International Journal of Forecasting 3 (1987) 159-170

North-Holland

159

FORECASTING THE VOLATILITY OF CURRENCY EXCHANGE RATES *

Stephen J. TAYLOR

Unioersity of Lmzcaster, Lancaster, LA1 4YX, UK

Abstract: Currency volatility is defined to be the standard deviation of day-to-day changes in the logarithm of the exchange rate. After a discussion of statistical models for exchange rates, the paper describes methods for choosing and assessing volatility forecasts using open, high, low and close prices. Results for DM/$ futures prices at the IMM in Chicago from 1977 to 1983 show high and low prices are valuable when seeking accurate volatility forecasts. The best forecasts are a weighted average of present and past high, low and close prices, with adjustments for weekend and holiday effects. The forecasts can be used to value currency options.

Keywords: Forecasting, Exchange rates, Volatility, Futures prices, Statistical models, Options.

1. Introduction

Finding a better forecast of a future exchange rate than the relevant forward rate or futures price is extremely difficult. This paper makes no attempt to forecast exchange rates. Instead we consider forecasts of volatility, i.e., forecasts of the rate of price changes but not of their direction. Volatility is the name given by option traders to the standard deviation of certain price changes. If we suppose exchange rates are recorded once per day, with Z, the price on day t, and if we also suppose that logarithmic price changes, X, = log( Z/Z,_ i), called returns, are independently and identically distributed (i.i.d.) then the volatility u is defined to be the standard deviation of X,. More complicated models are needed for financial prices since returns are not i.i.d. Suitable models are presented in section 2 with volatility defined as a conditional standard deviation.

This paper compares various volatility forecasts calculated from Chicago futures prices for the DM/$ rate. We consider daily open, high, low and close prices from 1977 to 1983 inclusive. In a recent book [Taylor (1986)] the author has investigated forecasts based upon closing prices alone. This paper seeks improved forecasts, primarily by using high and low prices since these prices are theoretically very helpful when estimating volatilities [Parkinson (1980), Garman and Klass (1980)]. Section 2 summarises some realistic price and volatility models, then section 3 presents theoretical forecasting results. The data is described in section 4. It is subdivided into two sets: prices from 1977 to 1981 are used to estimate parameters in section 5, then prices for 1982 and 1983 are used to assess various forecasts in section 6.

* This is an abbreviated and revised version of a paper presented at the Sixth International Symposium on Forecasting, Paris, June 1986. The longer paper is available upon request. Helpful comments by the editors and referees are gratefully

acknowledged.

160 S.J. Taylor / Forecasting the uolutility of currency exchange rates

Volatility forecasts have important practical applications for option traders. The purchaser of a currency option owns the right, but not an obligation, to make a currency transaction at a later date at a previously agreed rate. Most traders suppose an option’s price depends on the spot price, the exercise price, the time until the expiration date, domestic and foreign interest rates and the volatility a; they implicitly assume returns are i.i.d. Once u is specified, a fair option price can be calculated [Garman and Kohlhagen (1983) Gemmill (1986)]. Any trader possessing a superior forecast of u would expect to make a riskless profit trading mispriced options using some arbitrage strategy. This possibility provides an incentive to seek better volatility forecasts.

2. Modelling prices, returns and volatilities

2.1. Two assumptions

Recall the notation 2, for the price on trading day t, which will be supposed to be the closing price at a particular exchange, and the notation X, = log( Z/Z,_ i) for the return from day t - 1 to day t. Theoretically, expected returns E[X,] are zero for futures [Black (1976)] but depend on domestic and foreign interest rates for spot currency. We will only consider currency futures and will assume expected returns are zero.

International markets process information quickly and the well-known efficient market hypothesis states that past prices Zrpi, i > 0, are of no economic value when forecasting future prices Z,+,,

j > 0; only Z, is important. It will be assumed that prices follow a random walk so returns are uncorrelated, i.e., the correlation between X, and XI+, is zero for all integers t and all positive lags 7, as any small autocorrelation can be ignored when forecasting volatilities.

2.2. Discrete time models

Many researchers have observed that the standard deviation of returns appears to change with time. A suitable general model for futures returns is given by

x, = q-J> (1)

with V, a positive variable representing volatility and U, a normal variable having zero mean and unit variance. A random walk is obtained by assuming that the U, are independently and identically distributed. Also, I assume the processes {V,} and {U,} are stochastically independent.

In eq. (1) X,, V, and U, are random variables. By the end of day t, numerous traders will be responsible for particular outcomes x,, u, and u,. The actual volatility u, is a conditional standard deviation for X,. Given u,, the observed return x, is an observation from the normal distribution

N(O, $). A special case of eq. (1) is the model implicitly assumed by many option traders, defined by

supposing V, = 0, for all t. There is ample evidence that this special model is inadequate even within the lifetime of a futures contract [e.g., Taylor (1986, pp. 52, 106)]. More detailed models, consistent with the general model, eq. (1) have often been investigated [e.g., Clark (1973) Hsu (1977) Ali and Giaccotto (1982)]. In particular, Tauchen and Pitts (1983) have described an economic model for the reaction of individual traders to separate items of information and then V, is a function of the number of relevant information items during day t. Another approach has been developed in Engle (1982) and Engle and Bollerslev (1986). They have suggested ARCH models for which V, is a deterministic function of past returns X,_,, i > 0. It would then be possible to perfectly forecast v+i

S.J. Taylor / Forecasting the oolutility of currency exchange rates 161

using present and past returns although this is empirically disputable [Taylor (1986, ch. 4)]. Forecasts derived from ARCH models are similar to those presented here.

2.3. Overnight and open-market models

Prices will change overnight from the previous day’s close Z,_i to the price at which the market opens on day t, denoted Zp. Define Xi<, the overnight return, as log(Zp/Z,_,) and Xzl, the open-market return, as log(Z,/Z,‘). Then we assume

x,, = T/Q, i= 1,2, (2)

with U,, - N(O, e), u,, - N(0, 1 - e) and U,, independent of U,,. Clearly U, = XJ,, summing over i = I, 2. The term E represents the proportion of daily variance attributable to the hours when the

market is closed.

2.4. Continuous time models

Following common practise it will be assumed that prices change continually when the market is open and then follow a Brownian process. Let T be a real number representing trading time on a continuous scale. To model prices Z(T) throughout the day let

log[Z(T)] -log[Z:] = V,[B(T)-B(t-l)], t-l<T<t,

with {B(T), T real)} a non-standardised Brownian process for which B(T,) - B(T,) is normal with mean zero and variance (1 - E)(T* - T1) for any T2 > T1. Then U,, = B(t) - B(t - 1). The notation Z(T) is used for non-integer values of T. Daily high and low prices H, and L, are respectively defined by the maximum and minimum of { Zp, Z,, Z(T), t - 1 < T < t }.

2.5. Seasonal effects

Some returns are calculated from prices separated by more than 24 hours as markets close for weekends and holidays. Consequently the volatility V, may be seasonal. It will be supposed that a non-seasonal volatility V,* can be defined by

v,*=< when X, is a 24 hour return, (4)

= T/c/su otherwise,

for some constant s, > 1. The proportion of a return’s variance attributable to the hours when the market is closed is also seasonal. This proportion is now denoted er to emphasise its time-depen-

dence.

3. Forecasting volatility: theoretical results

3. I. In traducing related variables

It is impossible to calculate the exact volatility of prices on a particular day, then or later, because the volatility of prices changes frequently. Thus we cannot use current and past volatilities to forecast

162 S.J. Taylor / Forecasting the volatzhty of currency exchange rates

future volatilities u~+~, h > 0, because the series {u,} is not observable. This section argues that forecasts of u~+~ should be constructed from forecasts of related, observable variables. Two examples of related variables are absolute returns

M,=pg= v,pJl= v,lu,,+B(t)-B(t-l)l, (5)

and the daily range of log(price), i.e., log(high/low), defined as

Q = log( H, > - log( Z,)

=~,{max[B(T), t-l<T<t] -min[B(T), t-l<T<l]}. (6)

3.2. Relationships between the accuracy of forecasts

Suppose we try to forecast a random variable Al+h, which is related to the volatility Vt+h by

A t+h=Y+,,f(J,+h), Jt+,,= {$+/,> B(T), t+h-=T<t+h}.

Eqs. (5) and (6) illustrate two possible functions f. The accuracy of forecasts at+,, and q+h are assessed by their mean square errors

Let at+h be defined using information Z,, one possible example being the current and past values

{A,-,, i > 0). It will be assumed that any random variable constructed from Z, and V,,, is independent of any random variable constructed from .Zt+h. This is consistent with a perfectly efficient futures market. Then the best forecast of f( Jt+,,) using I, is simply its mean value, say P,,~+ h abbreviated* to pFLf.

Now suppose At+h and q+, are forecasts linked by

(9)

Then it can be shown, quickly, [cf. Taylor (1986, p. 99)] that

var(A,+,) - MSE(a,+,> =P? {var(Y+,) - MSE(?+,)}. (IO)

Consequently, if a,, h is the optimal forecast of Ar+h using Z,, then the optimal forecast of v+h using the same information is defined by eq. (9).

This result suggests the following strategy for forecasting volatility: forAsuitable information sets {I,} and an appropriate related series {A,} seek optimal forecasts { At+h} and hence optimal forecasts { c+, } using eq. (9). It is necessary to calculate CL,, which, in practise, requires assumptions that variables like Z-J,, and B(T,) - B(T,) have normal distributions.

3.3. Results for some stationary models

Now consider some results for special stationary models which will be helpful when assessing empirical estimates of autocorrelations and mean square errors. This is only possible for non-sea-

S.J. Taylor / Forecasting the volatility of currency exchange rates 163

sonal processes and involves technical complications which some readers may prefer to skip. Let V,* be the non-seasonal volatility, eq. (4), let f, * be a non-seasonal adaptation of f, =f(J,) and let A; = V,%* be the non-seasonal related variable. For example, f,* is simply f, when the related variable is the absolute return M, whilst f,* =f,/{m will b e used when considering the daily range 0,. Now suppose the processes { V,* } and { A* } are stationary. In an effort to keep the notation under control the stars are now suppressed for this subsection. Thus throughout the remainder of section 3.3 y refers to the non-seasonal volatility and likewise for A, and f,.

It can be shown that the autocorrelations p7,A = cor(A,, A,,,) are a constant times the autocorre- lations P,,~ = cor( y:,T/t+7):

P T,A = 01)

with Gf= Wi21/Nh1* 2 1. When A, is the absolute return M,, E[h2] = 1, E[f,]= /m and Gr= 1.57, whilst if A, is the range D, then E[f,*]=4 log,2, E[f,] = 2,/m and $r= 1.09 [Parkinson (1980, p. 62)]. Thus ranges would then be more highly autocorrelated than absolute returns.

The simplest appropriate stationary model for {V,} is given by supposing {log( V,)} is Gaussian with autocorrelations S#J~ and variance denoted by p* [Taylor (1986, chs. 3 and 4)]. Then {V,} has autocorrelations similar to +’ for low fi and

P T,A = V{exp( P’) - I}/{ 4, exp(P*) - 1).

Replacing = by = in eq. (12), the autocorrelation can be described by

(12)

P 7,A = KV, (13)

for a constant K depending on p and 4, with 0 -C K < 1. An ARMA (1, 1) process has the same autocorrelations as in eq. (13) for certain pacameter values. Thus it can be shown [cf. Taylor (1986, pp. 21-22, lOl-102)] that the best forecast A,,, linear in the variables A,, Ar_l, A,_2,. . . , is given

by

A,+, =pA + (G-8) f Bi(A,-i-pA), (14) i=O

with pA = E[A,] and f? the solution of the quadratic equation

82 - de + 1 = 0, d= {1+~*(1-2K)}/{~(1-K)},

having B < 1. Further algebra gives the following relative forecast q+, =a,+,/,,:

RMSE(i,+,) = MSE(&+l>/var(4+,) = 1 _:,:: e2,

RMSE(e+l) = MSE(c+,)/var(V,+,) = s.

(15)

MSE values for A”,+1 and the linked

06)

(17)

164 S.J. Tuylor / Forecasting the uolatrlity of currency exchange rutes

4. Data considered

We consider the prices of deutschemark futures at the International Monetary Market in Chicago from 3 January 1977 to 30 December 1983. With two exceptions, each March, June, September and December futures contract provides prices for the three months preceding its delivery month.

Thus, for example, the June 1977 contract provides prices for March, April and May 1977. The exceptions to this general pattern occur at the ends of the period: the March 1977 contract only gives prices for January and February 1977 whilst the March 1984 contract is used for December 1983 alone.

Futures prices are studied because reliable prices are published by the IMM, including the important high and low prices. Furthermore, changes in futures prices, unlike spot changes, theoretically have zero mean. Care is needed when the contract studied changes once a quarter but it is only necessary to ensure that all calculations of daily returns and overnight returns use two prices for the same futures contract. So, for example, the return for 1 March 1977 is calculated from the closing prices of the June 1977 contract on 28 February and 1 March 1977. Futures prices are related to spot prices by a well-known arbitrage equation involving domestic and foreign interest rates. It follows that the volatility of the spot price will be very similar to the futures volatility for every contract.

Closed-market and open-market returns, xit and xZr, have been calculated from appropriate closing prices, z,_i and zr, and an intermediate opening price zp by the formulae

xl, = log( zp/z,_ 1) and x2( = log( z/z:), (18)

for t = 1, 2.. . ,1762. Sample variances are presented on the next page for

(i) all returns and for (ii) ordinary Tuesdays to Fridays (z, recorded 24 hours after z,_i), (iii) ordinary Mondays (zt recorded 72 hours after z,_,), (iv) the rest (a holiday between the recording of z, and z,_i).

Adding the closed- and open-market variances gives the total variance. The estimate of E, the proportion of variance attributable to the closed-market hours, is the closed-market variance divided by the total variance. All the estimates exceed 0.5. It is obvious that the average volatility is greater on Mondays and holidays than on other days and the same conclusion applies to the proportion 6.

5. Constructing forecasts from prices 1977-1981

The seven year period (1977-1983) has been split into five years for constructing forecasts (1977-1981) and two years for assessing these forecasts ‘post-sample’ (1982-1983). We consider deseasonalised absolute returns rn: and ranges d:. These are defined using close, high and low

prices (z,, h,, I,) by

m, = Ilog(z,/z,-i) I) d, = log( h ,/I, > 3 (19)

m*=m f f’ d,? = d, for 24 hour returns, (2Oa)

mT = m,/s,, d: = d/s, for Mondays and holidays, (2Ob)

and I have used s, = 1.24 and sd = 1.11.

Category

S.J. Taylor / Forecasting the volatility of currency exchange rates 165

Sample Variance X 10 4 Estimate of c

size Closed mkt Open mkt

(i) all 1762 0.220 0.169 0.565

(ii) Tues-Fri 1386 0.184 0.157 0.540

(iii) Mondays 318 0.345 0.209 0.622

(iv) Holidays 58 0.393 0.248 0.613

5.1. Autocorrelations and parameter estimates

The autocorrelations of the observations rn: and d,* can be used to help construct forecasts. These forecasts should be reasonable if the processes generating the observations are stationary. The forecasts may, however, be seriously suboptimal if the processes are nonstationary. There are 1258 observations for each series up to the end of 1981 from which autocorrelations r, m and r, d have been calculated for lags r ranging from 1 to 50 days. All the autocorrelations are iositive and they generally decrease as r increases. The coefficients up to lag 5 and at various later lags are as shown in the table below. Clearly ranges are far more autocorrelated than absolute returns, as was predicted theoretically in the paragraph following eq. (11).

The sample autocorrelations r7,,, and r,,d are compatible with generating processes having theoretical autocorrelations KM+7 and K&I’ [cf. eqs. (12), (13)]. An appropriate common estimate of the autoregressive parameter is 4 = 0.986 and then k = 0.175 for series m* and k = 0.473 for series d*. Then using eq. (15) the estimates of the moving average parameter for an ARMA (1,l) model are 4 = 0.9245 for series m* and e= 0.8525 for series d*. More information about the estimation method is given in Taylor (1986, Section 3.9). It involves minimising the goodness-of-fit statistic 1258 Z(rT - I?$)‘, summing over r from 1 to 50. For the estimates presented here this statistic equals 52.7 for series m* and 42.4 for series d*.

5.2. Choosing a related variable

The methodology for forecasting volatility requires the selection of an observable related variable. Eq. (17) and the parameter estimates 4, 4 predict RMSE values for optimal volatility forecasts. These RMSE are respectively 0.314 and 0.174 for forecasts based on the deseasonalised quantities my and d: defined by eqs. (19) and (20). Consequently, the related variable used for subsequent forecast comparisons in this paper will be the deseasonalised daily range d:.

Lag Series m* Series d*

1 0.250 0.510

2 0.215 0.476

3 0.230 0.452

4 0.176 0.429

5 0.231 0.451

10 0.165 0.420

20 0.090 0.373

30 0.124 0.330

40 0.062 0.306

50 0.060 0.257

166 S.J. Taylor / Formating the volatility of currency exchange rates

5.3. Empirical RMSE

Suppose a forecast Jtth is made at time t for the quantity dF+,,, for times t = t,, . . , t,, and let d be the average value of d * up to the end of 1981. Then the empirical RMSE of the forecasts is defined to be

RMSE = s (d;C+, - a,+,,)‘/ ; (d,*,, - ci)*. (21)

The notation RMSE( j) is used for forecast method j. In this section h = 1, t, = 20 and t, = 1257.

5.4. Stationary forecasts

First, suppose d;C+, is forecast using the present and past values d;C_,, i > 0, and a stationary ARMA(l,l) model. Then from eq. (14)

J*+, =d+(rjAl) &‘(d;*_,-d). I=0

Inserting the estimates C$ = 0.986 and 6 = 0.8525 from section 5.1 and simplifying gives forecast l( FI )

Fl Jt+i= O.O14d+ 0.1335d: + 0.8525a,. (22)

Here & is the forecast of d: made at time t - 1. To begin the calculations I have let d0 = d. The first forecast gives RMSE(l) = 0.611. This empirical figure is very close to the theoretical prediction, 0.609, provided by eq. (16).

Second, suppose the information used is present and past values mT_,, i 2 0, and again an ARMA(l,l) model is assumed. Then an appropriate forecast is

cl t+l = (1 - +)Z+ ($I - e)rnT + 8h,,

for some constant b. Inserting a regression estimate 6 = 1.329 and the estimates appropriate to series m * ( C$ = 0.986, e^ = 0.9245) and then simplifying gives forecast 2

F2 d;+,= 0.0186Z + 0.0817mr + 0.9245d;. (23)

The second forecast gives RMSE(2) = 0.676. This is 6 per cent worse than the first forecast and confirms that daily ranges are more informative than absolute returns.

5.5. Non-stationary forecasts

Many researchers dislike an assumption that financial data has been generated by a stationary model. It is indeed possible that volatility and related variables have non-stationary mean values. As the estimate C$ is almost 1 appropriate forecasts avoiding the stationarity assumption are given by letting $I = 1 in the ARMA(l,l) forecasting equations. A value for 0 can be obtained by minimising

S.J. Taylor / Forecasting the volatility of currency exchange rofes 167

the empirical RMSE [cf. Taylor (1986, pp. 103-4)J. On this occasion 8 = 0.88 is appropriate. Then forecast 3 is an exponentially-weighted moving average of the d:

F3 a,+, = 0.12d: + O.SS&, (24)

with RMSE(3) = 0.618 marginally worse than RMSE(1) = 0.611. Likewise, forecast 4 is a weighted average of the my with a regression estimate 6 = 1.269 giving

F4 Jr+1 = 0.1523&r, + 0.88& (25)

with RMSE(4) = 0.686, again slightly worse than the comparable stationary forecast.

5.6. Combined forecasts

Garman and Klass (1980) presented theoretical results for volatility estimates computed from various functions of daily ranges, closed-market returns and open-market returns. They argued that all three types of price-changes should be used in volatility estimates. I have evaluated several combinations of forecasts but only minor improvements upon forecast 1 have been obtained.

Regression methods give the following best combinations of the stationary and non-stationary

forecasts

F5 = 0.831 Fl + 0.236 F2, (26)

F6 = 0,730 F3 + 0.348 F4, (27)

with RMSE(5) = 0.608 and RMSE(6) = 0.607 compared with RMSE(1) = 0.611.

5.7. Simple forecasts

The RMSE values describe the accuracy of a set of forecasts relative to the constant forecast defined by the historic mean d for a five-year period. Option traders do not use five years data to estimate volatility. A simple estimate of volatility suggested in textbooks is provided by the standard deviation of the twenty or so latest daily or weekly returns [e.g., Cox and Rubinstein (1985, pp. 255-7, 276-7)]. Consequently, we consider forecasts

for constants b and N. For N = 20 regression gives b = 0.987 as best for minimising RMSE and hence forecast 7 (the ‘state of the art’ forecast) is

F7 L = 0.987/[ !0mT3i/20] , (28)

for which RMSE(7) = 0.760. Optimising over both b and N defines forecast 8 with N = 15 and RMSE(8) = 0.757

F8 $+, = 0.987/[ ~0m.li/15] . (29)

168 S.J. Taylor / Forecasting the uolatrlrty of currency exchange rates

It may be argued that many option traders use high and low prices so we also consider a simple average of N ranges; firstly, N = 20 gives forecast 9 (the ‘advanced state of the art’ forecast [cf. Parkinson (1980) Cox and Rubinstein (1985, p. 277)])

F9 d^l+i = 5 d;C_,/20, (30) i=O

with RMsE(9) = 0.657 and, secondly, the best N defines forecast 10 with N = 10

FlO &+, = 5 d,*_JlO, r=O

(31)

and this final forecast has RMSE(10) = 0.639.

6. Forecasting results out-of-sample 1982-1983

The ten forecasts have been assessed ‘post-sample’ for two years and for forecast horizons h = 1, 5, 10 and 20 trading days. Post-sample RMSE figures are calculated using the same definition as before, i.e., eq. (21) but now with summation limits t, = 1259-h and t, = 1762-h.

6.1. Forecasting for the next day

The table presents RMSE(j) when h = 1 for the first five years and the final two years. F.5 and F6 were the best forecasts both ‘within-sample’ (1977-81) and ‘post sample’ (1982-83). Thus it is best to use both daily ranges d, and absolute returns m, when forecasting volatility, rather than only one of these variables. Of the two variables the ranges are clearly the more informative (RMSE(l) <

RMSE(2), etc.). Forecasts derived from stationary models gave slightly better predictions than their non-stationary

alternatives. Sophisticated forecasts (Fl to F6) would have been better than simple forecasts (F7 to FIO) and all the forecasts are better than the pre-1982 average d. Within-sample the best sophisti- cated and simple forecasts were F6 and FIO respectively, with RMSE(6)/RMSE(lO) = 0.95 both within- and post-sample. It is apparent that all the RMSE figures are higher post-sample than within-sample. This suggests the actual volatilities were less variable in the later period.

Forecast

1 Series d Stationary 2 Series m Stationary 3 Series d Non-stationary 4 Series M Non-stationary

5 Both d and M Stationary 6 Both d and m Non-stationary 7 Series m Simple, N = 20 8 Series d Simple, N = 20 9 Series m Simple, best N

10 Series d Simple, best N

Information Model

used tee

RMSE

1977-81 1982-83

0.611 0.744

0.676 0.793

0.618 0.756

0.686 0.880

0.608 0.735

0.607 0.738

0.760 0.900

0.657 0.802 0.757 0.919

0.639 0.781

S.J. Tuylor / Forecasting the oolatility of’currency exchange rates 169

Forecast RMSE, h

1 5 10 20

1 0.744 0.815 0.886 0.959

2 0.793 0.833 0.889 0.957

3 0.756 0.836 0.927 1.054

4 0.880 0.928 1.013 1.150

5 0.735 0.802 0.872 0.947

6 0.738 0.808 0.895 1.023

I 0.900 0.961 1.050 1.072

8 0.802 0.871 0.963 1.043

9 0.919 0.875 1.072 1.161

10 0.781 0.874 0.955 1.112

6.2. Forecasting for later days

Stationary forecasts d;+,, and d;+i, both made at time t, should satisfy

(32)

for the model considered. Eq. (32) applies to Fl, F2 and F5 with 4 = 0.986. As F3, F4 and F6 arise by letting 9 = 1 it is natural to then let JZ+h = a,+, and this has also been done for the simple forecasts F7 to FZO. The table presents RMSE( j) post-sample for h = 1, 5, 10 and 20.

F5 has the least RMSE for all the horizons h. This is a stationary forecast: it is based upon the assumption that the stochastic processes generating the d: and the rn: are stationary. As h

increases, the difference RMSE(6) - RMSE(5) between the accuracies of the best stationary and non-stationary forecasts increases. Thus it appears best to use + < 1 in eq. (32) and so to assume predicted volatilities regress towards a mean level as the horizon h increases [cf. Cox and Rubinstein (1985, p. 280) Taylor (1986, pp. lOS-llO)]. For h = 10 and 20 the three stationary forecasts are the three best forecasts. Also, for h = 20 only these forecasts have RMSE -C 1. Sophisticated forecasts are better than simple forecasts for all h, with RMSE(6)/RMSE(lO) between 0.92 and 0.94 for h > 1.

7. Concluding remarks

This paper has shown that volatility forecasts calculated from daily high and low prices are empirically better than forecasts calculated from daily closing prices. Beckers (1983) has also made this claim in support of the theoretical results of Parkinson (1980) and Garman and Klass (1980). The best forecasts use high, low and close prices. As the forecasting horizon increases it appears best to let the forecasts regress towards a mean value. It is possible that better forecasts can be constructed by considering further relevant information, in particular volatility figures implied by option prices in conjunction with certain option pricing models. I hope to discuss such forecasts in a later paper.

References

Ali, M.M. and C. Giaccotto, 1982, The identical distribution hypothesis for stock market prices - location and scale-shift alternatives, Journal of the American Statistical Association 77, 19-28.

170 S.J. Taylor / Forecasting the volatility of currency exchange rum

Beckers, S., 1983, Variances of security price returns based on high, low and closing prices, Journal of Business 56, 97-112. Black, F., 1976, The pricing of commodity contracts, Journal of Financial Economics 3, 167-179.

Clark, P.K., 1973, A subordinated stochastic process model with finite variance for speculative prices, Econometrica 41,

135-155.

Cox, J.C. and M. Rubinstein, 1985, Options markets (Prentice-Hall, Englewood Cliffs, NJ).

Engle, R.F., 1982, Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation, Econometrica

50, 987-1007.

Engle, R.F. and T. Bollerslev, 1986, Modelling the persistence of conditional variances, Econometric Reviews, forthcoming.

Garman, M.B. and M.J. Klass, 1980, On the estimation of security price volatilities from historical data, Journal of Business

53, 67-78.

Garman, M.B. and S.W. Kohlhagen, 1983, Foreign currency option values, Journal of International Money and Finance 2,

231-238.

Gemmill, G., 1986, A primer on the pricing of options on currencies and short-term interest rates, The Investment Analyst 81,

16-22.

Hsu, D.A. 1977, Tests for variance shift at an unknown time point, Applied Statistics 26, 279-284.

Parkinson, M., 1980, The extreme value method for estimating the variance of the rate of return. Journal of Business 53,

61-65.

Tauchen, G.E. and M. Pitts, 1983, The price variability-volume relationship on speculative markets, Econometrica 51,

485-505.

Taylor, S.J., 1986, Modelling financial time series (Wiley, Chichester).

Biography: Stephen J. TAYLOR, M.A., Ph. D. is a lecturer in Operational Research at the University of Lancaster. The author of Modelling Financial Time Series (Wiley, 1986) he is particularly interested in stock, currency and commodity prices, especially for futures and options contracts. Previous publications include articles in the Journals of the Royal Statistical Society, Journal of Financial and Quantitative Analysis, Journal of the Oper-

ational Research Society, Applied Economics and The Investment Analyst.