Currency Options and Exchange Rate Econo 9810226195, 9789810226190

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>:. 62, our estimates of Po are biased downward. This downward bias only strengthens our results in which (3Q estimates are significantly positive and close to one. 9 Table 5.4. Tests of the expectations hypothesis (changes in long-dated volatility). Currencies n and m 2 and 1

3 and 1

6 and l a

6 and 2 a

6 and 3

12 and l a

12 and 2 a

USD-GBP Ql

a

Ctl

ft

Ctl

ft

.081*

1.583

.094*

4.615

.282*

4.702

-.044*

4.807

.320

5.017

.389

4.335

.318

5.238

.376

-.040*

2.143

-.080*

1.973

.330*

3.520

-.197*

3.830

.288

3.953

.314

3.309

.303

4.266

.312

-.617

.130*

-.293

.274*

.621

.618

.219

1.847

.177

1.696

.200

1.533

.286

1.909

.177

-.703

.226*

.044

.520*

1.111

.712

1.105

.447*

3.567

.149

3.213

.151

2.780

.273

3.540

.143

-.331

.101*

3.255

.853

5.297

.727

.706

7.191

.235

6.947

.336

6.089

.540

8.140 8.371

-.488

-.156*

-.036

.094*

1.001

.284*

1.263

.251

.351

.217

.941

.929

.074

.316*

.393 .048*

.342

1.206

.207

-.157*

-.031

.302*

1.785

-405*

.274

.163*

.236

1.910

.173

1.782

.270

2.330

.163

-6.614

.233

-2.691

-1.111

-5.427

.073

.848

6.849

.875

6.049

1.315

7.266

.603

.159*

1.896

1.075

5.171

.522

6.343

.808

9.252

.419

6.702

.403

5.923

.428

8.601

.451

Vm,n-m

~ Vo,n = Ofi + ft (

-1.160

-7.817 6.624

12 and 6

ft

Ctl

USD-CHF

USD-JPY

-.075

2.509 12 and 3 a

ft

USD-DEM

-2.062

-.783*

[Vo,n ~ Vo,m] + Cm

\n — m J Uses the approximation Vm,n-m

— Vm,n-

Regression of changes in the square of the volatility quotes of the long-maturity option on the current spread between the square of the volatility quotes for the long and short maturity options. Daily observations for the period 1 December 1989 to 23 May 1995. The first column indicates the months to expiration of the long- and short-dated options. The standard errors are reported below the parameter estimates and have been corrected for overlapping observations using Hansen (1982). * indicates rejection of the null hypothesis (c*i = 0 or ft = 1) at the 5% level.

y

O u r inability to reject the null despite this downward bias may prove interesting as a contrast with the empirical evidence on the expectations hypothesis on the term structure of interest rates, where it is found that fio's are significantly less than one, and sometimes indistinguishable from zero.

86

5.5

Jose Manuel

Campa & P. H. Kevin

Chang

Overreact ions in Foreign E x c h a n g e Options

As a further test of the expectations hypothesis, we impose the condition c*o = 0 and 0Q = 1 in Eq. (5.7) and examine the relationship between information at time t and the residual expression £?Ji e. m = (1/*) E f - J ( V i m , ( l + i ) m - Vo, m ) - (V 0(fcm - V 0 , m ) •

(5.10)

Under the null hypothesis, and the approximation 6\ = 62, this residual term is white noise. Thus, no variables known at time t should have predictive power in a regression where this expression is the dependent variable. Even if we are unable to reject that 0o = 1, evidence of predictability of the forecast error based on information known at time t would constitute evidence against the expectations hypothesis. We test this restriction by regressing this residual term on the square of the current short-term volatility quote. This specification has the advantage of being interpreted as a test for overreactions of the current long volatility to changes in the current short volatility. If overreactions are present, then when Vo,m is high* Vo./tm is too high, leading to a negative prediction error. 10 Stein (1989) performs a similar test on implied volatilities of zero-to-one month and one-to-two month options on the S&P 500 index, detecting "overreactions" in the long rate. 1 1 Stein concludes that, although overreactions exist, the mispricing impact is small for shorter maturities but could lead to much more economically significant mispricings for longer maturities. With data on one-, two-, three-, sixand twelve-month options, we have a much richer term structure on which to test for overreactions. Furthermore, since we are examining longer maturities, given volatility errors would translate into larger absolute pricing errors in the option premium. 12 Table 5.5 reports the results of regressing the residuals in Eq. (5.10) against the short rate. 1 3 For the four currencies (USD-GBP, USD-DEM, USD-JPY, USD-CHF) and nine maturity pairs examined, 15 out of 36 coefficients on the short rate are 10

Relaxing the approximation 9\ = 62 indicates that our estimated coefficients are upward-biased and should be positive rather than zero under the null hypothesis. This decreases the likelihood that overreactions will be detected. We nonetheless report our findings for comparison with Stein (1989), which is characterised by the same bias. 11 Stein (1989) performs the test in implied volatilities instead of implied variances. This misspecification does not suffice, however, to explain the overreaction effects that he detects. 12 T w o other important differences exist between our data set and Stein's. T h e options in our sample trade with fixed time-to-expiration, therefore the term structure test is more exact; the S&P options examined by Stein (1989) have a fixed expiration data, resulting in options of varying maturities. Also, since implied volatilities are quoted, directly, the expectations hypothesis becomes more "explicit". In contrast, in markets where the premium is quoted, the expectations hypothesis may be less transparent. 13 A s stated earlier, under the expectations hypothesis, the left-hand side of the equation should have zero mean and be uncorrelated with any variables known at time t. As shown above, for some of the currecy-maturity pairs in our sample, the means differ from zero because of unequal mean implied volatilities for different maturities. To distinguish this violation from the "overreactions" phenomenon, we run these regressions including a constant on the right-hand side.

Learning from the Term Structure

of Implied

Volatility in .. .

87

positive, but none of them are significantly so. Of the 21 negative coefficients, only five are significant at the 5% level. Although we find slightly more negative coeffi­ cients than one would expect under the null hypothesis, especially given the positive bias inherent in the test, this does not provide evidence of systematic overreact ions. These results contrast with the findings reported in Stein (1989) which detects overreactions in options on the S&P 500. Table 5.5. Regression of the constrained residual on the short rate. Currencies

USD-GBP

Options 2 on 1

3 on 1

6 on 1

6 on 2

6 on 3

12 on 1

12 on 2

12 on 3

12 on 6

a

a

b

USD-JPY

USD-CHF

a

b

a

b

-6.212

.041

-13.385

.075

9.522

.094

13.770

.086

-12.260

.071

-6.419

10.380

.072

13.240

.048 .092

-18.700

.079

-5.578

.044

-.428

.004

-10.879

.074

18.450

.121

10.070

.065

6.606

.063

11.540

.069

-16.333

.092

4.514

-.028

9.593

-.112

1.906

-.009

15.110

.072

2.784

.018

8.909

.092

16.370

.098

-11.663

.073

9.938

-.058

17.383*

-.160*

13.278

-.104

12.080

.071

12.590

.083

7.958

.078

28.670

.179

-11.876

.074

17.808

-.146

23.239

-.209

-6.177

.002

8.138

.052

23.450

.159

15.160

.144

20.450

.061

-22.507

.076

-8.775

.016

13.552

-.154*

31.366

-.209*

25.320

.091

17.930

.065

13.430

.070

16.740

.067

-15.726

.052

-1.748

-.019

25.980

-.201*

15.026

-.100

22.440

.094

15.560

.061

13.590

.091

15.690

.077

15.833

-.122

12.364

-.098

15.404

-.122

42.982

-.266

16.670

.103

14.650

.066

13.460

.089

28.720

.187

10.027

-.090

41.302

-.285

36.974

-.309

56.183*

-.374*

24.850

.168

23.940

.150

22.860

.195

18.490

.131

K-\ (l/K)

b

USD-DEM

"I

] T [ V i T O | ( 4 + 1 ) m - V0,m] ~ [Vo.fcm - V0,m] 2=1 J

K-l = a + 6 [V0,m]

+ Y, 1=1

* «

* and ** respectively indicate significantly different from zero at the 5% and 10% level, 2-tailed tests. Tests for overreactions of the long rate to changes in the short rate. Under the null hypothesis, changes in the square of the shorter maturity volatility quotes minus the current spread between the square of long- and short-dated volatility quotes should be white noise. This difference is regressed on the square of the current short-term volatility quote. A negative sign indicates that the longterm volatility quote overreacts to changes in the short-term volatility quote. Daily observations for the period 1 December 1989 to 23 May 1995. The first column indicates the months to expiration of the long- and short-dated options. The standard errors are reported below the parameter estimates and have been corrected for overlapping observations using Hansen (1982).

88

5.6

Jose Manuel Cam-pa & P. H. Kevin

Chang

Out-of-Sample Forecasts of Future Implied Volatility

Our study of the expectations hypothesis in implied volatility could in principle also be helpful for forecasting future implied volatility. If the expectations hypothesis holds exactly, then the "forward volatility" is the best forecast of future implied volatility. The current long-short spread could prove helpful in predicting the change of the future spot rate, even if the expectations hypothesis does not hold exactly. In this case, the estimated regression coefficients, rather than the a = 0 and ft = 1 required by the expectations hypothesis, would be used to generate out-of-sample forecasts. Alternatively, if the current spread had zero contribution in predicting future implied volatility, the best forecast would be either current implied volatility, if volatility followed a random walk, or a measure that incorporated mean-reversion in the short rate without explicitly taking into consideration the long-short spread. To evaluate the out-of-sample predictive power resulting from our empirical assessments of the expectations hypothesis' realism, we re-estimate Eq. (5.9) us­ ing only the first two-thirds of our sample for maturity pairs 2-1, 6-3, and 12-6 months. 1 4 We then use the estimated parameters to forecast the short-term implied volatility (squared) for the remaining one third of the sample. We compare the root mean squared errors from this forecast with those obtained under four alternative specifications: (1) implied volatilities follow a random walk; (2) the expectations hypothesis always holds exactly (i.e. an = 0 and (5Q = 1); (3) implied volatilities (squared) follow an auto-regressive process of order 1; and (4) implied volatilities (squared) revert all the way to the long-run mean in one (observation) period. 15 As shown in Table 5.6, the relative performance of different forecast methods varies according to the time horizon being forecasted. For the European currencies, to forecast future one-month implied volatility (squared), the random walk proves best. For the dollar-yen, simply the long-run mean results in the lowest forecast errors. In forecasting three-month volatility (squared), again the random walk gen­ erates the lowest (or nearly equal to the lowest) forecast error for the European currencies, but an AR(1) proves best for the yen. To forecast six-month volatility (squared), the random walk outperforms the imposed expectations hypothesis (the forward rate) for all four currencies. The random walk has the lowest RMSE for the pound and the mark, and the second lowest RMSE for the yen. For the Swiss franc, however, the AR(1) and long-run mean both have lower RMSE's than the random 14

For the remaining maturity pairs with k different from two, the calculation of an out-of-sample forecast would have required information not available at time t. For example, using the 3-1 month pair to forecast one-month volatility two months hence would require knowledge of the one-month volatility one month hence, and vice versa. 15 T h e parameters for the AR(1) process and the associated long-run mean were also estimated using only the first two-thirds of the data.

Learning from the Term Structure

of Implied

Volatility in .. .

89

walk. For reference, root-mean squared forecast errors for variance in the range of 30-40 for the European currencies (lower for the yen) correspond to prediction errors in implied volatility in the range of 1.30 to 2.00 percentage points in volatility rates. Table 5.6. Root-mean squared errors of out-of-sample forecasts of short-term implied variance*. Forecast Model

RMSE of 1 month using 2 months

RMSE of 3 months using 6 months

RMSE of 6 months using 12 months

USBP Random Walk

26.22

29.04

41.81

Estimated Equation (7)*

33.52

40.67

52.23

Imposed Exp. Hypothesis

35.26

39.95

49.42

AR(1)

46.50

51.63

47.50

Long-run mean

82.60

62.03

48.10

USDM Random Walk

25.36

25.06

24.78

Estimated Equation (7)*

29.01

28.46

31.75

Imposed Exp. Hypothesis

30.94

25.60

27.20

AR(1)

32.30

30.73

26.11

52.14

35.41

26.23

Long-run mean

USJY Random Walk

51.90

34.08

21.36

Estimated Equation (7)*

47.73

32.82

20.85

Imposed Exp. Hypothesis

43.69

29.90

23.57

AR(1)

42.10

25.16

25.58

Long-run mean

39.36

30.24

32.86

USSF Random Walk

27.49

27.30

28.97

Estimated Equation (7)*

32.47

32.47

38.27

Imposed Exp. Hypothesis

32.84

27.29

31.13

AR(1)

35.39

32.56

26.98

Long-run mean

57.48

37.64

26.75

* ( l / * ) S j = 1 [(ff*Mi+l)m)2 _ ( ^ m ^ ]

=ao+pQ

[(a°>km)2

- (C7°' m ) 2 )] + Ef-^ttfcn

a

Implied variance is calculated as the square of implied volatility expressed in percentage points. The numbers are the root-mean squared errors of out-of-sample forcast of the short-term implied variance using the five methods indicated in the first column. Daily observations for the period 1 December 1989 to 30 July 1993 are used to forecast daily implied variances from 1 August 1993 to 23 May 1995. The random walk is the daily implied variance of the short-run option, the AR(1) is the forecasted implied variance from the estimated AR1 process on the time series of implied variances of the short-run option; the long-run mean is the sample mean of the implied variance of the short-run option for the period 1 December 1989 to 31 August 1992.

90

Jose Manuel Cam-pa & P. H. Kevin

Chang

In September 1992, implied volatilities in all the four markets rose precipitously, reflecting the uncertainty in foreign exchange markets that accompanied the crisis in the ERM. While implied volatilities in the preceding month had risen markedly, the term structure observed in August was relatively flat, and thus failed to pre­ dict the dramatic rise in implied volatility that occurred in September. In the week preceding the devaluation of the pound, short-run implied volatilities rose sharply, resulting in a steeply downward-sloping term structure, suggesting an im­ minent decline in short-run implied volatility. In fact, one-month implied volatility remained high and above that of longer horizons for the rest of the year. This failure suggests that this episode might best be explained by alternative models that incorporate the possibility of regime switching or peso problems, which we believe better characterise this particular period. This subject is discussed in the next chapter, which is based on Campa and Chang (1996). Because of the regime shift surrounding the September 1992 episode, we re-estimate the out-of-sample forecasting performance of alternative predictors of future implied volatility (squared) excluding the period 1 September 1992 to 31 December 1992. Daily observations from 1 December 1989 to 31 August 1992 are used to estimate coefficients for the estimated expectations hypothesis Eq. (5.7) and the AR(1) process. Out-of-sample forecasting ability is based on the period from 1 January 1993 to 23 May 1995. These results are summarised in Table 5.7. Excluding the last four months of 1992, we find slightly different implications for the best forecast of implied volatility (squared). At the three-month horizon, the AR(1) process generates the lowest RMSE for all currencies but the pound. Exclud­ ing the yen, both the random walk and the estimated expectations hypothesis are superior forecasts to the imposed expectations hypothesis. At the three-month hori­ zon, the results are mixed: the AR(1) forecast is best for the mark and Swiss franc, but the estimated expectations hypothesis for the pound and the imposed expecta­ tions hypothesis for the yen. In forecasting six-month implied volatility (squared), again the AR(1) process results in lower forecast errors than the forward rate for the three European currencies. For the yen, the forward rate performed better than all alternative forecasts except the imposed estimated expectations hypothesis, whose RMSE was slightly lower. Thus, excluding the last four months of 1992, the AR(1) process overall appears to dominate forward volatility as a forecast of future implied volatility (squared), especially for one-month and three-month volatilities. 5.7

Conclusion

After identifying a term structure of volatility based on implied volatilities in foreign exchange options, we test whether today's implied volatility on both longdated and short-dated options is consistent with future implied volatility of shortdated options, under the assumption of rational expectations. Using daily over-thecounter data for the dollar against the pound, mark, yen and Swiss franc, we are unable to reject the expectations hypothesis in the majority of cases. In sharp con­ trast to the literature on the term structure of interest rates, we conclude that for

Learning from the Term Structure of Implied Volatility in ...

91

all currencies and maturity pairs, current spreads between long-run and short-run volatility do predict the right direction of future short-rate and long-rate changes, even at horizons that are as brief as one month.

Table 5.7. Root-mean squared errors of out-of-sample forecasts of short-term implied variance* excluding the period 1 September 1992 to 31 December 1992. Forecast Model

RMSE of 1 month using 2 months

RMSE of 3 months using 6 months

RMSE of 6 months using 12 months

USBP Random Walk

36.29

35.21

37.53

Estimated Equation (7)*

35.70

31.59

38.53

Imposed Exp. Hypothesis

37.01

33.72

40.91

AR(1)

40.89

37.81

37.74

Long-run mean

59.30

46.86

38.89

USDM Random Walk

42.67

45.14

33.48

Estimated Equation (7)*

41.64

35.54

27.55

Imposed Exp. Hypothesis

42.97

34.81

27.13

AR(1)

39.47

32.56

24.76

Long-run mean

48.73

33.16

24.66

USJY Random Walk

57.94

47.05

36.42

Estimated Equation (7)*

53.98

42.40

33.33

Imposed Exp. Hypothesis

53.20

40.83

33.39

AR(1)

51.96

42.36

37.48

Long-run mean

60.47

44.54

40.54

USSF Random Walk

48.83

53.15

39.01

Estimated Equation (7)*

48.55

43.65

36.23

Imposed Exp. Hypothesis

50.13

42.91

35.61

AR(1)

48.39

42.33

32.06

Long-run mean

60.34

43.69

33.45

* (1/fc) E * " ! 1 [(