Table 3.7 Granger causality tests

In¬‚ation does not Granger cause z2t

F -Statistic

Variable Signi¬cance

0.03a

πt 5.8

z2t does not Granger cause in¬‚ation

F -Statistic

Variable Signi¬cance

0.03a

z2t 6.12

a

Means rejection at 5% level.

Modelling the Term Structure of Interest Rates 125

RESP OF PI RESP OF PI TO

SIZE = SIZE =

0.5

0.40

0.35

0.4

0.30

0.25 0.3

0.20

0.2

0.15

0.10

0.1

0.05

0.00 0.0

2 4 6 8 10 12 14 16 18 20

2 4 6 8 10 12 14 16 18 20

RESP OF Z2 RESP OF Z2 TO

SIZE = SIZE =

0.64

0.30

0.25 0.56

0.20

0.48

0.15

0.40

0.10

0.05 0.32

0.00

0.24

’0.05

0.16

’0.10

’0.15 0.08

2 4 6 8 10 12 14 16 18 20

2 4 6 8 10 12 14 16 18 20

Figure 3.17 Impulse-response functions and two standard error bands

Overall, considering the results for the factor loadings and the relationship between

the latent factors and the economic variables, the short-term interest rates are mostly

driven by the real interest rates. In parallel, the in¬‚ation expectations exert the most

important in¬‚uence on the long-term rates, as is usually assumed. Similar results con-

cerning the information content of the German term structure, namely in the longer

terms, regarding future changes in in¬‚ation rate were obtained in several previous papers,

namely Schich (1996), Gerlach (1995) and Mishkin (1991), using different samples and

testing procedures.53

53

The Mishkin (1991) and Jorion and Mishkin (1991) results on Germany are contradictory as, according

to Mishkin (1991), the short end of the term structure does not contain information on future in¬‚ation for all

OECD countries studied, except for France, the UK and Germany. Conversely, Jorion and Mishkin (1991)

conclude that the predictive power of the shorter rates about future in¬‚ation is low in the USA, Germany and

Switzerland.

Fama (1990) and Mishkin (1990a,b) present identical conclusions concerning the information content of

US term structure regarding future in¬‚ation and state that the US dollar short rates have information content

regarding future real interest rates and the longer rates contain information on in¬‚ation expectations. Mishkin

(1990b) also concludes that for several countries the information on in¬‚ation expectations is weaker than for

the United States. Mehra (1997) presents evidence of a cointegration relation between the nominal yield on a

10-year Treasury bond and the actual US in¬‚ation rate. Koedijk and Kool (1995), Mishkin (1991), and Jorion

and Mishkin (1991) supply some evidence on the information content of the term structure concerning in¬‚ation

rate in several countries.

126 Applied Quantitative Methods for Trading and Investment

Fleming and Remolona (1998) also found that macroeconomic announcements of the

CPI and the PPI affect mostly the long end of the term structure of interest rates, using

high frequency data. Nevertheless, it is also important to have in mind that, according

to Schich (1999), the information content of the term structure on future in¬‚ation is

time-varying and depends on the country considered.

The relationship between the factors and the referred variables is consistent with the

time-series properties of the factors and those variables. Actually, as observed in Clarida

et al. (1998), the I (1) hypothesis is rejected in Dickey“Fuller tests for the German in¬‚a-

tion rate and short-term interest rate, while, as previously stated, the factors are stationary.

3.8 CONCLUSIONS

The identi¬cation of the factors that determine the time-series and cross-section behaviour

of the term structure of interest rates is one of the most challenging research topics in

¬nance. In this chapter, it was shown that a two-factor constant volatility model describes

quite well the dynamics and shape of the German yield curve between 1986 and 1998.

The data supports the expectations theory with constant term premiums and thus the

term premium structure can be calculated and short-term interest rate expectations derived

from the adjusted forward rate curve. The estimates obtained for the term premium curve

are not inconsistent with the ¬gures usually conjectured. Nevertheless, poorer results are

obtained if the second factor is directly linked to the in¬‚ation rate, given that restrictions

on the behaviour of that factor are imposed, generating less plausible shapes and ¬gures

for the term premium curve.

We identi¬ed within the sample two periods of poorer model performance, both related

to world-wide gyrations in bond markets (Spring 1994 and 1998), which were char-

acterised by sharp changes in long-term interest rates while short-term rates remained

stable.

As to the evolution of bond yields in Germany during 1998, it seems that it is more

the (low) level of in¬‚ation expectations as compared to the level of real interest rates that

underlies the dynamics of the yield curve during that year. However, there still remains

substantial volatility in long bond yields to be explained. This could be related to the

spillover effects of international bond market developments on the German bond market

in the aftermath of the Russian and Asian crises.

It was also shown that one of those factors seems to be related to the ex-ante real

interest rate, while a second factor is linked to in¬‚ation expectations. This conclusion is

much in accordance with the empirical literature on the subject and is a relevant result

for modelling the yield curve using information on macroeconomic variables.

Therefore, modelling the yield curve behaviour, namely for VaR purposes, seems to be

reasonably approached by simulations of (ex-post) real interest rates and lagged in¬‚ation

rate. In addition, the results obtained suggest that a central bank has a decisive role

concerning the bond market moves, given that it in¬‚uences both the short and the long

ends of the yield curve, respectively by in¬‚uencing the real interest rate and the in¬‚ation

expectations. Accordingly, the second factor may also be used as an indicator of monetary

policy credibility.

REFERENCES

A¨t-Sahalia, Y. (1996), “Testing Continuous-Time Models of the Spot Interest Rate”, Review of

±

Financial Studies, 9, 427“470.

Modelling the Term Structure of Interest Rates 127

Babbs, S. H. and K. B. Nowman (1998), “An Application of Generalized Vasicek Term Structure

Models to the UK Gilt-edged Market: a Kalman Filtering analysis”, Applied Financial Economics,

8, 637“644.

Backus, D., S. Foresi, A. Mozumdar and L. Wu (1997), “Predictable Changes in Yields and For-

ward Rates”, mimeo.

Backus, D., S. Foresi and C. Telmer (1998), “Discrete-Time Models of Bond Pricing”, NBER

working paper no. 6736.

Balduzzi, P., S. R. Das, S. Foresi and R. Sundaram (1996), “A Simple Approach to Three Factor

Af¬ne Term Structure Models”, Journal of Fixed Income, 6 December, 43“53.

Bliss, R. (1997), “Movements in the Term Structure of Interest Rates”, Federal Reserve Bank of

Atlanta, Economic Review, Fourth Quarter.

Bolder, D. J. (2001), “Af¬ne Term-Structure Models: Theory and Implementation”, Bank of Canada,

working paper 2001-15.

Breeden, D. T. (1979), “An Intertemporal Asset Pricing Model with Stochastic Consumption and

Investment Opportunities”, Journal of Financial Economics, 7, 265“296.

Buhler, W., M. Uhrig-Homburg, U. Walter and T. Weber (1999), “An Empirical Comparison of

Forward-Rate and Spot-Rate Models for Valuing Interest-Rate Options”, The Journal of Finance,

LIV, 1, February.

Campbell, J. Y. (1995), “Some Lessons from the Yield Curve”, Journal of Economic Perspectives,

9, 3, 129“152.

Campbell, J. Y., A. W. Lo and A. C. MacKinlay (1997), The Econometrics of Financial Markets,

Princeton University Press, Princeton, NJ.

Cassola, N. and J. B. Lu´s (1996), “The Term Structure of Interest Rates: a Comparison of Alter-

±

native Estimation Methods with an Application to Portugal”, Banco de Portugal, working paper

no. 17/96, October 1996.

Cassola, N. and J. B. Lu´s (2003), “A Two-Factor Model of the German Term Structure of Interest

±

Rates”, Applied Financial Economics, forthcoming.

Chen, R. and L. Scott (1993a), “Maximum Likelihood Estimations for a Multi-Factor Equilibrium

Model of the Term Structure of Interest Rates”, Journal of Fixed Income, 3, 14“31.

Chen, R. and L. Scott (1993b), “Multi-Factor Cox“Ingersoll“Ross Models of the Term Structure:

Estimates and Test from a Kalman Filter”, working paper, University of Georgia.

Clarida, R. and M. Gertler (1997), “How the Bundesbank Conducts Monetary Policy”, in Romer,

C. D. and D. H. Romer (eds), Reducing In¬‚ation: Motivation and Strategy, NBER Studies in

Business Cycles, Vol. 30.

Clarida, R., J. Gali and M. Gertler (1998), “Monetary Policy Rules in Practice “ Some International

Evidence”, European Economic Review, 42, 1033“1067.

Cox, J., J. Ingersoll and S. Ross (1985a), “A Theory of the Term Structure of Interest Rates”,

Econometrica, 53, 385“407.

Cox, J., J. Ingersoll and S. Ross (1985b), “An Intertemporal General Equilibrium Model of Asset

Prices”, Econometrica, 53, 363“384.

Deacon, M. and A. Derry (1994), “Deriving Estimates of In¬‚ation Expectations from the Prices of

UK Government Bonds”, Bank of England, working paper 23.

De Jong, F. (1997), “Time-Series and Cross-section Information in Af¬ne Term Structure Models”,

Center for Economic Research.

Doan, T. A. (1995), RATS 4.0 User™s Manual, Estima, Evanston, IL.

Duf¬e, D. and R. Kan (1996), “A Yield Factor Model of Interest Rates”, Mathematical Finance,

6, 379“406.

Fama, E. F. (1975), “Short Term Interest Rates as Predictors of In¬‚ation”, American Economic

Review, 65, 269“282.

Fama, E. F. (1990), “Term Structure Forecasts of Interest Rates, In¬‚ation and Real Returns”, Jour-

nal of Monetary Economics, 25, 59“76.

Fleming, M. J. and E. M. Remolona (1998), “The Term Structure of Announcement Effects”,

mimeo.

Fung, B. S. C., S. Mitnick and E. Remolona (1999), “Uncovering In¬‚ation Expectations and Risk

Premiums from Internationally Integrated Financial Markets”, Bank of Canada, working paper

99-6.

128 Applied Quantitative Methods for Trading and Investment

Gerlach, S. (1995), “The Information Content of the Term Structure: Evidence for Germany”, BIS

working paper no. 29, September.

Geyer, A. L. J. and S. Pichler (1996), “A State-Space Approach to Estimate and Test Multifactor

Cox“Ingersoll“Ross Models of the Term Structure”, mimeo.

Gong, F. F. and E. M. Remolona (1997a), “A Three-factor Econometric Model of the US Term

Structure”, FRBNY Staff Reports, 19, January.

Gong, F. F. and E. M. Remolona (1997b), “In¬‚ation Risk in the U.S. Yield Curve: The Usefulness

of Indexed Bonds”, Federal Reserve Bank, New York, June.

Gong, F. F. and E. M. Remolona (1997c), “Two Factors Along the Yield Curve”, The Manchester

School Supplement, pp. 1“31.

Hamilton, J. D. (1994), Time Series Analysis, Princeton University Press, Princeton, NJ.

Jorion, P. and F. Mishkin (1991), “A Multicountry Comparison of Term-structure Forecasts at Long

Horizons”, Journal of Financial Economics, 29, 59“80.

Koedijk, K. G. and C. J. M. Kool (1995), “Future In¬‚ation and the Information in International

Term Structures”, Empirical Economics, 20, 217“242.

Litterman, R. and J. Scheinkman (1991), “Common Factors Affecting Bond Returns”, Journal of

Fixed Income, 1, June, 49“53.

Lu´s, J. B. (2001), Essays on Extracting Information from Financial Asset Prices, PhD thesis, Uni-

±

versity of York.

Mehra, Y. P. (1997), “The Bond Rate and Actual Future In¬‚ation”, Federal Reserve Bank of Rich-

mond, working paper 97-3, March.

Mishkin, F. (1990a), “What Does the Term Structure Tell Us About Future In¬‚ation”, Journal of

Monetary Economics, 25, 77“95.

Mishkin, F. (1990b), “The Information in the Longer Maturity Term Structure About Future In¬‚a-

tion”, Quarterly Journal of Economics, 55, 815“828.

Mishkin, F. (1991), “A Multi-country Study of the Information in the Shorter Maturity Term Struc-

ture About Future In¬‚ation”, Journal of International Money and Finance, 10, 2“22.

Nelson, C. R. and A. F. Siegel (1987), “Parsimonious Modelling of Yield Curves”, Journal of

Business, 60, 4.

Remolona, E., M. R. Wickens and F. F. Gong (1998), “What was the Market™s View of U.K.

Monetary Policy? Estimating In¬‚ation Risk and Expected In¬‚ation with Indexed Bonds”, FRBNY

Staff Reports, 57, December.

Ross, S. A. (1976), “The Arbitrage Theory of Capital Asset Pricing”, Journal of Economic Theory,

13, 341“360.

Schich, S. T. (1996), “Alternative Speci¬cations of the German Term Structure and its Information

Content Regarding In¬‚ation”, Deutsche Bundesbank, D.P. 8/96.

Schich, S. T. (1999), “What the Yield Curves Say About In¬‚ation: Does It Change Over Time?”,

OECD Economic Department Working Papers, No. 227.

Svensson, L. E. O. (1994), “Estimating and Interpreting Forward Interest Rates: Sweden 1992“4”,

CEPR Discussion Paper Series, No. 1051.

Vasicek, O. (1977), “An Equilibrium Characterisation of the Term Structure”, Journal of Financial

Economics, 5, 177“188.

Zin, S. (1997), “Discussion of Evans and Marshall”, Carnegie-Rochester Conference on Public

Policy, November.

4

Forecasting and Trading Currency Volatility:

An Application of Recurrent Neural

Regression and Model Combination—

CHRISTIAN L. DUNIS AND XUEHUAN HUANG

ABSTRACT

In this chapter, we examine the use of nonparametric Neural Network Regression (NNR)

and Recurrent Neural Network (RNN) regression models for forecasting and trading cur-

rency volatility, with an application to the GBP/USD and USD/JPY exchange rates. Both

the results of the NNR and RNN models are benchmarked against the simpler GARCH

alternative and implied volatility. Two simple model combinations are also analysed.

The intuitively appealing idea of developing a nonlinear nonparametric approach to

forecast FX volatility, identify mispriced options and subsequently develop a trading

strategy based upon this process is implemented for the ¬rst time on a comprehensive

basis. Using daily data from December 1993 through April 1999, we develop alternative

FX volatility forecasting models. These models are then tested out-of-sample over the

period April 1999“May 2000, not only in terms of forecasting accuracy, but also in terms

of trading ef¬ciency. In order to do so, we apply a realistic volatility trading strategy using

FX option straddles once mispriced options have been identi¬ed.

Allowing for transaction costs, most trading strategies retained produce positive returns.

RNN models appear as the best single modelling approach yet, somewhat surprisingly,

a model combination which has the best overall performance in terms of forecasting

accuracy fails to improve the RNN-based volatility trading results.

Another conclusion from our results is that, for the period and currencies considered,

the currency option market was inef¬cient and/or the pricing formulae applied by market

participants were inadequate.

4.1 INTRODUCTION

Exchange rate volatility has been a constant feature of the International Monetary System

ever since the breakdown of the Bretton Woods system of ¬xed parities in 1971“73. Not

surprisingly, in the wake of the growing use of derivatives in other ¬nancial markets, and

—

This chapter previously appeared under the same title in the Journal of Forecasting, 21, 317“354 (2002).

™ John Wiley & Sons, Ltd. Reproduced with permission.

Applied Quantitative Methods for Trading and Investment. Edited by C.L. Dunis, J. Laws and P. Na¨m

±

™ 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5

130 Applied Quantitative Methods for Trading and Investment

following the extension of the seminal work of Black“Scholes (1973) to foreign exchange

by Garman“Kohlhagen (1983), currency options have become an ever more popular way

to hedge foreign exchange exposures and/or speculate in the currency markets.

In the context of this wide use of currency options by market participants, having the

best volatility prediction has become ever more crucial. True, the only unknown variable

in the Garman“Kohlhagen pricing formula is precisely the future foreign exchange rate

volatility during the life of the option. With an “accurate” volatility estimate and knowing

the other variables (strike level, current level of the exchange rate, interest rates on both

currencies and maturity of the option), it is possible to derive the theoretical arbitrage-free

price of the option. Just because there will never be such thing as a unanimous agreement

on the future volatility estimate, market participants with a better view/forecast of the

evolution of volatility will have an edge over their competitors.

In a rational market, the equilibrium price of an option will be affected by changes

in volatility. The higher the volatility perceived by market participants, the higher the

option™s price. Higher volatility implies a greater possible dispersion of the foreign

exchange rate at expiry: all other things being equal, the option holder has logically an

asset with a greater chance of a more pro¬table exercise. In practice, those investors/market

participants who can reliably predict volatility should be able to control better the ¬nan-

cial risks associated with their option positions and, at the same time, pro¬t from their

superior forecasting ability.

There is a wealth of articles on predicting volatility in the foreign exchange market:

for instance, Baillie and Bollerslev (1990) used ARIMA and GARCH models to describe

the volatility on hourly data, West and Cho (1995) analysed the predictive ability of

GARCH, AR and nonparametric models on weekly data, Jorion (1995) examined the

predictive power of implied standard deviation as a volatility forecasting tool with daily

data, Dunis et al. (2001b) measured, using daily data, both the 1-month and 3-month

forecasting ability of 13 different volatility models including AR, GARCH, stochastic

variance and model combinations with and without the adding of implied volatility as an

extra explanatory variable.

Nevertheless, with the exception of Engle et al. (1993), Dunis and Gavridis (1997)

and, more recently, Laws and Gidman (2000), these papers evaluate the out-of-sample

forecasting performance of their models using traditional statistical accuracy criteria, such

as root mean squared error, mean absolute error, mean absolute percentage error, Theil-

U statistic and correct directional change prediction. Investors and market participants

however have trading performance as their ultimate goal and will select a forecasting

model based on ¬nancial criteria rather than on some statistical criterion such as root

mean squared error minimisation. Yet, as mentioned above, seldom has recently published

research applied any ¬nancial utility criterion in assessing the out-of-sample performance

of volatility models.

Over the past few years, Neural Network Regression (NNR) has been widely advocated

as a new alternative modelling technology to more traditional econometric and statistical

approaches, claiming increasing success in the ¬elds of economic and ¬nancial forecast-

ing. This has resulted in many publications comparing neural networks and traditional

forecasting approaches. In the case of foreign exchange markets, it is worth pointing out

that most of the published research has focused on exchange rate forecasting rather than

on currency volatility forecasts. However, ¬nancial criteria, such as Sharpe ratio, prof-

itability, return on equity, maximum drawdown, etc., have been widely used to measure

Forecasting and Trading Currency Volatility 131

and quantify the out-of-sample forecasting performance. Dunis (1996) investigated the

application of NNR to intraday foreign exchange forecasting and his results were evalu-

ated by means of a trading strategy. Kuan and Liu (1995) proposed two-step Recurrent

Neural Network (RNN) models to forecast exchange rates and their results were evalu-

ated using traditional statistical accuracy criteria. Tenti (1996) applied RNNs to predict the

USD/DEM exchange rate, devising a trading strategy to assess his results, while Franses

and Van Homelen (1998) use NNR models to predict four daily exchange rate returns

relative to the Dutch guilder using directional accuracy to assess out-of-sample forecast-

ing accuracy. Overall, it seems however that neural network research applied to exchange

rates has been so far seldom devoted to FX volatility forecasting.

Accordingly, the rationale for this chapter is to investigate the predictive power of

alternative nonparametric forecasting models of foreign exchange volatility, both from

a statistical and an economic point of view. We examine the use of NNR and RNN

regression models for forecasting and trading currency volatility, with an application to

the GBP/USD and USD/JPY exchange rates. The results of the NNR and RNN models are

benchmarked against the simpler GARCH (1,1) alternative, implied volatility and model

combinations: in terms of model combination, a simple average combination and the

Granger“Ramanathan (1984) optimal weighting regression-based approach are employed

and their results investigated.

Using daily data from December 1993 through April 1999, we develop alternative FX

volatility forecasting models. These models are then tested out-of-sample over the period

April 1999“May 2000, not only in terms of forecasting accuracy, but also in terms of

trading ef¬ciency. In order to do so, we apply a realistic volatility trading strategy using

FX option straddles once mispriced options have been identi¬ed.

Allowing for transaction costs, most trading strategies retained produce positive returns.

RNN models appear as the best single modelling approach but model combinations,

despite their superior performance in terms of forecasting accuracy, fail to produce

superior trading strategies.

Another conclusion from our results is that, for the period and currencies considered,

the currency option market was inef¬cient and/or the pricing formulae applied by market

participants were inadequate.

Overall, we depart from existing work in several respects.

Firstly, we develop alternative nonparametric FX volatility models, applying in par-

ticular an RNN architecture with a loop back from the output layer implying an error

feedback mechanism, i.e. we apply a nonlinear error-correction modelling approach to

FX volatility.

Secondly, we apply our nonparametric models to FX volatility, something that has not

been done so far. A recent development in the literature has been the application of

nonparametric time series modelling approaches to volatility forecasts. Gaussian kernel

regression is an example, as in West and Cho (1995). Neural networks have also been

found useful in modelling the properties of nonlinear time series. As mentioned above, if

there are quite a few articles on applications of NNR models to foreign exchange, stock

and commodity markets,1 there are rather few concerning ¬nancial markets volatility

1

For NNR applications to commodity forecasting, see, for instance, Ntungo and Boyd (1998) and Trippi and

Turban (1993). For applications to the stock market, see, amongst others, Deboeck (1994) and Leung et al.

(2000).

132 Applied Quantitative Methods for Trading and Investment

forecasting in general.2 It seems therefore that, as an alternative technique to more tra-

ditional statistical forecasting methods, NNR models need further investigation to check

whether or not they can add value in the ¬eld of foreign exchange volatility forecasting.

Finally, unlike previous work, we do not limit ourselves to forecasting accuracy but

extend the analysis to the all-important trading ef¬ciency, taking advantage of the fact

that there exists a large and liquid FX implied volatility market that enables us to apply

sophisticated volatility trading strategies.

The chapter is organised as follows. Section 4.2 describes our exchange rate and volatil-

ity data. Section 4.3 brie¬‚y presents the GARCH (1,1) model and gives the corresponding

21-day volatility forecasts. Section 4.4 provides a detailed overview and explains the pro-

cedures and methods used in applying the NNR and RNN modelling procedure to our

¬nancial time series, and it presents the 21-day volatility forecasts obtained with these

methods. Section 4.5 brie¬‚y describes the model combinations retained and assesses the

21-day out-of-sample forecasts using traditional statistical accuracy criteria. Section 4.6

introduces the volatility trading strategy using FX option straddles that we follow once

mispriced options have been identi¬ed through the use of our most successful volatility

forecasting models. We present detailed trading results allowing for transaction costs and

discuss their implications, particularly in terms of a quali¬ed assessment of the ef¬ciency

of the currency options market. Finally, Section 4.7 provides some concluding comments

and suggestions for further work.

4.2 THE EXCHANGE RATE AND VOLATILITY DATA

The motivation for this research implies that the success or failure to develop pro¬table

volatility trading strategies clearly depends on the possibility to generate accurate volatility

forecasts and thus to implement adequate volatility modelling procedures.

Numerous studies have documented the fact that logarithmic returns of exchange rate

time series exhibit “volatility clustering” properties, that is periods of large volatility

tend to cluster together followed by periods of relatively lower volatility (see, amongst

others, Baillie and Bollerslev (1990), Kroner et al. (1995) and Jorion (1997)). Volatility

forecasting crucially depends on identifying the typical characteristics of volatility within

the restricted sample period selected and then projecting them over the forecasting period.

We present in turn the two databanks we have used for this study and the modi¬cations

to the original series we have made where appropriate.

4.2.1 The exchange rate series databank and historical volatility

The return series we use for the GBP/USD and USD/JPY exchange rates were extracted

from a historical exchange rate database provided by Datastream. Logarithmic returns,

de¬ned as log(St /St’1 ), are calculated for each exchange rate on a daily frequency basis.

We multiply these returns by 100, so that we end up with percentage changes in the

exchange rates considered, i.e. st = 100 log(St /St’1 ).

2

Even though there are no NNR applications yet to foreign exchange volatility forecasting, some researchers

have used NNR models to measure the stock market volatility (see, for instance, Donaldson and Kamstra (1997)

and Bartlmae and Rauscher (2000)).

Forecasting and Trading Currency Volatility 133

Our exchange rate databank spans from 31 December 1993 to 9 May 2000, giving

us 1610 observations per exchange rate.3 This databank was divided into two separate

sets with the ¬rst 1329 observations from 31 December 1993 to 9 April 1999 de¬ned as

our in-sample testing period and the remaining 280 observations from 12 April 1999 to

9 May 2000 being used for out-of-sample forecasting and validation.

In line with the ¬ndings of many earlier studies on exchange rate changes (see, amongst

others, Engle and Bollerslev (1986), Baillie and Bollerslev (1989), Hsieh (1989), West

and Cho (1995)), the descriptive statistics of our currency returns (not reported here in

order to conserve space) clearly show that they are nonnormally distributed and heavily

fat-tailed. They also show that mean returns are not statistically different from zero.

Further standard tests of autocorrelation, nonstationarity and heteroskedasticity show that

logarithmic returns are all stationary and heteroskedastic. Whereas there is no evidence

of autocorrelation for the GBP/USD return series, some autocorrelation is detected at the

10% signi¬cance level for USD/JPY returns.

The fact that our currency returns have zero unconditional mean enables us to use

squared returns as a measure of their variance and absolute returns as a measure of their

standard deviation or volatility.4 The standard tests of autocorrelation, nonstationarity and

heteroskedasticity (again not reported here in order to conserve space) show that squared

and absolute currency returns series for the in-sample period are all nonnormally dis-

tributed, stationary, autocorrelated and heteroskedastic (except USD/JPY squared returns

which were found to be homoskedastic).

Still, as we are interested in analysing alternative volatility forecasting models and

whether they can add value in terms of forecasting realised currency volatility, we must

adjust our statistical computation of volatility to take into account the fact that, even if

it is only the matter of a constant, in currency options markets, volatility is quoted in

annualised terms. As we wish to focus on 1-month volatility forecasts and related trading

strategies, taking, as is usual practice, a 252-trading day year (and consequently a 21-

trading day month), we compute the 1-month volatility as the moving annualised standard

deviation of our logarithmic returns and end up with the following historical volatility

measures for the 1-month horizon:

√

t

1

σt = ( 252 — |st |)

21 t’20

where |st | is the absolute currency return.5 The value σt is the realised 1-month exchange

rate volatility that we are interested in forecasting as accurately as possible, in order to

see if it is possible to ¬nd any mispriced option that we could possibly take advantage of.

The descriptive statistics of both historical volatility series (again not reported here

in order to conserve space) show that they are nonnormally distributed and fat-tailed.

Further statistical tests of autocorrelation, heteroskedasticity and nonstationarity show

that they exhibit strong autocorrelation but that they are stationary in levels. Whereas

3

Actually, we used exchange rate data from 01/11/1993 to 09/05/2000, the data during the period 01/11/1993

to 31/12/1993 being used for the “pre-calculation” of the 21-day realised historical volatility.

4

Although the unconditional mean is zero, it is of course possible that the conditional mean may vary over

time.

5

The use of absolute returns (rather than their squared value) is justi¬ed by the fact that with zero unconditional

mean, averaging absolute returns gives a measure of standard deviation.

134 Applied Quantitative Methods for Trading and Investment

GBP/USD historical volatility is heteroskedastic, USD/JPY realised volatility was found

to be homoskedastic.

Having presented our exchange rate series databank and explained how we compute our

historical volatilities from these original series (so that they are in a format comparable

to that which prevails in the currency options market), we now turn our attention to the

implied volatility databank that we have used.

4.2.2 The implied volatility series databank

Volatility has now become an observable and traded quantity in ¬nancial markets, and par-

ticularly so in the currency markets. So far, most studies dealing with implied volatilities

have used volatilities backed out from historical premium data on traded options rather

than over-the-counter (OTC) volatility data (see, amongst others, Latane and Rendle-

man (1976), Chiras and Manaster (1978), Lamoureux and Lastrapes (1993), Kroner et al.

(1995) and Xu and Taylor (1996)).

As underlined by Dunis et al. (2000), the problem in using exchange data is that call

and put prices are only available for given strike levels and ¬xed maturity dates. The

corresponding implied volatility series must therefore be backed out using a speci¬c

option pricing model. This procedure generates two sorts of potential biases: material

errors or mismatches can affect the variables that are needed for the solving of the

pricing model, e.g. the forward points or the spot rate, and, more importantly, the very

speci¬cation of the pricing model that is chosen can have a crucial impact on the ¬nal

“backed out” implied volatility series.

This is the reason why, in this chapter, we use data directly observable on the market-

place. This original approach seems further warranted by current market practice whereby

brokers and market makers in currency options deal in fact in volatility terms and not in

option premium terms any more.6 The volatility time series we use for the two exchange

rates selected, GBP/USD and USD/JPY, were extracted from a market quoted implied

volatilities database provided by Chemical Bank for data until end-1996, and updated from

Reuters “Ric” codes subsequently. These at-the-money forward, market-quoted volatilities

are in fact obtained from brokers by Reuters on a daily basis, at the close of business

in London.

These implied volatility series are nonnormally distributed and fat-tailed. Further sta-

tistical tests of autocorrelation and heteroskedasticity (again not reported here in order

to conserve space) show that they exhibit strong autocorrelation and heteroskedasticity.

Unit root tests show that, at the 1-month horizon, both GBP/USD and USD/JPY implied

volatilities are stationary at the 5% signi¬cance level.

Certainly, as noted by Dunis et al. (2001b) and con¬rmed in Tables A4.1 and A4.3 in

Appendix A for the GBP/USD and USD/JPY, an interesting feature is that the mean level

of implied volatilities stands well above average historical volatility levels.7 This tendency

of the currency options market to overestimate actual volatility is further documented

6

The market data that we use are at-the-money forward volatilities, as the use of either in-the-money or out-of-

the-money volatilities would introduce a signi¬cant bias in our analysis due to the so-called “smile effect”, i.e.

the fact that volatility is “priced” higher for strike levels which are not at-the-money. It should be made clear

that these implied volatilities are not simply backed out of an option pricing model but are instead directly

quoted from brokers. Due to arbitrage they cannot diverge too far from the theoretical level.

7

As noted by Dunis et al. (2001b), a possible explanation for implied volatility being higher than its historical

counterpart may be due to the fact that market makers are generally options sellers (whereas end users are

Forecasting and Trading Currency Volatility 135

by Figures A4.1 and A4.2 which show 1-month actual and implied volatilities for the

GBP/USD and USD/JPY exchange rates. These two charts also clearly show that, for

each exchange rate concerned, actual and implied volatilities are moving rather closely

together, which is further con¬rmed by Tables A4.2 and A4.4 for both GBP/USD and

USD/JPY volatilities.

4.3 THE GARCH (1,1) BENCHMARK VOLATILITY

FORECASTS

4.3.1 The choice of the benchmark model

As the GARCH model originally devised by Bollerslev (1986) and Taylor (1986) is well

documented in the literature, we just present it very brie¬‚y, as it has now become widely

used, in various forms, by both academics and practitioners to model conditional variance.

We therefore do not intend to review its many different variants as this would be outside

the scope of this chapter. Besides, there is a wide consensus, certainly among market

practitioners, but among many researchers as well that, when variants of the standard

GARCH (1,1) model do provide an improvement, it is only marginal most of the time.

Consequently, for this chapter, we choose to estimate a GARCH (1,1) model for both

the GBP/USD and USD/JPY exchange rates as it embodies a compact representation

and serves well our purpose of ¬nding an adequate benchmark for the more complex

NNR models.

In its simple GARCH (1,1) form, the GARCH model basically states that the conditional

variance of asset returns in any given period depends upon a constant, the previous

period™s squared random component of the return and the previous period™s variance.

In other words, if we denote by σt2 the conditional variance of the return at time t and

2

µt’1 the squared random component of the return in the previous period, for a standard

GARCH (1,1) process, we have:

σt2 = ω + ±µt’1 + βσt’1

2 2

(4.1)

Equation (4.1) yields immediately the 1-step ahead volatility forecast and, using recursive

substitution, Engle and Bollerslev (1986) and Baillie and Bollerslev (1992) give the n-step

ahead forecast for a GARCH (1,1) process:

σt+n = ω[1 + (± + β) + · · · + (± + β)n’2 ] + ω + ±µt2 + βσt2

2

(4.2)

This is the formula that we use to compute our GARCH (1,1) n-step ahead out-of-

sample forecast.

4.3.2 The GARCH (1,1) volatility forecasts

If many researchers have noted that no alternative GARCH speci¬cation could consistently

outperform the standard GARCH (1,1) model, some such as Bollerslev (1987), Baillie

more often option buyers): there is probably a tendency among option writers to include a “risk premium”

when pricing volatility. Kroner et al. (1995) suggest another two reasons: (i) the fact that if interest rates are

stochastic, then the implied volatility will capture both asset price volatility and interest rate volatility, thus

skewing implied volatility upwards, and (ii) the fact that if volatility is stochastic but the option pricing formula

is constant, then this additional source of volatility will be picked up by the implied volatility.

136 Applied Quantitative Methods for Trading and Investment

and Bollerslev (1989) and Hsieh (1989), amongst others, point out that the Student-t

distribution ¬ts the daily exchange rate logarithmic returns better than conditional nor-

mality, as the former is characterised by fatter tails. We thus generate GARCH (1,1) 1-step

ahead forecasts with the Student-t distribution assumption.8 We give our results for the

GBP/USD exchange rate:

log(St /St’1 ) = µt

µt |•t’1 ∼ N(0, σt2 )

σt2 = 0.0021625 + 0.032119µt’1 + 0.95864σt’1

2 2

(0.0015222) (0.010135) (0.013969) (4.3)

where the ¬gures in parentheses are asymptotic standard errors. The t-values for ± and

2 2

β are highly signi¬cant and show strong evidence that σt2 varies with µt’1 and σt’1 . The

coef¬cients also have the expected sign. Additionally, the conventional Wald statistic for

testing the joint hypothesis that ± = β = 0 clearly rejects the null, suggesting a signi¬cant

GARCH effect.

The parameters in equation (4.3) were used to estimate the 21-day ahead volatility

forecast for the USD/GBP exchange rate: using the 1-step ahead GARCH (1,1) coef-

¬cients, the conditional 21-day volatility forecast was generated each day according to

equation (4.2) above. The same procedure was followed for the USD/JPY exchange rate

volatility (see Appendix B4.3).

Figure 4.1 displays the GARCH (1,1) 21-day volatility forecasts for the USD/GBP

exchange rate both in- and out-of-sample (the last 280 observations, from 12/04/1999 to

GARCH Vol. Forecast Realised Vol.

18 18

16 16

14 14

12 12

10 10

8 8

.

6 6

4 4

2 2

0 0

05/01/94

05/05/94

05/09/94

05/01/95

05/05/95

05/09/95

05/01/96

05/05/96

05/09/96

05/01/97

05/05/97

05/09/97

05/01/98

05/05/98

05/09/98

05/01/99

05/05/99

05/09/99

05/01/00

05/05/00

Figure 4.1 GBP/USD GARCH (1,1) volatility forecast (%)

8

Actually, we modelled conditional volatility with both the normal and the t-distribution. The results are only

slightly different. However, both the Akaike and the Schwarz Bayesian criteria tend to favour the t-distribution.

We therefore selected the results from the t-distribution for further tests (see Appendix B for the USD/GBP

detailed results).

Forecasting and Trading Currency Volatility 137

09/05/2000). It is clear that, overall, the GARCH model ¬ts the realised volatility rather

well during the in-sample period. However, during the out-of-sample period, the GARCH

forecasts are quite disappointing. The USD/JPY out-of-sample GARCH (1,1) forecasts

suffer from a similar inertia (see Figure C4.1 in Appendix C).

In summary, if the GARCH (1,1) model can account for some statistical properties of

daily exchange rate returns such as leptokurtosis and conditional heteroskedasticity, its

ability to accurately predict volatility, despite its wide use among market professionals,

is more debatable. In any case, as mentioned above, we only intend to use our GARCH

(1,1) volatility forecasts as a benchmark for the nonlinear nonparametric neural network

models we intend to apply and test whether NNR/RNN models can produce a substantial

improvement in the out-of-sample performance of our volatility forecasts.

4.4 THE NEURAL NETWORK VOLATILITY FORECASTS

4.4.1 NNR modelling

Over the past few years, it has been argued that new technologies and quantitative sys-

tems based on the fact that most ¬nancial time series contain nonlinearities have made

traditional forecasting methods only second best. NNR models, in particular, have been

applied with increasing success to economic and ¬nancial forecasting and would constitute

the state of the art in forecasting methods (see, for instance, Zhang et al. (1998)).

It is clearly beyond the scope of this chapter to give a complete overview of arti¬cial

neural networks, their biological foundation and their many architectures and poten-

tial applications (for more details, see, amongst others, Simpson (1990) and Hassoun

(1995)).9

For our purpose, let it suf¬ce to say that NNR models are a tool for determining

the relative importance of an input (or a combination of inputs) for predicting a given

outcome. They are a class of models made up of layers of elementary processing units,

called neurons or nodes, which elaborate information by means of a nonlinear transfer

function. Most of the computing takes place in these processing units.

The input signals come from an input vector A = (x [1] , x [2] , . . . , x [n] ) where x [i] is

the activity level of the ith input. A series of weight vectors Wj = (w1j , w2j , . . . , wnj )

is associated with the input vector so that the weight wij represents the strength of the

connection between the input x [i] and the processing unit bj . Each node may additionally

have also a bias input θj modulated with the weight w0j associated with the inputs. The

total input of the node bj is formally the dot product between the input vector A and the

weight vector Wj , minus the weighted input bias. It is then passed through a nonlinear

transfer function to produce the output value of the processing unit bj :

n

bj = f x [i] wij ’ w0j θj = f (Xj ) (4.4)

i=1

9

In this chapter, we use exclusively the multilayer perceptron, a multilayer feedforward network trained by

error backpropagation.

138 Applied Quantitative Methods for Trading and Investment

In this chapter, we have used the sigmoid function as activation function:10

1

f (Xj ) = (4.5)

1 + e’Xj

Figure 4.2 allows one to visualise a single output NNR model with one hidden layer and

two hidden nodes, i.e. a model similar to those we developed for the GBP/USD and the

USD/JPY volatility forecasts. The NNR model inputs at time t are xt[i] (i = 1, 2, . . . , 5).

[j ]

The hidden nodes outputs at time t are ht (j = 1, 2) and the NNR model output at time

˜

t is yt , whereas the actual output is yt .

At the beginning, the modelling process is initialised with random values for the

weights. The output value of the processing unit bj is then passed on to the single

output node of the output layer. The NNR error, i.e. the difference between the NNR

forecast and the actual value, is analysed through the root mean squared error. The latter

is systematically minimised by adjusting the weights according to the level of its deriva-

tive with respect to these weights. The adjustment obviously takes place in the direction

that reduces the error.

As can be expected, NNR models with two hidden layers are more complex. In general,

they are better suited for discontinuous functions; they tend to have better generalisation

capabilities but are also much harder to train. In summary, NNR model results depend

crucially on the choice of the number of hidden layers, the number of nodes and the type

of nonlinear transfer function retained.

In fact, the use of NNR models further enlarges the forecaster™s toolbox of available

techniques by adding models where no speci¬c functional form is a priori assumed.11

Following Cybenko (1989) and Hornik et al. (1989), it can be demonstrated that speci¬c

NNR models, if their hidden layer is suf¬ciently large, can approximate any continuous

x t[1]

ht[1]

x t[2] Σ «

∼

yt

Σ «

x t[3]

Σ «

ht[2] yt

x t[4]

x t[5]

Figure 4.2 Single output NNR model

10

Other alternatives include the hyperbolic tangent, the bilogistic sigmoid, etc. A linear activation function is

also a possibility, in which case the NNR model will be linear. Note that our choice of a sigmoid implies

variations in the interval ]0, +1[. Input data are thus normalised in the same range in order to present the

learning algorithm with compatible values and avoid saturation problems.

11

Strictly speaking, the use of an NNR model implies assuming a functional form, namely that of the trans-

fer function.

Forecasting and Trading Currency Volatility 139

function.12 Furthermore, it can be shown that NNR models are equivalent to nonlinear

nonparametric models, i.e. models where no decisive assumption about the generating

process must be made in advance (see Cheng and Titterington (1994)).

Kouam et al. (1992) have shown that most forecasting models (ARMA models, bilin-

ear models, autoregressive models with thresholds, nonparametric models with kernel

regression, etc.) are embedded in NNR models. They show that each modelling procedure

can in fact be written in the form of a network of neurons.

Theoretically, the advantage of NNR models over other forecasting methods can there-

fore be summarised as follows: as, in practice, the “best” model for a given problem

cannot be determined, it is best to resort to a modelling strategy which is a generalisation

of a large number of models, rather than to impose a priori a given model speci¬cation.

This has triggered an ever-increasing interest for applications to ¬nancial markets (see,

for instance, Trippi and Turban (1993), Deboeck (1994), Rehkugler and Zimmermann

(1994), Refenes (1995) and Dunis (1996)).

Comparing NNR models with traditional econometric methods for foreign exchange rate

forecasting has been the topic of several recent papers: Kuan and Liu (1995), Swanson

and White (1995) and Gen¸ ay (1996) show that NNR models can describe in-sample

c

data rather well and that they also generate “good” out-of-sample forecasts. Forecasting

accuracy is usually de¬ned in terms of small mean squared prediction error or in terms of

directional accuracy of the forecasts. However, as mentioned already, there are still very

few studies concerned with ¬nancial assets volatility forecasting.

4.4.2 RNN modelling

RNN models were introduced by Elman (1990). Their only difference from “regular”

NNR models is that they include a loop back from one layer, either the output or the

intermediate layer, to the input layer. Depending on whether the loop back comes from

the intermediate or the output layer, either the preceding values of the hidden nodes or the

output error will be used as inputs in the next period. This feature, which seems welcome

in the case of a forecasting exercise, comes at a cost: RNN models will require more

connections than their NNR counterparts, thus accentuating a certain lack of transparency

which is sometimes used to criticise these modelling approaches.

Using our previous notation and assuming the output layer is the one looped back, the

RNN model output at time t depends on the inputs at time t and on the output at time

t ’ 1:13

yt = F (xt , yt’1 )

˜ ˜ (4.6)

There is no theoretical answer as to whether one should preferably loop back the intermedi-

ate or the output layer. This is mostly an empirical question. Nevertheless, as looping back

the output layer implies an error feedback mechanism, such RNN models can successfully

be used for nonlinear error-correction modelling, as advocated by Burgess and Refenes

12

This very feature also explains why it is so dif¬cult to use NNR models, as one may in fact end up ¬tting

the noise in the data rather than the underlying statistical process.

13

With a loop back from the intermediate layer, the RNN output at time t depends on the inputs at time t

and on the intermediate nodes at time t ’ 1. Besides, the intermediate nodes at time t depend on the inputs

at time t and on the hidden layer at time t ’ 1. Using our notation, we have therefore: yt = F (xt , ht’1 ) and

˜

ht = G(xt , ht’1 ).

140 Applied Quantitative Methods for Trading and Investment

x t[1]

ht[1]

Σ «

x t[2]

∼

yt

Σ «

x t[3] Σ «

ht[2] yt

∼

r t[1] = yt’1 ’ yt’1

Figure 4.3 Single output RNN model

(1996). This is why we choose this particular architecture as an alternative modelling

strategy for the GBP/USD and the USD/JPY volatility forecasts. Our choice seems fur-

ther warranted by claims from Kuan and Liu (1995) and Tenti (1996) that RNN models

are superior to NNR models when modelling exchange rates.

Figure 4.3 allows one to visualise a single output RNN model with one hidden layer

and two hidden nodes, again a model similar to those developed for the GBP/USD and

the USD/JPY volatility forecasts.

4.4.3 The NNR/RNN volatility forecasts

4.4.3.1 Input selection, data scaling and preprocessing

In the absence of an indisputable theory of exchange rate volatility, we assume that a

speci¬c exchange rate volatility can be explained by that rate™s recent evolution, volatil-

ity spillovers from other ¬nancial markets, and macroeconomic and monetary policy

expectations.

In the circumstances, it seems reasonable to include, as potential inputs, exchange rate

volatilities (including that which is to be modelled), the evolution of important stock and

commodity prices, and, as a measure of macroeconomic and monetary policy expectations,

the evolution of the yield curve.14

As explained above (see footnote 10), all variables were normalised according to our

choice of the sigmoid activation function. They had been previously transformed in log-

arithmic returns.15

Starting from a traditional linear correlation analysis, variable selection was achieved

via a forward stepwise neural regression procedure: starting with both lagged historical

and implied volatility levels, other potential input variables were progressively added,

keeping the network architecture constant. If adding a new variable improved the level of

explained variance over the previous “best” model, the pool of explanatory variables was

updated. If there was a failure to improve over the previous “best” model after several

14

On the use of the yield curve as a predictor of future output growth and in¬‚ation, see, amongst others, Fama

(1990) and Ivanova et al. (2000).

15

Despite some contrary opinions, e.g. Balkin (1999), stationarity remains important if NNR/RNN models are

to be assessed on the basis of the level of explained variance.

Forecasting and Trading Currency Volatility 141

attempts, variables in that model were alternated to check whether no better solution could

be achieved. The model chosen ¬nally was then kept for further tests and improvements.

Finally, conforming with standard heuristics, we partitioned our total data set into three

subsets, using roughly 2/3 of the data for training the model, 1/6 for testing and the

remaining 1/6 for validation. This partition in training, test and validation sets is made

in order to control the error and reduce the risk of over¬tting. Both the training and

the following test period are used in the model tuning process: the training set is used to

develop the model; the test set measures how well the model interpolates over the training

set and makes it possible to check during the adjustment whether the model remains valid

for the future. As the ¬ne-tuned system is not independent from the test set, the use of

a third validation set which was not involved in the model™s tuning is necessary. The

validation set is thus used to estimate the actual performance of the model in a deployed

environment.

In our case, the 1329 observations from 31/12/1993 to 09/04/1999 were considered

as the in-sample period for the estimation of our GARCH (1,1) benchmark model. We

therefore retain the ¬rst 1049 observations from 31/12/1993 to 13/03/1998 for the training

set and the remainder of the in-sample period is used as test set. The last 280 observations

from 12/04/1999 to 09/05/2000 constitute the validation set and serve as the out-of-sample

forecasting period. This is consistent with the GARCH (1,1) model estimation.

4.4.3.2 Volatility forecasting results

We used two similar sets of input variables for the GBP/USD and USD/JPY volatilities,

with the same output variable, i.e. the realised 21-day volatility. Input variables

included the lagged actual 21-day realised volatility (Realised21t’21 ), the lagged

implied 21-day volatility (IVOL21t’21 ), lagged absolute logarithmic returns of the

exchange rate (|r|t’i , i = 21, . . . , 41) and lagged logarithmic returns of the gold price

(DLGOLDt’i , i = 21, . . . , 41) or of the oil price (DLOILt’i , i = 21, . . . , 41), depending

on the currency volatility being modelled.

In terms of the ¬nal model selection, Tables D4.1a and D4.1b in Appendix D give the

performance of the best NNR and RNN models over the validation (out-of-sample) data

set for the USD/GBP volatility. For the same input space and architecture (i.e. with only

one hidden layer), RNN models marginally outperform their NNR counterparts in terms

of directional accuracy. This is important as trading pro¬tability crucially depends on

getting the direction of changes right. Tables D4.1a and D4.1b also compare models with

only one hidden layer and models with two hidden layers while keeping the input and

output variables unchanged: despite the fact that the best NNR model is a two-hidden

layer model with respectively ten and ¬ve hidden nodes in each of its hidden layers, on

average, NNR/RNN models with a single hidden layer perform marginally better while

at the same time requiring less processing time.

The results of the NNR and RNN models for the USD/JPY volatility over the val-

idation period are given in Tables D4.2a and D4.2b in Appendix D. They are in line

with those for the GBP/USD volatility, with RNN models outperforming their NNR

counterparts and, in that case, the addition of a second hidden layer rather deteriorating

performance.

Finally, we selected our two best NNR and RNN models for each volatility, NNR

(44-10-5-1) and RNN (44-1-1) for the GBP/USD and NNR (44-1-1) and RNN (44-5-1)

142 Applied Quantitative Methods for Trading and Investment

for the USD/JPY, to compare their out-of-sample forecasting performance with that of

our GARCH (1,1) benchmark model. This evaluation is conducted on both statistical and

¬nancial criteria in the following sections. Yet, one can easily see from Figures E4.1

and E4.2 in Appendix E that, for both the GBP/USD and the USD/JPY volatilities, these

out-of-sample forecasts do not suffer from the same degree of inertia as was the case for

the GARCH (1,1) forecasts.

4.5 MODEL COMBINATIONS AND FORECASTING

ACCURACY

4.5.1 Model combination

As noted by Dunis et al. (2001a), today most researchers would agree that individual

forecasting models are misspeci¬ed in some dimensions and that the identity of the “best”

model changes over time. In this situation, it is likely that a combination of forecasts

will perform better over time than forecasts generated by any individual model that is

kept constant.

Accordingly, we build two rather simple model combinations to add to our three existing

volatility forecasts, the GARCH (1,1), NNR and RNN forecasts.16

The simplest forecast combination method is the simple average of existing forecasts.

As noted by Dunis et al. (2001b), it is often a hard benchmark to beat as other methods,

such as regression-based methods, decision trees, etc., can suffer from a deterioration of

their out-of-sample performance.

We call COM1 the simple average of our GARCH (1,1), NNR and RNN volatility

forecasts with the actual implied volatility (IVOL21). As we know, implied volatility is

itself a popular method to measure market expectations of future volatility.

Another method of combining forecasts suggested by Granger and Ramanathan (1984)

is to regress the in-sample historical 21-day volatility on the set of forecasts to obtain

appropriate weights, and then apply these weights to the out-of-sample forecasts: it is

denoted GR. We follow Granger and Ramanathan™s advice to add a constant term and not

to constrain the weights to add to unity. We do not include both ANN and RNN forecasts in

the regression as they can be highly collinear: for the USD/JPY, the correlation coef¬cient

between both volatility forecasts is 0.984.

We tried several alternative speci¬cations for the Granger“Ramanathan approach. The

parameters were estimated by ordinary least squares over the in-sample data set. Our best

model for the GBP/USD volatility is presented below with t-statistics in parentheses, and

the R-squared and standard error of the regression:

Actualt,21 = ’5.7442 + 0.7382RNN44t,21 + 0.6750GARCH (1, 1)t,21 + 0.3226IVOLt,21

(’6.550) (7.712) (8.592) (6.777)

R 2 = 0.2805 S.E. of regression = 1.7129 (4.7a)

16

More sophisticated combinations are possible, even based on NNR models as in Donaldson and Kamstra

(1996), but this is beyond the scope of this chapter.

Forecasting and Trading Currency Volatility 143

For the USD/JPY volatility forecast combination, our best model was obtained using the

NNR forecast rather than the RNN one:

Actualt,21 = ’9.4293 + 1.5913NNR44t,21 + 0.06164GARCH (1, 1)t,21 + 0.1701IVOLt,21

(’7.091) (7.561) (0.975) (2.029)

R 2 = 0.4128 S.E. of regression = 4.0239 (4.7b)

As can be seen, the RNN/NNR-based forecast gets the highest weight in both cases,

suggesting that the GR forecast relies more heavily on the RNN/NNR model forecasts

than on the others. Figures F4.1 and F4.2 in Appendix F show that the GR and COM1

forecast combinations, as the NNR and RNN forecasts, do not suffer from the same

inertia as the GARCH (1,1) out-of-sample forecasts do. The Excel ¬le “CombGR JPY”

on the accompanying CD-Rom documents the computation of the two USD/JPY volatility

forecast combinations and that of their forecasting accuracy.

We now have ¬ve volatility forecasts on top of the implied volatility “market fore-

cast” and proceed to test their out-of-sample forecasting accuracy through traditional

statistical criteria.

4.5.2 Out-of-sample forecasting accuracy

As is standard in the economic literature, we compute the Root Mean Squared Error

(RMSE), the Mean Absolute Error (MAE) and Theil U-statistic (Theil-U). These measures

have already been presented in detail by, amongst others, Makridakis et al. (1983),

Pindyck and Rubinfeld (1998) and Theil (1966), respectively. We also compute a “correct

directional change” (CDC) measure which is described below.

ˆ

Calling σ the actual volatility and σ the forecast volatility at time „ , with a forecast

period going from t + 1 to t + n, the forecast error statistics are respectively:

t+n

RMSE = (σ„ ’ σ„ )2

ˆ

(1/n)

„ =t+1

t+n

MAE = (1/n) | σ„ ’ σ„ |

ˆ

„ =t+1

®

t+n t+n t+n

° (1/n) σ„2 »

Theil-U = (σ„ ’ σ„ )2

ˆ σ„2 +

ˆ

(1/n) (1/n)

„ =t+1 „ =t+1 „ =t+1

t+n

CDC = (100/n) D„

„ =t+1

where D„ = 1 if (σ„ ’ σ„ ’1 )(σ„ ’ σ„ ’1 ) > 0 else D„ = 0

ˆ

The RMSE and the MAE statistics are scale-dependent measures but give us a basis

to compare our volatility forecasts with the realised volatility. The Theil-U statistic is

144 Applied Quantitative Methods for Trading and Investment

independent of the scale of the variables and is constructed in such a way that it necessarily

lies between zero and one, with zero indicating a perfect ¬t.

For all these three error statistics retained the lower the output, the better the forecasting

accuracy of the model concerned. However, rather than on securing the lowest statistical

forecast error, the pro¬tability of a trading system critically depends on taking the right

position and therefore getting the direction of changes right. RMSE, MAE and Theil-U

are all important error measures, yet they may not constitute the best criterion from a

pro¬tability point of view. The CDC statistic is used to check whether the direction given

by the forecast is the same as the actual change which has subsequently occurred and,

for this measure, the higher the output the better the forecasting accuracy of the model

concerned. Tables 4.1 and 4.2 compare, for the GBP/USD and the USD/JPY volatility

respectively, our ¬ve volatility models and implied volatility in terms of the four accuracy

measures retained.

These results are most interesting. Except for the GARCH (1,1) model (for all criteria

for the USD/JPY volatility and in terms of directional change only for the GBP/USD

volatility), they show that our ¬ve volatility forecasting models offer much more precise

indications about future volatility than implied volatilities. This means that our volatility

forecasts may be used to identify mispriced options, and a pro¬table trading rule can

possibly be established based on the difference between the prevailing implied volatility

and the volatility forecast.

The two NNR/RNN models and the two combination models predict correctly direc-

tional change at least over 57% of the time for the USD/JPY volatility. Furthermore,

for both volatilities, these models outperform the GARCH (1,1) benchmark model on all

Table 4.1 GBP/USD volatility models forecasting accuracy

GBP/USD Vol. RMSE MAE Theil-U CDC

IVOL21 1.98 1.63 0.13 49.64

GARCH (1,1) 1.70 1.48 0.12 48.57

NNR (44-10-5-1) 1.69 1.42 0.12 50.00

RNN (44-1-1) 1.50 1.27 0.11 52.86

COM1 1.65 1.41 0.11 65.23

GR 1.67 1.37 0.12 67.74

USD/JPY volatility models forecasting accuracy17

Table 4.2

USD/JPY Vol. RMSE MAE Theil-U CDC

IVOL21 3.04 2.40 0.12 53.21

GARCH (1,1) 4.46 4.14 0.17 52.50

NNR (44-1-1) 2.41 1.88 0.10 59.64

RNN (44-5-1) 2.43 1.85 0.10 59.29

COM1 2.72 2.29 0.11 56.79

GR 2.70 2.13 0.11 57.86

17

The computation of the COM1 and GR forecasting accuracy measures is documented in the Excel ¬le

“CombGR’ JPY” on the accompanying CD-Rom.

Forecasting and Trading Currency Volatility 145

evaluation criteria. As a group, NNR/RNN models show superior out-of-sample forecast-

ing performance on any statistical evaluation criterion, except directional change for the

GBP/USD volatility for which they are outperformed by model combinations. Within this

latter group, the GR model performance is overall the best in terms of statistical fore-

casting accuracy. The GR model combination provides the best forecast of directional

change, achieving a remarkable directional forecasting accuracy of around 67% for the

GBP/USD volatility.

Still, as noted by Dunis (1996), a good forecast may be a necessary but it is certainly

not a suf¬cient condition for generating positive trading returns. Prediction accuracy is not

the ultimate goal in itself and should not be used as the main guiding selection criterion

for system traders. In the following section, we therefore use our volatility forecasting

models to identify mispriced foreign exchange options and endeavour to develop pro¬table

currency volatility trading models.

4.6 FOREIGN EXCHANGE VOLATILITY TRADING MODELS

4.6.1 Volatility trading strategies

Kroner et al. (1995) point out that, since expectations of future volatility play such a

critical role in the determination of option prices, better forecasts of volatility should

lead to a more accurate pricing and should therefore help an option trader to identify

over- or underpriced options. Therefore a pro¬table trading strategy can be established

based on the difference between the prevailing market implied volatility and the volatility

forecast. Accordingly, Dunis and Gavridis (1997) advocate to superimpose a volatility

trading strategy on the volatility forecast.

As mentioned previously, there is a narrow relationship between volatility and the

option price. An option embedding a high volatility gives the holder a greater chance of

a more pro¬table exercise. When trading volatility, using at-the-money forward (ATMF)

straddles, i.e. combining an ATFM call with an ATFM put with opposite deltas, results in

taking no forward risk. Furthermore, as noted, amongst others, by Hull (1997), both the

ATMF call and put have the same vega and gamma sensitivity. There is no directional bias.

If a large rise in volatility is predicted, the trader will buy both call and put. Although

this will entail paying two premia, the trader will pro¬t from a subsequent movement

in volatility: if the foreign exchange market moves far enough either up or down, one

of the options will end deeply in-the-money and, when it is sold back to the writing

counterparty, the pro¬t will more than cover the cost of both premia. The other option will

expire worthless. Conversely, if both the call and put expire out-of-the-money following

a period of stability in the foreign exchange market, only the premia will be lost.

If a large drop in volatility is predicted, the trader will sell the straddle and receive the

two option premia. This is a high-risk strategy if his market view is wrong as he might

theoretically suffer unlimited loss, but, if he is right and both options expire worthless,

he will have cashed in both premia.

4.6.2 The currency volatility trading models

The trading strategy adopted is based on the currency volatility trading model proposed

by Dunis and Gavridis (1997). A long volatility position is initiated by buying the 1-month

146 Applied Quantitative Methods for Trading and Investment

ATMF foreign exchange straddle if the 1-month volatility forecast is above the prevailing

1-month implied volatility level by more than a certain threshold used as a con¬rmation

¬lter or reliability indicator. Conversely, a short ATMF straddle position is initiated if the

1-month volatility forecast is below the prevailing implied volatility level by more than

the given threshold.

To this effect, the ¬rst stage of the currency volatility trading strategy is, based on

the threshold level as in Dunis (1996), to band the volatility predictions into ¬ve classes,

namely, “large up move”, “small up move”, “no change”, “large down move” and “small

down move” (Figure 4.4). The change threshold de¬ning the boundary between small and

large movements was determined as a con¬rmation ¬lter. Different strategies with ¬lters

ranging from 0.5 to 2.0 were analysed and are reported with our results.

The second stage is to decide the trading entry and exit rules. With our ¬lter rule,

a position is only initiated when the 1-month volatility forecast is above or below the

prevailing 1-month implied volatility level by more than the threshold. That is:

• If Dt > c, then buy the ATMF straddle

• If Dt < ’c, then sell the ATFM straddle

where Dt denotes the difference between the 1-month volatility forecast and the prevailing

1-month implied volatility, and c represents the threshold (or ¬lter).

In terms of exit rules, our main test is to assume that the straddle is held until expiry

and that no new positions can be initiated until the existing straddle has expired. As, due

to the drop in time value during the life of an option, this is clearly not an optimal trading

strategy, we also consider the case of American options which can be exercised at any

time until expiry, and thus evaluate this second strategy assuming that positions are only

held for ¬ve trading days (as opposed to one month).18

As in Dunis and Gavridis (1997), pro¬tability is determined by comparing the level

of implied volatility at the inception of the position with the prevailing 1-month realised

historical volatility at maturity.

It is further weighted by the amount of the position taken, itself a function of the

difference between the 1-month volatility forecast and the prevailing 1-month implied

volatility level on the day when the position is initiated: intuitively, it makes sense to

assume that, if we have a “good” model, the larger |Dt |, the more con¬dent we should

be about taking the suggested position and the higher the expected pro¬t. Calling G this

gearing of position, we thus have:19

G = |Dt |/|c| (4.8)

C = change threshold

’C C

0.0

Large down Small down Small up Large up

No change

Figure 4.4 Volatility forecasts classi¬cation

18

For the “weekly” trading strategy, we also considered closing out European options before expiry by taking

the opposite position, unwinding positions at the prevailing implied volatility market rate after ¬ve trading

days: this strategy was generally not pro¬table.

19

Laws and Gidman (2000) adopt a similar strategy with a slightly different de¬nition of the gearing.

Forecasting and Trading Currency Volatility 147

Pro¬tability is therefore de¬ned as a volatility net pro¬t (i.e. it is calculated in volatility

points or “vols” as they are called by options traders20 ). Losses are also de¬ned as

a volatility loss, which implies two further assumptions: when short the straddle, no

stop-loss strategy is actually implemented and the losing trade is closed out at the then

prevailing volatility level (it is thus reasonable to assume that we overestimate potential

losses in a real world environment with proper risk management controls); when long

the straddle, we approximate true losses by the difference between the level of implied

volatility at inception with the prevailing volatility level when closing out the losing trade,

whereas realised losses would only amount to the premium paid at the inception of the

position (here again, we seem to overestimate potential losses). It is further assumed that

volatility pro¬ts generated during one period are not reinvested during the next. Finally,

in line with Dunis and Gavridis (1997), transaction costs of 25 bp per trade are included

in our pro¬t and loss computations.

4.6.3 Trading simulation results

The currency volatility trading strategy was applied from 31 December 1993 to 9 May

2000. Tables 4.3 and 4.4 document our results for the GBP/USD and USD/JPY monthly

trading strategies both for the in-sample period from 31 December 1993 to 9 April 1999

and the out-of-sample period from 12 April 1999 to 9 May 2000. The evaluation discussed

below is focused on out-of-sample performance.

For our trading simulations, four different thresholds ranging from 0.5 to 2.0 and two

different holding periods, i.e. monthly and weekly, have been retained. A higher threshold

level implies requiring a higher degree of reliability in the signals and obviously reduces

the overall number of trades.

The pro¬tability criteria include the cumulative pro¬t and loss with and without gearing,

the total number of trades and the percentage of pro¬table trades. We also show the average

gearing of the positions for each strategy.

Firstly, we compare the performance of the NNR/RNN models with the benchmark

GARCH (1,1) model. For the GBP/USD monthly volatility trading strategy in Table 4.3,

the GARCH (1,1) model generally produces higher cumulative pro¬ts not only in-sample

but also out-of-sample. NNR/RNN models seldom produce a higher percentage of prof-

itable trades in-sample or out-of-sample, although the geared cumulative return of the

strategy based on the RNN (44-1-1) model is close to that produced with the bench-

mark model. With NNR/RNN models predicting more accurately directional change than

the GARCH model, one would have intuitively expected them to show a better trading

performance for the monthly volatility trading strategies.

This expected result is in fact achieved by the USD/JPY monthly volatility trading

strategy, as shown in Table 4.4: NNR/RNN models clearly produce a higher percentage

of pro¬table trades both in- and out-of-sample, with the best out-of-sample performance

being that based on the RNN (44-5-1) model. On the contrary, the GARCH (1,1) model-

based strategies produce very poor trading results, often recording an overall negative

cumulative pro¬t and loss ¬gure.

20

In market jargon, “vol” refers to both implied volatility and the measurement of volatility in percent per

annum (see, amongst others, Malz (1996)). Monetary returns could only be estimated by comparing the actual

pro¬t/loss of a straddle once closed out or expired against the premium paid/received at inception, an almost

impossible task with OTC options.

148 Applied Quantitative Methods for Trading and Investment

Table 4.3 GBP/USD monthly volatility trading strategy

1 Threshold = 0.5 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR

P/L without gearing 58.39% 54.47% 81.75% 54.25% 81.40% 5.77% 5.22% 12.90% 10.35% 11.98%

P/L with gearing 240.44% 355.73% 378.18% 182.38% 210.45% 16.60% 35.30% 42.91% 16.41% 19.44%

Total trades 59 59 61 58 61 12 12 11 10 10

Profitable trades 67.80% 70.00% 77.05% 70.69% 83.61% 50.00% 58.33% 72.73% 70.00% 80.00%

Average gearing 2.83 4.05 3.87 2.35 2.13 1.55 2.17 2.39 1.46 1.44

2 Threshold = 1.0 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR

P/L without gearing 61.48% 57.52% 82.61% 64.50% 60.24% 7.09% 12.74% 12.08% 8.33% 9.65%

P/L with gearing 134.25% 190.80% 116.99% 88.26% 8.65% 20.23% 20.79% 10.53% 11.03%

211.61%

Total trades 51 51 45 47 7 8 9 5 6

58

Profitable trades 72.55% 69.09% 80.00% 78.72% 85.71% 87.50% 66.67% 80.00% 83.33%

84.48%

Average gearing 1.68 2.21 2.08 1.53 1.32 1.18 1.61 1.48 1.19 1.13

3 Threshold = 1.5 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR

P/L without gearing 53.85% 61.13% 66.24% 62.26% 39.22% 9.26% 8.67% 8.93% 10.36% 8.69%

P/L with gearing 74.24% 114.88% 113.04% 80.65% 49.52% 11.16% 11.75% 11.32% 11.00% 9.92%

Total trades 40 40 52 31 24 4 6 6 3 2

Profitable trades 80.00% 71.43% 80.77% 83.87% 83.33% 100.00% 83.33% 83.33% 100% 100.00%

Average gearing 1.28 1.62 1.43 1.24 1.19 1.12 1.31 1.22 1.05 1.14

4 Threshold = 2.0 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR

P/L without gearing 69.03% 63.04% 60.57% 48.35% 20.97% 4.39% 7.80% 10.54% 4.39% -

P/L with gearing 103.33% 98.22% 85.19% 63.05% 24.25% 5.37% 8.71% 11.29% 4.70% -

Total trades 24 33 31 16 8 1 4 4 1 0

Profitable trades 82.76% 78.57% 79.49% 94.12% 88.89% 100.00% 100.00% 100.00% 100% -

Average gearing 1.35 1.40 1.24 1.25 1.16 1.22 1.09 1.06 1.06 -

Note: Cumulative P/L figures are expressed in volatility points.

Secondly, we evaluate the performance of model combinations. It is quite disappointing

as, for both monthly volatility trading strategies, model combinations produce on average

much lower cumulative returns than alternative strategies based on NNR/RNN models for

the USD/JPY volatility and on either the GARCH (1,1) or the RNN (44-1-1) model for

the GBP/USD volatility. As a general rule, the GR combination model fails to clearly

outperform the simple average model combination COM1 during the out-of-sample period,

something already noted by Dunis et al. (2001b).

Overall, with the monthly holding period, RNN model-based strategies show the

strongest out-of-sample trading performance: in terms of geared cumulative pro¬t, they

come ¬rst in four out of the eight monthly strategies analysed, and second best in the

remaining four cases. The strategy with the highest return yields a 106.17% cumulative

pro¬t over the out-of-sample period and is achieved for the USD/JPY volatility with the

RNN (44-5-1) model and a ¬lter equal to 0.5.

The results of the weekly trading strategy are presented in Tables G4.1 and G4.2 in

Appendix G. They basically con¬rm the superior performance achieved through the use

of RNN model-based strategies and the comparatively weak results obtained through the

use of model combination.

Forecasting and Trading Currency Volatility 149

Table 4.4 USD/JPY monthly volatility trading strategy

1 Threshold = 0.5 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR

’26.42% ’8.57%

P/L without gearing 31.35% 16.42% 19.15% 19.73% 16.21% 20.71% 16.19% 3.21%

’4.42%

P/L with gearing 151.79% 144.52% 6.93% 152.36% 82.92% 63.61% 106.17% 75.78% 11.50%

Total trades 62 62 60 60 61 13 13 12 12 13

Profitable trades 54.84% 51.61% 38.33% 53.33% 60.66% 76.92% 84.62% 50.00% 66.67% 61.54%

Average gearing 3.89 3.98 3.77 2.41 2.36 2.86 3.91 5.73 2.63 1.86

2 Threshold = 1.0 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR

’9.71% ’1.94%

P/L without gearing 25.83% 44.59% 40.60% 64.35% 20.70% 21.32% 13.53% 26.96%

P/L with gearing 67.66% 105.01% 76.72% 122.80% 52.81% 45.81% 42.14% 21.41% 46.53%

43.09%

Total trades 58 58 51 52 12 12 12 11 12

57

Profitable trades 62.07% 56.90% 58.82% 69.23% 83.33% 83.33% 41.67% 63.64% 83.33%

36.84%

Average gearing 2.01 2.16 1.97 1.58 1.55 1.97 1.94 3.03 1.41 1.66

3 Threshold = 1.5 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR

P/L without gearing 47.07% 23.62% 40.93% 46.63% 84.17% 19.65% 25.54% 1.87% 5.09% 23.80%

P/L with gearing 92.67% 75.69% 86.60% 75.70% 109.49% 40.49% 73.19% 31.31% 10.65% 32.91%

Total trades 51 51 46 37 42 10 10 12 6 10

Profitable trades 64.71% 55.77% 52.17% 59.46% 73.81% 80.00% 80.00% 33.33% 50% 90.00%

Average gearing 1.71 1.72 1.60 1.33 1.35 1.54 1.94 2.21 1.37 1.41

4 Threshold = 2.0 Trading days = 21

In-sample Out-of-sample

Sample

Observation (1-1329) (1330-1610)

Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)

Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR

P/L without gearing 52.52% 28.98% 39.69% 27.89% 75.99% 34.35% 33.70% 0.11% 7.74% 3.09%

’5.11% 12.35%

P/L with gearing 202.23% 72.21% 49.77% 35.77% 94.30% 60.33% 64.76% 5.48%

Total trades 37 37 41 21 26 10 10 12 3 7

Profitable trades 59.46% 61.54% 56.10% 61.90% 80.77% 90.00% 90.00% 33% 100% 71%

Average gearing 1.67 1.59 1.39 1.25 1.26 1.54 1.68 1.54 1.54 1.23

Note: Cumulative P/L figures are expressed in volatility points.