A study of diffusion processes for foreign exchange rates

A Study of Diffusion Processes Exchange Rates

for Foreign

ELTOX SCOTT*

Florida Statr Cniucrsit_v, Tallahassrr, FL 32306,

C’S_-1

The nature of the stochastic process generating the time path of foreign exchange rates plays an important role in dynamic theories of international financial economics. An important consideration in this stochastic process is the relationship between currency return variances and eschange rate levels. Using five years of daily data separated into quarterly intervals, this srudv demonstrates that currencv return

variances depend on eschange rate levels and the dependency is.unstable intertemporall~. Thus, these empirical results contradict the assumption of stable log-normal distributions and the more general assumption oi constant elasticities ofvariances. For elasticity coefficients ordinarv least squares estimates are compared to maximum likelihood estimates; the maximum likelihood estimator clearly is superior. Implications of these results for models of foreign exchange rates are discussed.

Stochastic processes that generate time paths of security prices critically int-luence dynamic theories of financial economics. For example, for multiperiod consumption and investment decisions knowledge of the diffusion process underlying securities and portfolios leads to knowledge of the diffusion process of an individual’s wealth, enabling the individual to select the utility maximizing portfolio (hlerton, 1971). The form of the diffusion process of security prices also is critical in option pricing models. Black and Scholes (1973) assume a iog-normal process where the variance of security returns is time invariant. Cos (19-j), Cos and Ross (1976), and Emanuel and MacBeth (1982) assume a constant elasticity of variance process where return variances depend on price levels according to a specified elasticity coefficient .l To this point these models commonly have been applied to returns from common stocks with limited applications to other securities. This study compares the nature of returns on foreign exchange to the nature of returns that is assumed for these models. * The Peterson,

authors thank Stewart L. Brown, E. Ray Canterbery, and two anonymous referees for helpful comments.

0261-j606;8~,OJ,O~Gj-l~~~~.00

0

198- Butterworth

& Co (Publishers)

David

Ltd

R. Peterson,

Pamela

P.

406

Ili//ii.iiuu l’rmssrs

far I-on~~qttE.~rl,ar~qr Kdrrr

The nature of the stochastic process generating the time exchange rates p1ai.s an important role in dynamic international For instance, the consumption-based international asset pricing (1981) assumes both commoditv prices and eschange rates follow lto process. Hence, the observe2 process characterizing erchange consequences for testing empirically Stulz’s fundamental pricing as his propositions that:

path of foreign financial models. model of St& a non-stationarl rate changes has equation, as xvell

1. The espected excess real return of a risky asset is proportional to the covariance between the home currency rate of return on that asset and changes in the uorld real consumption rate. 2. Forward eschange rates are inversely proportional to the covariance between changes in eschange rates and changes in world real consumption rates. The process characterizing eschange rate changes also plays a critical role in currency futures and currency options pricing. For instance, the currencl- option pricing model of Biger and Hull (1983) and Garman and Kohlhage: (1983) assumes rates are described by a (stationary) log-normal process. In general, the assumption of log-normality in continuous-time international financial models is sufficient to allow the same analysis as in static mean-variance models but without the objectional assumptions of quadratic utility or normality of the distribution of absolute eschange rate changes. Also at the theoretical level, the assumption of a non-stationary Ito process describing eschange rate changes is sufficient for deriving equilibrium pricing relationships. However, such an assumption wreaks havoc for empirical tests. Identification of the state variables driving changes in means and variances proves difficult at best. Farber rl al. (lW7), Westerfield (1977), h[cFarland rt a/. (1982). and others investigate empirically the diffusion process characterizing exchange rate changes through time. Without esception, these authors report severe departures from normality, especially for the pre-1973 fixed-rate period. All of these works emplo! the research technique of Fama and Roll (1968), namely quantifying the characteristic esponent of a stable-Paretian distribution fitted to a time series of (here) exchange rates. Results indicate that distributions of eschange rates belong to the stable, infinite-variance class. For example, Westerfield tests chronological eschange rate sequences and finds the characteristic esponents to be reasonabl) stable for sums of between one and ten weekly observations; characteristic esponents rise with the length of the sequence but reach a maximum of 1.69. McFarland et al. (1982) report similar results using daily eschange rate data for both chronological and randomized sequences. This study also investigates diffusion processes characterizing (daily) eschange rate changes but employs a different technique from that of Fama and Roll. Specifically, processes characterizing foreign exchange rates are investigated b> direct estimation of the parameters of the constant elasticity of variance diffusion process (CEV process). The CEV process incorporates changing return variances while allowing exchange rate dynamics to be represented by a stochastic process with a continuous sample path. The finding that characteristic exponents of the distribution of the intertemporal sum for currency returns rise with the sum size Further, empirically obserl-ed current) violates the stable variance assumption. return distributions are not as outlier-prone as a non-normal stable law hypothesis would imply, which suggests a bounded return variance. The constant elasticity of

ALAN L.?'UCKER AND ELTON Sco-r-r

467

variance diffusion process can produce bounded but non-stationary return variances. Direct estimation of parameters for the CEi’ process can provide a basis for tests of alternative candidate processes. For example, the log-normal, square root, and absolute diffusion processes can be compared by simply investigating alternative values of elasticity of variance coefficients relating returns on foreign currency positions and eschange rate levels. This study employs the above estimation technique to investigate the nature of the diffusion process of spot prices for the eschange of US dollars for six major trading currencies: the British pound, Canadian dollar, Deutsche mark, French the nature franc, Japanese yen, and the Swiss franc.2 Further, this study investigates of the diffusion process for a portfolio consisting of equally-weighted returns on holdings of these six foreign currencies. a The parameters of the constant elasticity of variance diffusion process are estimated by two methods: 1. Direct estimation of the parameters employing ordinary 2. hIaximum likelihood estimation, which is a more efticient by Christie (1982).

least squares. technique developed

The stability of the individual and portfolio exchange rate elasticity coefficients also is investigated. Results indicate that currency return variances depend on exchange rate levels and this dependency is unstable through time. This study is developed in the following sections. Section I presents the constant elasticity of variance model of the diffusion process for foreign eschange rates. The methodology, including the estimation equations and data, is presented in Section II. Section III reports empirical results and discusses the implications of the findings. The summary and conclusions are presented in Section IV.

I. The Diffusion Process A log-normal be expressed

diffusion as

process

of the stochastic

time path of currency

returns

can

dS - = /ldf +adZ,

Cl>

s

where S=the spot foreign eschange rate; ~1=the mean return; G =the instantaneous standard deviation of return; t=time; dZ =the differential of a Wiener process. Equation
<2>

dS - = pdt +&Y’-‘dZ,

S

where 6 =a scale coefficient; p=the elasticity of the standard deviation of the exchange rate with respect to the exchange rate. Equation <2>, developed by Cos (1975) and Cox and Ross (1976), contains the log-normal process as a special case when p =l and 6=cs. Otherwise, d =6S”-‘. Hence, the constant elasticity of variance process includes the log-normal process as a possibility but atlows the standard deviation of return to vary with exchange rate levels (and consequently to be time variant).

-KS3

Dc/jirsiort

Prorrssts rbr Fortiiqt~E.~cbatzqr Kattj

The value of the parameter ~1 is needed to model the diffusion process. If p is equal to one, then, as an exchange rare \-aries, the currency risk (measured by o) does not change. Currency risk exhibiting homogeneity of degree one in the exchange rate provides an eloquent simpliticntion in currency option pricing. Specifically, p = 1 allows the pricing of currency options in terms of exercise priceexchange rate and option price-eschange rate relatives, eliminating one variable, the exchange rate.4 However, when ~1 is less (greater) than one, the variance of returns decreases (increases) a-hen eschange rates rise.5 When rates fall, ifp is less (greater) than one, the variance of return increases (decreases). h p significantly different from one necessitates a more general and complcs currency option pricing model analogous to the Cos and Ross (19-6) or Emanuel and AlacBeth (1982) models.6 i\lso, a value of fj different from one complicates the problem of asset valuation and multiperiod international investment decisions.’ The empirical analysis employed here estimates values and assesses the stability of p for each of the sis eschange rates. International investors topically hold portfolios of currencies to reduce risk through diversification (secSolnik, 19--lb: or Jacquillat and Solnik, 1978). If p # 1 for individual currencies, then randomly selected portfolios generally will eshibit elasticity coefficients different from one. .%ng and Peterson (19Wb) generate the following general espression for portfolio elasticity, p,, :$

uhere N is the number of securities in the portfolio and r. is the correlation coefficient between securities i and -1. From equation (3), portfolio elasticit) coefficients are comples combinations of individual securitv elasticity coefficients. In general, portfolio elasticity coefficients may rise or with ,\;-. This stud! of p. for a portfolio provides estimates of /I,, and an analvsis of the stability consisting of equally-weighted returns on holdings of the sis foreign currencies.

hi1

II. Empirical

Design

Direct estimation of the parameters of the constant elasticity of variance diffusion process is accomplished employing ordinary least squares (OLS) and a more efficient maximum likelihood estimation (SILE). Equation (2) can be rewritten as

Squaring both sides of equation <4), taking logarithms, letting At equal one, and adding a disturbance term transforms (A> into the regression equation 1

ln

(- 1 ‘+-/I _=

x, +r,

ln(S,)+r,,

where x,, and X, are the intercept and slope coefficients equal to 2 In(d) and 2(p - l), respectively, and t refers to a time series of observations on currencv or portfolio rates and returns.

AL.+s

L.. TUCKER

ASD

ELTOS

SCOTT

469

with OLS estimation of regression equation Two problems occur, however, (5). First, errors may be introduced in the coefficients when the differencing interval dt is not small because the transformation (5) invokes Ito’s multiplication rule that (dZ)’ =&. This rule only holds to an approximation when discrete time however, estimation errors sampling is employed. If& and ln(S,) are uncorrelated, of the slope coefficient should be small. Second, OLS provides inefficient estimates relative to maximum likelihood estimates since the error term Y, is not normally distributed. The error term is distributed as the logarithm of a Gamma variable adjusted to have a zero mean. Thus, the usual tests of significance and contidence intervals on individual estimates of coefficients do not apply. Of course the OLS estimates are unbiased and consistent. Hence, provided the differencing interval is small, OLS estimates of p, the coefficient of interest, may exhibit little difference from estimates generated employing maximum likelihood estimation.g Christie (1982) derives the relevant loglikelihood function used for maximum likelihood estimation of the diffusion coefficients. i\ssuming the mean ,LLis zero in equation (2), the loglikelihood function for a sample of size T is

(6)

L = -T/2

ln(2rt) -T/Z

ln(6’) -0

2 in(J) -&J-z

1 es -?I),

where the summations are overt= [l, T], r, =ASjS, A represents discrete intervals, and 0 =p - 1,I’) Hence, if 0 is significantlv different from zero, then the log-normal process holds. Standard Newton-Raphson iterative procedures, as discussed br Rao (1973), provide estimates of 6 and 0 by masimizing equation (6). If the returns in equation (2) are normally distributed, then hlLE parameter estimates are both asvmptotically efficient and asymptotically normally distributed. .Data for this study are obtained from the Inttrnatioual Alorwtag~ L\farkrt l-tar Books for the period 1980 through 1984. Data are the daily interbank last spot exchange rates (bid side) as quoted by the Continental Illinois Sational Bank and Trust Co. of Chicago. The six rates are expressed as US dollars per unit of foreign exchange. The total sample consists of 1260 daily rates for each of the six foreign currencies. The five-year period is divided into twenty quarterly subperiods. Thus, it is assumed that every currencv follows a constant elasticity of variance diffusion process for every quarter. i\lthough the partitioning of the subperiods is arbitrary, presumably inferences can be drawn concerning the intertemporal stability of each currency’s diffusion coefficients. l1 The value of &. ,S,, is measured as the currency return for each day. The value of/i is the average currency return over the quarter. For the portfolio, every currency’s return is equally-weighted, implying rebalancing at the end of each trading day. These daily changes establish an index that represents S,. The indes value is set at 1.0 at the beginning of the first day, and then incremented each dav. bv. the portfolio return over the dav. The initial value of the index selected will not effect the estimated elasticitv coefficient, but 6 is determined arbitrarily. However, since p(o) is the coefricient of interest, the arbitrary value of S, IS immaterial. III.

Empirical

Results

and Implications

Twenty elasticity coefticients (e), and twenty scale coefticients (a), one of each for every quarter, are estimated for every foreign currency using both OLS and blLE procedures. Further, these procedures are employed to estimate coefficients quarterly for the equally-weighted portfolio, and over all 1260 observations for

1-o

Dc/jiuion

Procr~~rs/Or I’orr@yr~Ex?hqr

R‘1tr.c

each of the six currencies and the portfolio. The DFP algorithm in the program GQOPT, from Princeton University, is used to maximize equation (6). The initial coefficient estimates used to commence the iterative process are 6, 0 = -0.5. The convergence criterion employed is CO\‘(ln S, rfS -“) = 0 k 0.0 )(101 .I2 TabIe 1 reports the elasticity coefficient (0) and its standard error for every currency and the portfolio using the AILE procedure. The first twenty rows correspond to the quarters 19801 through 1984IV, while the Iast row corresponds to all 1260 observations over the entire five-year period. Only the AlLE elasticity coefficients are reported. The blLE scale coefficients are, in every case, equal to the square root Of(I;‘T)~ rjS-“, where 0 is the corresponding hILE elasticity coefficient for the period.‘:’ The OLS elasticity coefficients are similar to their corresponding masimum likelihood estimates, as eshibited in the plots provided in Figure 1 .‘-I h \Yilcoson signed rank test we performed rejects the null of no difference in OLS and I\ILE elasticity coefficients (at 5 per cent) for just the Deutsche mark and Japanese yen; 0 and p -1 are insignificantly different for the remaining four currencies and the equally-weighted portfolio over the twenty quarters. As anticipated, the standard errors of the estimates are smaller for the bILE procedure; across all elasticitv estimates the standard error of 0 under the bILE procedure is approximately one-third of the standard error employing OLS. ‘-1 signed rank test rejects the null of no difference in OLS and MLE elasticity coefficient standard errors at the 0.001 level both for every currency individually and for the portfolio. Hence, the masimum likelihood estimator clearly is more efticient. The last row of Table 1 shows that for five of the sis currencies, as well as the equally-weighted portfolio, the diffusion process over the entire five-year period is not log-normal. hll of the elasticity coefficients, with the exception of 0 for the Japanese yen, are statistically different from zero at the 1 per cent lel-el. Escept for log-normality, the Japanese ,yen, which eshibits each of the currencies and the portfolio eshibit a statistically significant and direct association betu-een return variance and value levels.lj The small standard errors suggest that this result is robust. For sterling, for instance, the null of log-normality at traditional significance le\-els would still be rejected with a much greater standard error; approsimately a sis-fold increase in the pound’s standard error of 0 would be required to not reject the null. Further, it is interesting to note that under OLS estimation, the null of log-normality cannot be rejected (at even 10 per cent) for the British pound, the Swiss franc, and the portfolio, as well as the Japanese yen.16 Although log-normality can be rejected for all but the Japanese yen over the entire five-vear period, the first twenty rows of Table 1 show that all currencies and the equally-weighted portfolio exhibit periods of log-normality as well as periods when return variance is negativelv associated with value levels. For instance, for the Deutsche mark there are eleven quarters when 0 is insignificantly different from zero (at 5 per cent), four quarters when 0 is significantly less than zero, and five quarters when 8 is significantly greater than zero. This result suggests that currency elasticities may be unstable intertemporally. The panels in Figure 1 graphically indicate this instability. Intertemporal instability of elasticity coefficients implies that a constant elasticity of variance diffusion process may not adequately model eschange rate behavior over estended periods of time. Any such intertemporal instability may, at first, suggest the inability to employ a constant elasticity of variance model for, say, the accurate pricing of currency options. Even if exchange rate elasticitv coefficients

1983111 19831V 19841 198411 1984111 19841V I9801 19841V

I9801 10801I 19801I I 19801 v 19811 198111 19811II 19811V 19821 198211 1982111 19821V 19831 198311

Period

1.284

- 14.714 1.861

7.082 -2.182 30.038 - 6.206 0.747 - 1.498 -5.859 -3.170 4.059 -2.392

I.022

-6.477 -7.311 -4.370

-2.055 -0.873 - 2.989 -3.102

0

British pound

TAiu.ri

4.562 4.533 3.467 3.685 0. IO5

4.304 4.632 3.466 4.902 7.272 3.604 3.315 4.238 5.966 2.346

5.623 2.522 1.969

I I m.3

4.222 3.457

0

-13.301 -29.763 -15.014 2.580

45.574 - 56 .‘J97 -52.790

6.498 -31.570 -25.854 - 12.700 -28.974 -56.992 -21.786

16.984 0.578

5.428 6.313 4.307 7.674 7.759 34.134 28.510 59.628 26.239 5.564 9.282 13.649

7.434 6.511 33.158 8.015 15.616 19.293 8.423

s.e. (0)

Canadian dollar

coefficients

- 12.578 -3.368 -3.120 -6.242 12.175 54.725 -13,264

lilasticity

s.e. (0)

I.

their

s.e. (0)

3.972 -0.307 - 15.459 5.394 1.230

- 10.58’) -2.2’)7 - 3.762 -4.217 -.1.126 1.450 5.543 1.614 -0.811 16.630 7.409 - 11.393 - 13.448 10.569 8.598

3.362 2.334 0.138

6.526 5.553 3.486 7.125 1.755 6.599 7.349 1.306 4.034 2.845 3.122

3.550 6.740 3.430 2.661 2.180 2.260

0

French franc

errc~rs

4.575 1.132

3.511 -0.932 -14.412

-1.514 1.805 5.230 0.114 -7.725 IO.806 9.147 -3.810 -13.267 - 8.299 6.391

-11.017 - 15.470 -0.514 - 7.458 -3.536

slancl;‘rd

11.443 2.496

0

Deutsche mark

(0) and

1.373 3.374 0.430 4.731 0.08 I

2.827 4.201 x.229 2.858 3.086 1.139 2.334 9.150 2.699 1.878 0.718 4.238 6.835 0.678 4.231 3.930

s.e. (0)

over

the

0.431 0.841 3.601 3.614 0.444 0.154 0.754 0.706 0.271 0.279 0.304 1.419 0.308 0.192 0.256 0.310

-2.192 -2.103 2.6% (I.471 -2.338 -3.151 2.818 0.020

2.293 O.‘J65 0.2x3 0.391 0.296

-2.755 2.311 I.525 -0.074 -2.909 3. I04 0.311 - 1.657

-0.067 --3.l’J5 3.431 2.134 - 3.893

0

ye’1 s.e. (0)

quarters

Japanese

twenty Swiss franc

-2.004 9.908 7.077 -1.371 - 14.478 1 1.042 1.342

0.858 -11.211) 1.252 -2.031 15.576 2.807 - 0.994 -8.592

2.000 2.567 3.225 2.491 5.989 3.208 4.171 6.027 6.655 5.284 1.602 3.527 3.981 4.86 I 0.183

3.6’JO 3.020 3.410

I .8’J8 3.575 5.YJ4

s.e. (0)

l!J841\‘.

-5.773 -6.921 1.312 -8.395 - 3.488 -1.694

0

10801

5.183 3.474 2,‘J.N) 3.488 J.‘J36 5.073 3.13’) 7.350 S.‘J70 0.010 8.6’J5 8.503 5.791 4.904 4.509 5.725 6.086 O.I?X

- I .824 -8.088 9.282 -4.755 IO.220 5.842 -5.(X4 -20.352 - I I. I82 11.118 7.175 -0.64 I -220.296 6.712 1.365

I O.-I‘Kl

-2.680 -2.701

I

3.101 3.431

S.C. (II)

-4.885

-0.x7

--‘).SY)

--x.IHI(,

0

liqually-wcightcd portfolio

(‘1 British pound Canadian dollar Deutsche mark French franc J,Rp;re;r;,e,,i;n U’ ‘ Equally-Lveighted

‘I’..“._J

-0.1 12 0.107 0.200 0.172 -0.122

o.lj-~r.ll~j -lJ..T93 -0.363 -lJ.o-3 -0.216 -0.716

0.1)77 0.1w

portfolio

i-<)

j 1

i’2

‘;l..ll. _ /

G

(-I) I’,!.I):

.(I; _;

ll.~M2 -11.2Oj --~I.203 -0.33-l -0.118 -0.j26 -0.526

(3 RL

_,

-0.16-I 0.120 ll.133 l,.33() I~.591 CI.O-j ~1.0-5

(1.21 1 0.112 0.6( 16 0.312 0.121 0.244 ~~24-1

,Yotus: Columns (I) through (4) are the correlntion cocit?cients betuccn elastlcirv cokticicnt cstim.ltcs and their four lagged estimates. Column (3) is the coefficient ofdetcrmination ixtuwn ckticity cociiicient rstimarcs .lnd xn equ.~llyaeighted index oi estitxtes.

tend

to be highly

so that shorter

volatile

a constant horizons.

over

time,

however,

elasticity the four

coefficient estimates and their lags i= 1, 2, 3 and 1. Columns

through

~,l~o,_,,respectively,

portfolio.

Oldfield

would

-0.175,

be consistent

correlation and

statistically

argue

significant

and the equally-weighted

if a securitv’s

the presence

coefticients (15) values. for

respectively. at the

price

tend

of jump

to be small

cent

of

the

level.

28

is described

intertemporally, Thus, a finding

and split

of

However,

evenlv

between

and the portfolio, the are (0.1~~~3, -0.253,

correlation

Hence,

historical values of 0 may be used to accurately predict To evaluate the extent of covariabilitv in the elasticity equally-weighted every quarter.

behavior

components.

Across all currencies the four lag structures

None

5 per

patterns, rates over between

lagged estimates, I’.,:. ,,,_,, are calculated for (1) through (4) of Table 2 present r:,,,,,,

that

with

coefficients

0.208,

repeating

and if successive jumps are correlated will eshibit correlation through rime.

the 28 reported correlation positive (13) and negative average

may follow

mav adcquatelv model correlation coefficients

for each of the sis currencies

ct al. (1977

solely by a jump process then elasticity coefficients correlation

they

elasticitv of variance process To investigate this possibility,

it does

coefficients not

are

appear

that

future 0 values. coefticients. an index

of

elasticity coefficients of the sis currencies was constructed for Each currency elasticity coefficient and the portfolio elasticit)

coefficient is then regressed against this indes over the twenty quarters. coefficient of determination represents the proportion of the variability attributable to the variability in average 0s. This indicates the estent to which exists a ‘market effect’ in the fluctuation of currencv elasticity coefricients

The in 0 there over

time. The coefficient of determination (R’) is reported for every currency and the portfolio in column (5) of Table 2. The average R’ is 0.307. ;\ll of the slope coefficients are positive.‘; These covariance measures have implications for empirical measurement of eschange risk and international nominal interest rate differentials. For instance, Solnik’s (1974a) model of eschange risk shows that riskless nominal interest rate differentials between nations are equal to espected changes in eschange rates plus a term dependent relatively large coefticients of determination contemporaneous

correlation

between

currency

on eschange in Table elasticity

rate corariances. 2 retlect the coefficients.

The large

Table

3

473

,IL.ISL. TC.CKEK ASD ELTOS SCOTT

40-

a

Pound Sterhng

b

Conodtan

e Japanese

4.

I

Dollar

Yen

/

I

I

,

I

1

I

I

,

,

/

,

,

,

,

,

,

I4

I5

I8

Franc f SWISS

IOO-

-20 x .Z .s 5 w

4. C

Deutsche

t , , 4. 9 Equally

Mark

1 -weIghted

portfolm

20-

-401

0

,

,

/

2

4

6 20

,

,

8

IO

Quarters

12 1980

-

OLS

regression

- - - -

Log

likelihood

1

I - 1984 IV

funcbon

-2o-3o-

-40 0

I

I

2

4

I

I

I

I

I

I

I

6 8 IO 12 14 16 18 20 Quarters 1980 I-1984 IV FIGI.KE 1. Elasticity

presents the contemporaneous the equally-weighted portfolio average correlation coefficient

coefficients.

correlation coefficients between each currency and elasticity coefficients over all twenty quarters. The is 0.416. As anticipated, currencies of the European

hlonetary System (the Deutsche mark and French franc) and highly positively correlated. Interestingly, the Canadian dollar coefficient of The final

- 0.165. esamination

of

I!? is

a

correlation

analysis

the Swiss franc are exhibits an average of

8

with

the

contemporaneous equally-weighted

moments portfolio.

moments may adequately If the constant elasticity omits higher moments, However, correlation portfolio skewness, None level.

the average and kurtosis

of the return Jarrow and approximate of variance

more comples security price distributions. diffusion process is misspecit‘ied because

then the calculated value of 0 may be statistically coefticients are small. Across all sis currencies correlation are 0.068,

coefficients of 0 uith -ij.033, -0.024, and

of the correlation coefficients Thus, there is insignificant

are statistically linear association

coefficients

and

the

moments

of

correlations

with

the

third

fourth

absolute value) which recognizes The

empirical

evidence

presented

the

return moments

indicate moments. thus

these results of variances

contradict since there

the is

Relativelv

reassuring

inappropriateness

far suggests

1980 through 1981. of stable log-normal

at rhe 3 per cent currcncv elasticity

distribution. are

that

since

smail large

of equation OS are.

on

(in <2),

average,

portfolio and ail currencies of the yen there appears a and value lel-els over the

This result distributions.

more general no evidence

it

biased. ‘and the

the mean. x.-ariance, -0.055, respectively.

significant between

than zero for the equallv-weighted yen. That is, with the esception between currency return variance

entire five-year period modeling assumption perspective, elasticities

and

correlations would only the first two

numerically greater except the Japanese positive association

distribution for every currencv and the Rudd (1982) find that the krst four

contradicts \‘ieu-ed

rhe popular in a broader

assumption of constant that currencv elasticit

coefficients are stable intertemporally. Currency OS are uncorrelatcd x;irh past &. Further evidence indicates that 0s ofdiffcrent currencies move together over time, with approsimatelv 30 per cent ofthe variance ofthe changes in (1 being accounted for by changes in an index of OS. Elasticitv coefficients correlation with the first four moments df the return appears to be no association between elimination

eshibit insigniricant linear distribution. Finally, there of unsvstemacic risk and

susceptibilit)of portfolio variance changes to value level changes. For the nineteen the equally-weighted portfolio 8 and G’ changes between the twenty quarters, change in opposite directions nine times. The correlation coefficient between these measures is just 0.051. Further study of diffusion processes of exchange rates is needed for better modeling of volatility changes in currencies and, hence, better valuation of international financial assets. The frequent \.-iolations of log-normality reported in Table

1 make

models

based

on stationary

Ito

processes

suspect.

Severrhrless,

as

ALAN L.

TUCKER

AND

ELTON

SCCKI-

475

noted by Adler and Dumas (1983), the conditional distribution of instantaneous (not discrete sample) rates of return could still be normal, in which case, models based on non-stationary processes, such as that of Stulz (1981), would apply. State variables which are required for these models, however, makes their practical application difficult at best. This study demonstrates that the inclusion of the eschange rate level as a state variable is useful in modeling the process generating currency returns. The frequent violations of log-normality reported in Table 1 also raise questions about the currency option pricing model of Biger and Hull (1783) and Garman and Kohlhagen (1983). Shastri and Tandon (1986) and Tucker (1985) report mispricing when applying this model to value call currency options traded at the Philadelphia Exchange. This mispricing ma)- be attributable, at least partially, to violations of the assumption of currency return variance stationarit)-.‘*’

IV. Summary

and

Conclusions

This study analyzes diffusion processes of foreign exchange rates. Specificall!-, daily eschange rate data for the twenty quarters 19801 through 198JI\’ are used to estimate elasticity coefficients relating return variance and value levels for sis currencies and a portfolio of these six currencies. Coefficients are estimated using ordinary least squares and maximum likelihood estimation procedures. Both procedures yield similar coefticients; as anticipated, the elasticity coefficients generated emploving a masimum likelihood function erhibit substakiailv smaller standard errors. Over all twenty quarters, the overall portfolio and five of the sis currencies individuallv exhibit a significantly positive association between return variance and value le\:els when the more efficient AILE procedure is used. For all currencies escept the Japanese yen, the null of log-normality is rejected. Elasticit! coefficients also are unstable intertemporally. These empiiical results, therefore, contradict the assumption of stable log-normal distributions and the more general assumption of constant elasticities of variances. Results reported here suggest that rates ma]; be normally distributed with changing variance. ,A process in which the variance of returns changes randoml) through time appears to characterize eschange rate changes better than a stationary stable-paretian process. Notes I. ‘Ample empirical evidence exists concerning the type oi process ch&lmcterizing indlviduzl common stocks. Beckers (19S~l), MacBcrh and 5Ierv1lle (198lI), Emanuel and SIacBerh jl982), and Ang and Peterson (1984a) find that the log-normal process does not describe common stock price behavior as well as the constant elasticity oi variance process. In general, these authors report an inverse association bet\veen return variance and srock price levels. Christie ~1982) attributes the negative elasticity oi variance with respect to equity value to financial Icveragtl. Emanuel and AlacBerh, and ,ing and Peterson also find that the constant elasticiry oi variance process may be inadequate. They report that equity security elasticitv coefficients are unstable intertemporally. .-Ing and Peterson (1984b) report similar results for the elasricit!; coetiicients oi equity portfolios. Further, .\ng and Peterson tind frequent intertemporallv negatlre correlations betueen portfolio elasticity coefficients and portfolio variances, implying investors ml? trade unsystematic risk for ‘elasticity risk’ as they vary portfolio size. 2. These six foreign exchange rates are chosen for the assumed domestic KS investor for txvo

reasons: 1. 2.

The six foreign The Philadelphia

nations are leading industrial US trade partners. and Chicago Board Option exchanges both trade options

on these six rates.

Dirjilsiort Procrssrrfbr

476

Forr&

E.xxharxr

Rofrs

3. .\n equally-Lveightcd

return Index is chosen as a representative Index of foreign rschJnge rates. Currency indexes .tbound. The form ofthe diffusion process of such indexes has Implicxlons for the pricing of currency Index options and currency index futures. For example, the Philadelphia Exchange recently began options trading on the European Currency Lnlt. The Seu York Cotton Exchange recentlv began futures trading on an index nearly identical to the Federal Reserve Bo.~rd’s trade-Lveighted dollar index. 4. See Blger and Hull (1983) and Garman and liohlhagen (1983) for an adaptation of the Black and Scholes (19’3) model to price foreign currency options. 5. A rationale for why p may be less than one for stocks is provided by Black (19-6). who notes that if the standard deviation ot a firm is a function of its leverage ratio, then as securit!- prices increase (decrease), the leverage ratio decreases (increases), decreasing (increasing) the risk ofthe stock. At the theoretical level there are several reasons why /) mav be different from one for a currency. For instance, if the standard deviation of a nation is a function of its trade deficit, then as the nation’s currency appreciates (depreciates), the deficit decreases (increases), decreasing (increasing) the risk of the currencv and thus implying a p of less than one. ‘Asset’ models of exchange rate

6.

7. 8. 9. IO.

Il. 12.

13. 14. 15.

16.

17.

determination often model rates as a function of trade deficits (see, for esampie. Hooper and Alorton, 1982). The Emanuel and 1lacBeth (1982) model is applicable to underlying securities that exhibit continuous proportional dividends and when p is greater than one. Since foreign exchange exhibits continuous proportional interest where the proportion factor is the foreign riskless rate of interest (see Biger and Hull, 1983). the Emanuel and AlarBeth model is easily adaptable to value currency options. The Cox and Ross (19-G) model assumes p =0.5 (the square root model) and is applicable to underlying securities that exhibit continuous dividends of the form /1(X. I) = i.r +r. where b is the dividend, i is the proportion factor, and r is a constant. The Coz and Ross model also is readily ndnptablc to value currency options by setting L‘equal to zero and i equal to the foreign riskless rate of interest. For example, ifp is not equal to one, thecurrency beta \vill likely change with exchange rates since the covariance of the currency with the foreign exchange market will likely change. See .\ng and Peterson (19841~) equation (5). Christie (1982) reports that common stock elasticity estimates from least squares procedures arc very similar to those emanating from maximum likelihood estimation. See’the appendix of Christie (1982) for the derivation of equation (6). The resrrlction that the mean /I is zero is of no practical importance in estimation of the other parameters as, empirically, the mean only affects the variance in the fourth significant digit (see Ohlson and Penman, 1985). Quarterly subperiods also coincide with the initial maturity ofthe shortest-lived currency options traded at the Chicago Board Option and Philadelphia exchanges. In general, convergence was quite slow. For example, the mean number of iterations over the changing the initial coefficient t\renty quarters for the British pound was 34.85. However, estimates and convergence criterion made only trivial differences to the number of iterations. See Christie (1982). equation (‘43). Of course, every p is adjusted by subtracting one. i\n interesting hypothesis is that the Japanese yen exhibits log-normality because ofexchange rate smoothing associated \vlth frequent Japanese central bank intervention in the foreign eschange market. This tinding may have implications for the extant empirical literature concermng the diffusion process of common stock. hng and Peterson (198+&a, 198qb) provide the only comprehensive studies of the properties ofthe elasticity coefficient of equity. They use monthly darn over 53 Tears for up to 1235 firms. ;\lthough they find a generally inverse association between return variance and stock price levels, they cannot reject the log-normal process as representatix-e of stock price behavior. Holvever, due to the size of their sample, Ang and Peterson employ OLS, rather than more expensive ,\ILE, procedures. If the magnitude of the differences in standard errors under the two procedures is similar for stock as reported here for currency, then perhaps more exact stntements can be made concerning stock price behavior using maximum likelihood estimation. Hence, a reexamination of the equity diffusion process may prove fruitful. These results are qualitatively consistent with the positive covariability in stock elasticity coefficients reported by Emanuel and 51acBeth (1982) and Ang and Peterson (1981a). Quantitatively, however, the variability in individual currency 0s accounted for by an index of currency OS is approximately three times as great at the 10 per cent of variability m individual

,\L.w

L. TLXCER

.CD

ELTOS

SCOTT

477

stock Us accounted for by an index of stock t)s, as reported b!- Emanuel 3nd Macl3rth. .md .ing and Peterson. 18. Distributions that violate the log-normality assumption do not necessarily indicate that rhe Black and Scholes model should be discarded, Ifa model predicts reasonably, violations ofass.;mptions are relatively unimportant. Cnfortunately, empirical tests ofthe Black and Scholes model applied to currency options show systematic, significant, mispricing (see Shasrri and Tandon. 1986; or Tucker, 1985). In further research on option pricing models, \ve have used estimates of implicit standard deviations of returns -based on the Black and Scholes model and actual currency option prices as predictors of future standard deviations of currency returns. The results show tk implicit standard deviations are clearly superior to historical standard deviations; thus the Bl.tck and Scholes model, while imperfect as a pricing model, can provide useful informar:on about distributions of currency returns. In another part of this research we are developing methods for estimating implicit elasticity coefficients using the CE\’ model and actual currencr Aces. ‘-1 . forthcoming paper reporting these results will be axailable from the authors.

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