ctus rcw.ew.h •• ca,r to show du1 mt)' arc 1hc mrrra ona to p:-t unbiucd atimalts.. Ullnc the lllhlyi rhto«m . •lid fact that X1 +- + x. has dmtiry 1)1whm.l'l are indcpeMlcntdra ws from a1t1NW·d uponmual distnbuuon. #fhttl • 1f .$ hat aNnlttCUIUlattft fu.ncrionf( x); Ihm f{S) folows I Naftdar4 Mn1formchatnbMhon,and the of tlic ,-di ... 11ntvaJ.c out o(" - I of I uniformdilfflh.rion i1 ,,,,, .

~ ••-•!(,,-

economically accepted that many extreme stock market returns arc clustered in time and affected by the same factors. Hence, standard errors will be ilJusorily too low if one assumes tha t the dara arc independent. There is no consensus procedure to overcome that problem . In practice, applied papers often repon the Hill or OLS estimato~ together with a caveat that the observations are not necessarily independent, 10 that the nominal standard errors probably underestimate the true standard errors. Moreover, sometimes a lognormal firs better. lndee~ since early on, some have attacked the fit of the Pareto law (see Pcnk y 1992). The ruson , broadly, is that adding more param-

eters (e.g., a curvature), as a lognormal permits, can only improve the fit. However,

the Pareto law hu survived the test of rime: It 6" still qu ite well. The extra degree of frttdom allowed by a lognonnal might be a distraction from the essence of the phenomenon. 7.2. Testing With an infin itely large empirical data set, one can re;ca any nontautolog ical theory. Hence, the main question of empirical work should be how well a theory fits, rather than whether it 6" perfectly (i.e., within the standard errors). Leamer 6c Lcvinsohn ( 199S) argue that, in the context of empirical researc h in international trad e, too much energy is spent seeing if a theory firs exactly . Rather, researchers shouJd aim at broad , although ncccnariJy nonabsolutc, regularities. In other words, ..cstimarc, don't rest ." lriji 6c Simon (1964, p. 78) remarked tha t Ga lileo's law of the inc lined plane , wh ich stares that the distance traveled by a baU rolling down the plane increases with the square of the timct docs ignore variables that may be important under various circumstances: irregularities in the ball or the plane, rolli ng friction, air resistance, possible elecrrial or magnetic: fields if the ball is metal, variations in the gravitational field and so on, ad infinitum . The enormous progrcu that physics has made in three ttnruries may be partly attributed to its willingness to ignore for a time discrepancies from theories that are in some sense substantilllly correct.

N

OJ

ConstJtcnt with these suggestions_ some of the debate on Zipfr law should be ast in terms of how wel~ or poorly, it fit$, rather than whether it can berejected . The empirical rucarch cmiblishcs that the data are typically well described by a PL with exponent CE j0.8, 1.2): This pattern catalyzes a search for an underlyi ng mechanism . Nonetheless , it is useful to have a test, so what is a test for the fit of a PL? Many papers in practice do not provide such a test. Some authors (Clausct et al. 2008) advocate the Kolmogorov-Smimov test. Gabaix & tbcagimov (2008b) provided a simple tell using the O1..Sregression framework of the previo us subsection. We define s. run the 01.S regression, In(, - ½) ~ eon.stant- ClnS w + 4 ( lnS {ij-

ii

- (11o ~f.1.~) and i-ti...J,)

s,)'+noise,

utimat ·e the values t and 4. The term (lnS; - ,, ) 2 captures a quadnoc deviation from an cx:acr PL, and the coc:ffidcnt s. reccmcn the quadratic term . With this n:ccntcring, 1hcestimate of the PL uponcnts C is the same regardless of the inclusion of th e quad 1.0

ratic term. The rest of the PL is to reject the null of an exact PL if and o nly if 14 /C21> l.9S ·(2n)- ' 1'.

8. CONCLUSION Al the hi.story of science sho~ trying to solve apparently na rro w, but sharp ly po sed, nontrivial problems iJ a fruitful way to make subscantial progress. As Schumpeter p. lSS ) noted for PL.s,studying such questions may • tay the founda tions for an entirel y novel type of theory .• PL.s have forced ecooomisrs to write new theo ries, e.g., on the origins of cities, firm5yimunational trade, CEO pay, or o f extreme move.mcncsin stock market fluctuations. Accordingly , J list some open quCStions m the Future Issues scaion . The rime is ripe for economists to use those Pl.s to investigate old and new regularities with renewed models and data, continuing rhc uad ition of Gtbrat, Champemownc , Mandelbroc, and Simon .

re•

of •It~mosr rcn ckabl• ~e in {rll,n lhou ld not only' gen

(s-X}+[x--x-]-D . 0

(I+ r)'

(5.9)

Assuming for a moment th at no dividends arc expected before expiration (D = 0), th e price of the American call option mus t be greater than the intrinsk value, S0 X, because for a positive interest mtc, Xis greate r than X/(1 + r)T In other words, without any expected dividends, th= is no incentive to exercise the American call option early because selling the option to someone else givesthe investor greater value than exercising it and receiving on ly the intrinsk value. Early exercise of an in-the -money call option may be desirable, however, ju st before a rdativdy large cash distribution on the underlying security. Indiv idual stock options in the United States are not dividend protceted, meaning that the strike price does not automatically adjust for the natural drop in stock price on the ex-dividend date. We can gain some insight into possible early exercise of an in-the-mo ney American call option when dividends arc expected by using the put'-Call par ity relationship in Equation 5.8. Rearranging th e Europe•n put-call par ity rdation ship gives

(S0 -XJ-eo =

D-[x--x-]-Po • (I + ,)'

(5.10)

The intrinsic value of a European call option, S0 -X, will exceed the European call price, , 0 , if D > X - Xl(l + r)T + p0 • If the cxpecrcd dividend is large enough, it mi ght be desirable to exercise the optio n in order to captu re the intrin sic value. In other words, early exercise of a call option may be optima l 02013 The ResearchFoundation of CFAInstitute

71

Fundamentalsof Futureland Options if a pending dividend on the underl ying stock exceeds the ti me value component, composed of the intere st opportunity cost, X - X/(1 + r)': and the price of a European put op tion . This smement suggests the possibility of desirabl e early exercise, but it is only an approximation because we have used European options , not American options, to derive the insight. For example, the numeria.l illustration in the preceding section started with a ca ll opti on that was S5 in th e money. The call option price wa s S7.75, with an exerc ise value of $5.00 and a tim e value component of $2.75. The time value included an interest oppor tunity cost of 10. 15 and an insurance value of the call opt ion (approxima ted by the price of a European put option) ofS2.60. Thus, if this stock were going to pay a cash dividend of $2.75 per sh2re or higher , early exerc ise of the call option migh t be opti m al jus t before the ex-divide nd date. Intuiti vely, the dividend needed to trigge r the early exercise of a call opt ion needs to be higher if prevailing interest rates arc higher. Fo r examp le, if the interest rate were 5.0% instead of 1.8%, the time value component of th e S5 in -the -money call opti on wo uld be S2.92 in stea d of S2.75. We can also gain some insight about the possibility of early exercise for an in-the-money American put option from th e put-call parity relation ship for European options. Rearranging th e rdationship in Equation 5.8 gives

(X -Sol - Po

=[x--x-]-(eo (l+r)'

+D).

(5.11)

The exercise value of a put option, X - S0 , will exceed the European put price , p 0, if c0 + D < X - Xl(l + r)1:In other words, early exercise of an Americ an put op tion might be d esirable if it is so deeply in the money that a European call option with the same st rike price plus the present value of expected dividend s is less than the present value of the interest opportunity cost. Again, this statement is only an approximation because we h ave used European options and not American opt ions to derive the insight. For examp le, early exercise wou ld have been profi table in rhe preceding numer ical examp le for a put option with a strike price of SlOOon a stock that had fallen to $80. Specifical ly, the insurance value of the put option (approximated by the price of a European call option) was worth on ly S0.10 but the interest opporrunity cost on the strike price for one month was S0.15. In fact, sensitivi ty analysis for a volatility estimate of 40.0% and the Bla ck-Sc ho les option-pricing formul a (to be discu ssed later in thi s chapter) show that early put option exercise could be optimal wi th these param eter values for any stock pri ce below about $81. Of course, given higher interes t rates or more time to exp iration, the interest opportun ity cost would be higher and the put optio n might n ot need to be so far in the money . For example, sensiti vity analysis

72

020 13 The ResearchFoundationof CFAlnsbtute

OptionContracts:PricingRelationships shows that if annualized interest rates were at 5.0%, instead ofl.8%, then the breakeven early exercise stock price would be about S85 instead ofS8 1. We note again that the numerica l examples of early exercise of American options have only been approximate because American option prices do not

Fundamentals of Futuresand Opcions

where c, represents the value of the European call option at time / when the

American put is exercised and D, represents the value of th e dividends paid by the security to that point. This payoff would be positive, which suggests that the initial value of the portfolio would also have to be positive whether the

strictly conform to the European put-call parity condition. Specifically, the prices of American option s arc affected by the potential for early exercise even when they are currently out of the money. Our simple analysis suggests that

put optio n was exercised early or not.

exercising American options early may be advantageous, but the exact tim-

Americ an call and put options:

ing for early exercise is beyond the scope of the analysis here and genera lly requires the use of a specific option -pricing model for Ameri can options. The important concepts are that (1) it may be desirable to exercise an Ameri can call option early if an expected dividend is large and (2) it may be desirable to exercise an American put option early if it is deep enough in the money .

C0 -fli > S0 -X - D. (5.14) To derive the upper bound for the difference between American call and put options, we begin with the put-cal l parity relationship for European options as given in Equation 5.8. lf no dividends are expected robe paid before expiration, the American calloption is worth the same as the European call, C0 = , 0• We also know that the American put option is worth at least as much as the European put, P0 p., so with a potentially larger American put price we have

Put- Call Parity Boundsfor AmericanOptions Alth ough the exact put-call parity relationship does not apply to American opt ions, we can use the relationship to derive upper and lower bounds . To

Rearranging terms in the inequality for the initial portfolio value and using the fact that C, c0 gives a lower bound for the difference between th e

Co(w/odividends)-P 0 (w/o dividends),;

der ive the lower bound, cons ider the payoff for a portfolio that co ntaini.: a

European call option, cash equal to X + D, shorting the underlying security, and selling an American put. Consider first the case if the American put option is held to expiration. If the put opt ion is not exercised early, the contingency table for the portfolio gives a fixed positive payoff as follows.

s ,x

s ,x

Europeancall option Ctsh • X+D

0

sT-x

(X • D)(I • ,)'

-Security

-Sr-D(hr)T

(X • D)(l +r)' -ST-D(hr )T

-American put option Tot1Jpayoff

-(X - S,)

X(I • r)'-X

S0

-__!!___,. (l+r)

(5.15)

With expected dividends, the lower bound for the American call price is smaller, and the lower bound for the put price is larger than without divi• dends. Con sequently, the inequality without dividends will hold even with expected dividends, so the upper bound for the difference between Ameri can call and put opt ions with or without expected dividends is C0 -P 0 ,; S0 -_x __ (l+r{

(5.16)

Therefore, the upper and lower bounds for th e Americ an option put-call pa rity relationship using Equations 5.14 and 5.16 arc

0 X(l. ,)T-X

S0 -X-D

< C0 -P 0

So-~

(l + r)T

-

(5.17)

With a .fixed tor.I payoff at expiration and a positive interest rate, the initial

Note, howev er, that the relationship is not an exact equality as it is for

value of th e portfolio would have to be positive to avoid a risk.less retwn with no initi al investment:

European options. The pricing relationships for put and call option s covered so far in this chapter can be summarized as follows.

(5.12)

If the American put option is exercised early at some time t, the value of th e position would be

c, +[(X + DXl+r)

1

- S, -D , -

(X-s,)]

(5.13)

= c, +x[(1+r( - 1]+[ D(l+r)' - D,]. 02 0 13 The RestarchFoundationof CFAInstitute

73

Call Option

PutOpti on

Americanintrinsknluc

Co• max(O,So-X)

P0 = max(O,X- S0)

Europeanlowerbound

c0 > S0 -

Po > Xl(l+rl

Americanlowerbound

Co CO(cqualicyforD • O)

74

X /( I+ r}7- D

+ D-S

0

Pa~Po

C20 13 The ResearchFoundationof CFAlnsbtute

OptionContracts : PricingRelan onships

The put-call parity relationship s may be summarize d as follows.

Fundamentals of Futuresand Options

price" world. Assuming tha t the "up" p rice of the stock, call option exercise price,

Europe,nput-