examined thepredictive power of implied standard deviation as a volatility forecasting tool with dailydata, Dunis et al. (2001b) measured, using daily data, both the 1-month and 3-monthforecasting ability of 13 different volatility models including AR, GARCH, stochasticvariance and model combinations with and without the adding of implied volatility as anextra 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-sampleforecasting performance of their models using traditional statistical accuracy criteria, suchas root mean squared error, mean absolute error, mean absolute percentage error, TheilU statistic and correct directional change prediction. Investors and market participantshowever have trading performance as their ultimate goal and will select a forecastingmodel based on ﬁnancial criteria rather than on some statistical criterion such as rootmean squared error minimisation. Yet, as mentioned above, seldom has recently publishedresearch applied any ﬁnancial utility criterion in assessing the out-of-sample performanceof volatility models.Over the past few years, Neural Network Regression (NNR) has been widely advocatedas a new alternative modelling technology to more traditional econometric and statisticalapproaches, claiming increasing success in the ﬁelds of economic and ﬁnancial forecasting. This has resulted in many publications comparing neural networks and traditionalforecasting approaches. In the case of foreign exchange markets, it is worth pointing outthat most of the published research has focused on exchange rate forecasting rather thanon currency volatility forecasts. However, ﬁnancial criteria, such as Sharpe ratio, profitability, return on equity, maximum drawdown, etc., have been widely used to measure
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and quantify the out-of-sample forecasting performance. Dunis (1996) investigated theapplication of NNR to intraday foreign exchange forecasting and his results were evaluated by means of a trading strategy. Kuan and Liu (1995) proposed two-step RecurrentNeural Network (RNN) models to forecast exchange rates and their results were evaluated using traditional statistical accuracy criteria. Tenti (1996) applied RNNs to predict theUSD/DEM exchange rate, devising a trading strategy to assess his results, while Fransesand Van Homelen (1998) use NNR models to predict four daily exchange rate returnsrelative to the Dutch guilder using directional accuracy to assess out-of-sample forecasting accuracy. Overall, it seems however that neural network research applied to exchangerates has been so far seldom devoted to FX volatility forecasting.Accordingly, the rationale for this chapter is to investigate the predictive power ofalternative nonparametric forecasting models of foreign exchange volatility, both froma statistical and an economic point of view. We examine the use of NNR and RNNregression models for forecasting and trading currency volatility, with an application tothe GBP/USD and USD/JPY exchange rates. The results of the NNR and RNN models arebenchmarked against the simpler GARCH (1,1) alternative, implied volatility and modelcombinations: in terms of model combination, a simple average combination and theGranger–Ramanathan (1984) optimal weighting regression-based approach are employedand their results investigated.Using daily data from December 1993 through April 1999, we develop alternative FXvolatility forecasting models. These models are then tested out-of-sample over the periodApril 1999–May 2000, not only in terms of forecasting accuracy, but also in terms oftrading efﬁciency. In order to do so, we apply a realistic volatility trading strategy usingFX option straddles once mispriced options have been identiﬁed.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 producesuperior trading strategies.Another conclusion from our results is that, for the period and currencies considered,the currency option market was inefﬁcient and/or the pricing formulae applied by marketparticipants were inadequate.Overall, we depart from existing work in several respects.Firstly, we develop alternative nonparametric FX volatility models, applying in particular an RNN architecture with a loop back from the output layer implying an errorfeedback mechanism, i.e. we apply a nonlinear error-correction modelling approach toFX volatility.Secondly, we apply our nonparametric models to FX volatility, something that has notbeen done so far. A recent development in the literature has been the application ofnonparametric time series modelling approaches to volatility forecasts. Gaussian kernelregression is an example, as in West and Cho (1995). Neural networks have also beenfound useful in modelling the properties of nonlinear time series. As mentioned above, ifthere are quite a few articles on applications of NNR models to foreign exchange, stockand commodity markets,1 there are rather few concerning ﬁnancial markets volatility1For NNR applications to commodity forecasting, see, for instance, Ntungo and Boyd (1998) and Trippi andTurban (1993). For applications to the stock market, see, amongst others, Deboeck (1994) and Leung et al.(2000).
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forecasting in general.2 It seems therefore that, as an alternative technique to more traditional statistical forecasting methods, NNR models need further investigation to checkwhether or not they can add value in the ﬁeld of foreign exchange volatility forecasting.Finally, unlike previous work, we do not limit ourselves to forecasting accuracy butextend the analysis to the all-important trading efﬁciency, taking advantage of the factthat there exists a large and liquid FX implied volatility market that enables us to applysophisticated volatility trading strategies.The chapter is organised as follows. Section 4.2 describes our exchange rate and volatility data. Section 4.3 brieﬂy presents the GARCH (1,1) model and gives the corresponding21-day volatility forecasts. Section 4.4 provides a detailed overview and explains the procedures and methods used in applying the NNR and RNN modelling procedure to ourﬁnancial time series, and it presents the 21-day volatility forecasts obtained with thesemethods. Section 4.5 brieﬂy describes the model combinations retained and assesses the21-day out-of-sample forecasts using traditional statistical accuracy criteria. Section 4.6introduces the volatility trading strategy using FX option straddles that we follow oncemispriced options have been identiﬁed through the use of our most successful volatilityforecasting models. We present detailed trading results allowing for transaction costs anddiscuss their implications, particularly in terms of a qualiﬁed assessment of the efﬁciencyof the currency options market. Finally, Section 4.7 provides some concluding commentsand suggestions for further work.
4.2 THE EXCHANGE RATE AND VOLATILITY DATAThe motivation for this research implies that the success or failure to develop proﬁtablevolatility trading strategies clearly depends on the possibility to generate accurate volatilityforecasts and thus to implement adequate volatility modelling procedures.Numerous studies have documented the fact that logarithmic returns of exchange ratetime series exhibit “volatility clustering” properties, that is periods of large volatilitytend to cluster together followed by periods of relatively lower volatility (see, amongstothers, Baillie and Bollerslev (1990), Kroner et al. (1995) and Jorion (1997)). Volatilityforecasting crucially depends on identifying the typical characteristics of volatility withinthe 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 modiﬁcationsto the original series we have made where appropriate.4.2.1 The exchange rate series databank and historical volatilityThe return series we use for the GBP/USD and USD/JPY exchange rates were extractedfrom a historical exchange rate database provided by Datastream. Logarithmic returns,deﬁned 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 theexchange rates considered, i.e. st = 100 log(St /St−1 ).2Even though there are no NNR applications yet to foreign exchange volatility forecasting, some researchershave used NNR models to measure the stock market volatility (see, for instance, Donaldson and Kamstra (1997)and Bartlmae and Rauscher (2000)).
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Our exchange rate databank spans from 31 December 1993 to 9 May 2000, givingus 1610 observations per exchange rate.3 This databank was divided into two separatesets with the ﬁrst 1329 observations from 31 December 1993 to 9 April 1999 deﬁned asour in-sample testing period and the remaining 280 observations from 12 April 1999 to9 May 2000 being used for out-of-sample forecasting and validation.In line with the ﬁndings of many earlier studies on exchange rate changes (see, amongstothers, Engle and Bollerslev (1986), Baillie and Bollerslev (1989), Hsieh (1989), Westand Cho (1995)), the descriptive statistics of our currency returns (not reported here inorder to conserve space) clearly show that they are nonnormally distributed and heavilyfat-tailed. They also show that mean returns are not statistically different from zero.Further standard tests of autocorrelation, nonstationarity and heteroskedasticity show thatlogarithmic returns are all stationary and heteroskedastic. Whereas there is no evidenceof autocorrelation for the GBP/USD return series, some autocorrelation is detected at the10% signiﬁcance level for USD/JPY returns.The fact that our currency returns have zero unconditional mean enables us to usesquared returns as a measure of their variance and absolute returns as a measure of theirstandard deviation or volatility.4 The standard tests of autocorrelation, nonstationarity andheteroskedasticity (again not reported here in order to conserve space) show that squaredand absolute currency returns series for the in-sample period are all nonnormally distributed, stationary, autocorrelated and heteroskedastic (except USD/JPY squared returnswhich were found to be homoskedastic).Still, as we are interested in analysing alternative volatility forecasting models andwhether they can add value in terms of forecasting realised currency volatility, we mustadjust our statistical computation of volatility to take into account the fact that, even ifit is only the matter of a constant, in currency options markets, volatility is quoted inannualised terms. As we wish to focus on 1-month volatility forecasts and related tradingstrategies, taking, as is usual practice, a 252-trading day year (and consequently a 21trading day month), we compute the 1-month volatility as the moving annualised standarddeviation of our logarithmic returns and end up with the following historical volatilitymeasures for the 1-month horizon:1  √( 252 × |st |)21 t−20t
σt =
where |st | is the absolute currency return.5 The value σt is the realised 1-month exchangerate volatility that we are interested in forecasting as accurately as possible, in order tosee if it is possible to ﬁnd any mispriced option that we could possibly take advantage of.The descriptive statistics of both historical volatility series (again not reported herein order to conserve space) show that they are nonnormally distributed and fat-tailed.Further statistical tests of autocorrelation, heteroskedasticity and nonstationarity showthat they exhibit strong autocorrelation but that they are stationary in levels. Whereas3Actually, we used exchange rate data from 01/11/1993 to 09/05/2000, the data during the period 01/11/1993to 31/12/1993 being used for the “pre-calculation” of the 21-day realised historical volatility.4Although the unconditional mean is zero, it is of course possible that the conditional mean may vary overtime.5The use of absolute returns (rather than their squared value) is justiﬁed by the fact that with zero unconditionalmean, averaging absolute returns gives a measure of standard deviation.
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GBP/USD historical volatility is heteroskedastic, USD/JPY realised volatility was foundto be homoskedastic.Having presented our exchange rate series databank and explained how we compute ourhistorical volatilities from these original series (so that they are in a format comparableto that which prevails in the currency options market), we now turn our attention to theimplied volatility databank that we have used.4.2.2 The implied volatility series databankVolatility has now become an observable and traded quantity in ﬁnancial markets, and particularly so in the currency markets. So far, most studies dealing with implied volatilitieshave used volatilities backed out from historical premium data on traded options ratherthan over-the-counter (OTC) volatility data (see, amongst others, Latane and Rendleman (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 calland put prices are only available for given strike levels and ﬁxed maturity dates. Thecorresponding implied volatility series must therefore be backed out using a speciﬁcoption pricing model. This procedure generates two sorts of potential biases: materialerrors or mismatches can affect the variables that are needed for the solving of thepricing model, e.g. the forward points or the spot rate, and, more importantly, the veryspeciﬁcation of the pricing model that is chosen can have a crucial impact on the ﬁnal“backed out” implied volatility series.This is the reason why, in this chapter, we use data directly observable on the marketplace. This original approach seems further warranted by current market practice wherebybrokers and market makers in currency options deal in fact in volatility terms and not inoption premium terms any more.6 The volatility time series we use for the two exchangerates selected, GBP/USD and USD/JPY, were extracted from a market quoted impliedvolatilities database provided by Chemical Bank for data until end-1996, and updated fromReuters “Ric” codes subsequently. These at-the-money forward, market-quoted volatilitiesare in fact obtained from brokers by Reuters on a daily basis, at the close of businessin London.These implied volatility series are nonnormally distributed and fat-tailed. Further statistical tests of autocorrelation and heteroskedasticity (again not reported here in orderto 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 impliedvolatilities are stationary at the 5% signiﬁcance level.Certainly, as noted by Dunis et al. (2001b) and conﬁrmed in Tables A4.1 and A4.3 inAppendix A for the GBP/USD and USD/JPY, an interesting feature is that the mean levelof implied volatilities stands well above average historical volatility levels.7 This tendencyof the currency options market to overestimate actual volatility is further documented6The market data that we use are at-the-money forward volatilities, as the use of either in-the-money or out-ofthe-money volatilities would introduce a signiﬁcant 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 clearthat these implied volatilities are not simply backed out of an option pricing model but are instead directlyquoted from brokers. Due to arbitrage they cannot diverge too far from the theoretical level.7As noted by Dunis et al. (2001b), a possible explanation for implied volatility being higher than its historicalcounterpart may be due to the fact that market makers are generally options sellers (whereas end users are
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by Figures A4.1 and A4.2 which show 1-month actual and implied volatilities for theGBP/USD and USD/JPY exchange rates. These two charts also clearly show that, foreach exchange rate concerned, actual and implied volatilities are moving rather closelytogether, which is further conﬁrmed by Tables A4.2 and A4.4 for both GBP/USD andUSD/JPY volatilities.
4.3 THE GARCH (1,1) BENCHMARK VOLATILITYFORECASTS4.3.1 The choice of the benchmark modelAs the GARCH model originally devised by Bollerslev (1986) and Taylor (1986) is welldocumented in the literature, we just present it very brieﬂy, as it has now become widelyused, 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 outsidethe scope of this chapter. Besides, there is a wide consensus, certainly among marketpractitioners, but among many researchers as well that, when variants of the standardGARCH (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 boththe GBP/USD and USD/JPY exchange rates as it embodies a compact representationand serves well our purpose of ﬁnding an adequate benchmark for the more complexNNR models.In its simple GARCH (1,1) form, the GARCH model basically states that the conditionalvariance of asset returns in any given period depends upon a constant, the previousperiod’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 and2εt−1 the squared random component of the return in the previous period, for a standardGARCH (1,1) process, we have:22+ βσt−1σt2 = ω + αεt−1
(4.1)
Equation (4.1) yields immediately the 1-step ahead volatility forecast and, using recursivesubstitution, Engle and Bollerslev (1986) and Baillie and Bollerslev (1992) give the n-stepahead forecast for a GARCH (1,1) process:2σt+n= ω[1 + (α + β) + · · · + (α + β)n−2 ] + ω + αεt2 + βσt2
(4.2)
This is the formula that we use to compute our GARCH (1,1) n-step ahead out-ofsample forecast.4.3.2 The GARCH (1,1) volatility forecastsIf many researchers have noted that no alternative GARCH speciﬁcation could consistentlyoutperform the standard GARCH (1,1) model, some such as Bollerslev (1987), Bailliemore 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 arestochastic, then the implied volatility will capture both asset price volatility and interest rate volatility, thusskewing implied volatility upwards, and (ii) the fact that if volatility is stochastic but the option pricing formulais constant, then this additional source of volatility will be picked up by the implied volatility.
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and Bollerslev (1989) and Hsieh (1989), amongst others, point out that the Student-tdistribution ﬁts the daily exchange rate logarithmic returns better than conditional normality, as the former is characterised by fatter tails. We thus generate GARCH (1,1) 1-stepahead forecasts with the Student-t distribution assumption.8 We give our results for theGBP/USD exchange rate:log(St /St−1 ) = εtεt |ϕt−1 ∼ N(0, σt2 )22σt2 = 0.0021625 + 0.032119εt−1+ 0.95864σt−1
(0.0015222)
(0.010135)
(0.013969)
(4.3)
where the ﬁgures in parentheses are asymptotic standard errors. The t-values for α and22β are highly signiﬁcant and show strong evidence that σt2 varies with εt−1and σt−1. Thecoefﬁcients also have the expected sign. Additionally, the conventional Wald statistic fortesting the joint hypothesis that α = β = 0 clearly rejects the null, suggesting a signiﬁcantGARCH effect.The parameters in equation (4.3) were used to estimate the 21-day ahead volatilityforecast for the USD/GBP exchange rate: using the 1-step ahead GARCH (1,1) coefﬁcients, the conditional 21-day volatility forecast was generated each day according toequation (4.2) above. The same procedure was followed for the USD/JPY exchange ratevolatility (see Appendix B4.3).Figure 4.1 displays the GARCH (1,1) 21-day volatility forecasts for the USD/GBPexchange rate both in- and out-of-sample (the last 280 observations, from 12/04/1999 toGARCH Vol. Forecast
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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 onlyslightly 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/GBPdetailed results).
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09/05/2000). It is clear that, overall, the GARCH model ﬁts the realised volatility ratherwell during the in-sample period. However, during the out-of-sample period, the GARCHforecasts are quite disappointing. The USD/JPY out-of-sample GARCH (1,1) forecastssuffer 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 ofdaily exchange rate returns such as leptokurtosis and conditional heteroskedasticity, itsability 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 networkmodels we intend to apply and test whether NNR/RNN models can produce a substantialimprovement in the out-of-sample performance of our volatility forecasts.
4.4 THE NEURAL NETWORK VOLATILITY FORECASTS4.4.1 NNR modellingOver the past few years, it has been argued that new technologies and quantitative systems based on the fact that most ﬁnancial time series contain nonlinearities have madetraditional forecasting methods only second best. NNR models, in particular, have beenapplied with increasing success to economic and ﬁnancial forecasting and would constitutethe 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 artiﬁcialneural networks, their biological foundation and their many architectures and potential applications (for more details, see, amongst others, Simpson (1990) and Hassoun(1995)).9For our purpose, let it sufﬁce to say that NNR models are a tool for determiningthe relative importance of an input (or a combination of inputs) for predicting a givenoutcome. 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 transferfunction. 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] isthe 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 theconnection between the input x [i] and the processing unit bj . Each node may additionallyhave also a bias input θj modulated with the weight w0j associated with the inputs. Thetotal input of the node bj is formally the dot product between the input vector A and theweight vector Wj , minus the weighted input bias. It is then passed through a nonlineartransfer function to produce the output value of the processing unit bj :
bj = f
 n
x wij − w0j θj[i]
= f (Xj )
(4.4)
i=1
9In this chapter, we use exclusively the multilayer perceptron, a multilayer feedforward network trained byerror backpropagation.
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In this chapter, we have used the sigmoid function as activation function:10f (Xj ) =
11 + e−Xj
(4.5)
Figure 4.2 allows one to visualise a single output NNR model with one hidden layer andtwo hidden nodes, i.e. a model similar to those we developed for the GBP/USD and theUSD/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 timet is ỹt , whereas the actual output is yt .At the beginning, the modelling process is initialised with random values for theweights. The output value of the processing unit bj is then passed on to the singleoutput node of the output layer. The NNR error, i.e. the difference between the NNRforecast and the actual value, is analysed through the root mean squared error. The latteris systematically minimised by adjusting the weights according to the level of its derivative with respect to these weights. The adjustment obviously takes place in the directionthat 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 generalisationcapabilities but are also much harder to train. In summary, NNR model results dependcrucially on the choice of the number of hidden layers, the number of nodes and the typeof nonlinear transfer function retained.In fact, the use of NNR models further enlarges the forecaster’s toolbox of availabletechniques by adding models where no speciﬁc functional form is a priori assumed.11Following Cybenko (1989) and Hornik et al. (1989), it can be demonstrated that speciﬁcNNR models, if their hidden layer is sufﬁciently large, can approximate any continuous
x t[1]x t[2]
Σ
∫
ht[1]Σ
x t[3]
Σ
∫
ht[2]
∫
y∼tyt
x t[4]x t[5]
Figure 4.2
Single output NNR model
10Other alternatives include the hyperbolic tangent, the bilogistic sigmoid, etc. A linear activation function isalso a possibility, in which case the NNR model will be linear. Note that our choice of a sigmoid impliesvariations in the interval ]0, +1[. Input data are thus normalised in the same range in order to present thelearning algorithm with compatible values and avoid saturation problems.11Strictly speaking, the use of an NNR model implies assuming a functional form, namely that of the transfer function.
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function.12 Furthermore, it can be shown that NNR models are equivalent to nonlinearnonparametric models, i.e. models where no decisive assumption about the generatingprocess must be made in advance (see Cheng and Titterington (1994)).Kouam et al. (1992) have shown that most forecasting models (ARMA models, bilinear models, autoregressive models with thresholds, nonparametric models with kernelregression, etc.) are embedded in NNR models. They show that each modelling procedurecan in fact be written in the form of a network of neurons.Theoretically, the advantage of NNR models over other forecasting methods can therefore be summarised as follows: as, in practice, the “best” model for a given problemcannot be determined, it is best to resort to a modelling strategy which is a generalisationof a large number of models, rather than to impose a priori a given model speciﬁcation.This has triggered an ever-increasing interest for applications to ﬁnancial 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 rateforecasting has been the topic of several recent papers: Kuan and Liu (1995), Swansonand White (1995) and Gençay (1996) show that NNR models can describe in-sampledata rather well and that they also generate “good” out-of-sample forecasts. Forecastingaccuracy is usually deﬁned in terms of small mean squared prediction error or in terms ofdirectional accuracy of the forecasts. However, as mentioned already, there are still veryfew studies concerned with ﬁnancial assets volatility forecasting.4.4.2 RNN modellingRNN 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 theintermediate layer, to the input layer. Depending on whether the loop back comes fromthe intermediate or the output layer, either the preceding values of the hidden nodes or theoutput error will be used as inputs in the next period. This feature, which seems welcomein the case of a forecasting exercise, comes at a cost: RNN models will require moreconnections than their NNR counterparts, thus accentu