ating a certain lack of transparencywhich is sometimes used to criticise these modelling approaches.Using our previous notation and assuming the output layer is the one looped back, theRNN model output at time t depends on the inputs at time t and on the output at timet − 1:13ỹt = F (xt , ỹt−1 )(4.6)There is no theoretical answer as to whether one should preferably loop back the intermediate or the output layer. This is mostly an empirical question. Nevertheless, as looping backthe output layer implies an error feedback mechanism, such RNN models can successfullybe used for nonlinear error-correction modelling, as advocated by Burgess and Refenes12This very feature also explains why it is so difﬁcult to use NNR models, as one may in fact end up ﬁttingthe noise in the data rather than the underlying statistical process.13With a loop back from the intermediate layer, the RNN output at time t depends on the inputs at time tand on the intermediate nodes at time t − 1. Besides, the intermediate nodes at time t depend on the inputsat time t and on the hidden layer at time t − 1. Using our notation, we have therefore: ỹt = F (xt , ht−1 ) andht = G(xt , ht−1 ).
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Applied Quantitative Methods for Trading and Investmentx t[1]Σ
x t[2]
∫
ht[1]Σ
x t[3]
Σ
∫
ht[2]
∫
y∼t
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 modellingstrategy for the GBP/USD and the USD/JPY volatility forecasts. Our choice seems further warranted by claims from Kuan and Liu (1995) and Tenti (1996) that RNN modelsare superior to NNR models when modelling exchange rates.Figure 4.3 allows one to visualise a single output RNN model with one hidden layerand two hidden nodes, again a model similar to those developed for the GBP/USD andthe USD/JPY volatility forecasts.4.4.3 The NNR/RNN volatility forecasts4.4.3.1 Input selection, data scaling and preprocessingIn the absence of an indisputable theory of exchange rate volatility, we assume that aspeciﬁc exchange rate volatility can be explained by that rate’s recent evolution, volatility spillovers from other ﬁnancial markets, and macroeconomic and monetary policyexpectations.In the circumstances, it seems reasonable to include, as potential inputs, exchange ratevolatilities (including that which is to be modelled), the evolution of important stock andcommodity prices, and, as a measure of macroeconomic and monetary policy expectations,the evolution of the yield curve.14As explained above (see footnote 10), all variables were normalised according to ourchoice of the sigmoid activation function. They had been previously transformed in logarithmic returns.15Starting from a traditional linear correlation analysis, variable selection was achievedvia a forward stepwise neural regression procedure: starting with both lagged historicaland implied volatility levels, other potential input variables were progressively added,keeping the network architecture constant. If adding a new variable improved the level ofexplained variance over the previous “best” model, the pool of explanatory variables wasupdated. If there was a failure to improve over the previous “best” model after several14On the use of the yield curve as a predictor of future output growth and inﬂation, see, amongst others, Fama(1990) and Ivanova et al. (2000).15Despite some contrary opinions, e.g. Balkin (1999), stationarity remains important if NNR/RNN models areto be assessed on the basis of the level of explained variance.
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attempts, variables in that model were alternated to check whether no better solution couldbe achieved. The model chosen ﬁnally was then kept for further tests and improvements.Finally, conforming with standard heuristics, we partitioned our total data set into threesubsets, using roughly 2/3 of the data for training the model, 1/6 for testing and theremaining 1/6 for validation. This partition in training, test and validation sets is madein order to control the error and reduce the risk of overﬁtting. Both the training andthe following test period are used in the model tuning process: the training set is used todevelop the model; the test set measures how well the model interpolates over the trainingset and makes it possible to check during the adjustment whether the model remains validfor the future. As the ﬁne-tuned system is not independent from the test set, the use ofa third validation set which was not involved in the model’s tuning is necessary. Thevalidation set is thus used to estimate the actual performance of the model in a deployedenvironment.In our case, the 1329 observations from 31/12/1993 to 09/04/1999 were consideredas the in-sample period for the estimation of our GARCH (1,1) benchmark model. Wetherefore retain the ﬁrst 1049 observations from 31/12/1993 to 13/03/1998 for the trainingset and the remainder of the in-sample period is used as test set. The last 280 observationsfrom 12/04/1999 to 09/05/2000 constitute the validation set and serve as the out-of-sampleforecasting period. This is consistent with the GARCH (1,1) model estimation.4.4.3.2 Volatility forecasting resultsWe 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 variablesincluded the lagged actual 21-day realised volatility (Realised21t−21 ), the laggedimplied 21-day volatility (IVOL21t−21 ), lagged absolute logarithmic returns of theexchange 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), dependingon the currency volatility being modelled.In terms of the ﬁnal model selection, Tables D4.1a and D4.1b in Appendix D give theperformance of the best NNR and RNN models over the validation (out-of-sample) dataset for the USD/GBP volatility. For the same input space and architecture (i.e. with onlyone hidden layer), RNN models marginally outperform their NNR counterparts in termsof directional accuracy. This is important as trading proﬁtability crucially depends ongetting the direction of changes right. Tables D4.1a and D4.1b also compare models withonly one hidden layer and models with two hidden layers while keeping the input andoutput variables unchanged: despite the fact that the best NNR model is a two-hiddenlayer model with respectively ten and ﬁve hidden nodes in each of its hidden layers, onaverage, NNR/RNN models with a single hidden layer perform marginally better whileat the same time requiring less processing time.The results of the NNR and RNN models for the USD/JPY volatility over the validation period are given in Tables D4.2a and D4.2b in Appendix D. They are in linewith those for the GBP/USD volatility, with RNN models outperforming their NNRcounterparts and, in that case, the addition of a second hidden layer rather deterioratingperformance.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)
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for the USD/JPY, to compare their out-of-sample forecasting performance with that ofour GARCH (1,1) benchmark model. This evaluation is conducted on both statistical andﬁnancial criteria in the following sections. Yet, one can easily see from Figures E4.1and E4.2 in Appendix E that, for both the GBP/USD and the USD/JPY volatilities, theseout-of-sample forecasts do not suffer from the same degree of inertia as was the case forthe GARCH (1,1) forecasts.
4.5 MODEL COMBINATIONS AND FORECASTINGACCURACY4.5.1 Model combinationAs noted by Dunis et al. (2001a), today most researchers would agree that individualforecasting models are misspeciﬁed in some dimensions and that the identity of the “best”model changes over time. In this situation, it is likely that a combination of forecastswill perform better over time than forecasts generated by any individual model that iskept constant.Accordingly, we build two rather simple model combinations to add to our three existingvolatility forecasts, the GARCH (1,1), NNR and RNN forecasts.16The 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 oftheir out-of-sample performance.We call COM1 the simple average of our GARCH (1,1), NNR and RNN volatilityforecasts with the actual implied volatility (IVOL21). As we know, implied volatility isitself 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 obtainappropriate weights, and then apply these weights to the out-of-sample forecasts: it isdenoted GR. We follow Granger and Ramanathan’s advice to add a constant term and notto constrain the weights to add to unity. We do not include both ANN and RNN forecasts inthe regression as they can be highly collinear: for the USD/JPY, the correlation coefﬁcientbetween both volatility forecasts is 0.984.We tried several alternative speciﬁcations for the Granger–Ramanathan approach. Theparameters were estimated by ordinary least squares over the in-sample data set. Our bestmodel for the GBP/USD volatility is presented below with t-statistics in parentheses, andthe 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)R = 0.28052
(7.712)
(8.592)
S.E. of regression = 1.7129
(6.777)(4.7a)
16More sophisticated combinations are possible, even based on NNR models as in Donaldson and Kamstra(1996), but this is beyond the scope of this chapter.
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For the USD/JPY volatility forecast combination, our best model was obtained using theNNR forecast rather than the RNN one:Actualt,21 = −9.4293 + 1.5913NNR44t,21 + 0.06164GARCH (1, 1)t,21 + 0.1701IVOLt,21(−7.091)R = 0.41282
(7.561)
(0.975)
(2.029)
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 forecaststhan on the others. Figures F4.1 and F4.2 in Appendix F show that the GR and COM1forecast combinations, as the NNR and RNN forecasts, do not suffer from the sameinertia as the GARCH (1,1) out-of-sample forecasts do. The Excel ﬁle “CombGR JPY”on the accompanying CD-Rom documents the computation of the two USD/JPY volatilityforecast combinations and that of their forecasting accuracy.We now have ﬁve volatility forecasts on top of the implied volatility “market forecast” and proceed to test their out-of-sample forecasting accuracy through traditionalstatistical criteria.4.5.2 Out-of-sample forecasting accuracyAs 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 measureshave already been presented in detail by, amongst others, Makridakis et al. (1983),Pindyck and Rubinfeld (1998) and Theil (1966), respectively. We also compute a “correctdirectional change” (CDC) measure which is described below.Calling σ the actual volatility and σ̂ the forecast volatility at time τ , with a forecastperiod going from t + 1 to t + n, the forecast error statistics are respectively:t+nRMSE = (1/n)(σ̂τ − στ )2τ =t+1
MAE = (1/n)
t+n
|σ̂τ − στ |
τ =t+1
 t+nt+nt+n(1/n)Theil-U = (1/n)(σ̂τ − στ )2σ̂ 2 + (1/n)σ 2τ
τ =t+1
CDC = (100/n)
t+n
τ =t+1
τ
τ =t+1
Dτ
τ =t+1
where Dτ = 1 if (στ − στ −1 )(σ̂τ − στ −1 ) > 0 else Dτ = 0The RMSE and the MAE statistics are scale-dependent measures but give us a basisto compare our volatility forecasts with the realised volatility. The Theil-U statistic is
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independent of the scale of the variables and is constructed in such a way that it necessarilylies between zero and one, with zero indicating a perfect ﬁt.For all these three error statistics retained the lower the output, the better the forecastingaccuracy of the model concerned. However, rather than on securing the lowest statisticalforecast error, the proﬁtability of a trading system critically depends on taking the rightposition and therefore getting the direction of changes right. RMSE, MAE and Theil-Uare all important error measures, yet they may not constitute the best criterion from aproﬁtability point of view. The CDC statistic is used to check whether the direction givenby 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 modelconcerned. Tables 4.1 and 4.2 compare, for the GBP/USD and the USD/JPY volatilityrespectively, our ﬁve volatility models and implied volatility in terms of the four accuracymeasures retained.These results are most interesting. Except for the GARCH (1,1) model (for all criteriafor the USD/JPY volatility and in terms of directional change only for the GBP/USDvolatility), they show that our ﬁve volatility forecasting models offer much more preciseindications about future volatility than implied volatilities. This means that our volatilityforecasts may be used to identify mispriced options, and a proﬁtable trading rule canpossibly be established based on the difference between the prevailing implied volatilityand the volatility forecast.The two NNR/RNN models and the two combination models predict correctly directional 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 allTable 4.1
GBP/USD volatility models forecasting accuracy
GBP/USD Vol.
RMSE
MAE
Theil-U
CDC
IVOL21GARCH (1,1)NNR (44-10-5-1)RNN (44-1-1)COM1GR
1.981.701.691.501.651.67
1.631.481.421.271.411.37
0.130.120.120.110.110.12
49.6448.5750.0052.8665.2367.74
Table 4.2
USD/JPY volatility models forecasting accuracy17
USD/JPY Vol.
RMSE
MAE
Theil-U
CDC
IVOL21GARCH (1,1)NNR (44-1-1)RNN (44-5-1)COM1GR
3.044.462.412.432.722.70
2.404.141.881.852.292.13
0.120.170.100.100.110.11
53.2152.5059.6459.2956.7957.86
17The computation of the COM1 and GR forecasting accuracy measures is documented in the Excel ﬁle“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 forecasting performance on any statistical evaluation criterion, except directional change for theGBP/USD volatility for which they are outperformed by model combinations. Within thislatter group, the GR model performance is overall the best in terms of statistical forecasting accuracy. The GR model combination provides the best forecast of directionalchange, achieving a remarkable directional forecasting accuracy of around 67% for theGBP/USD volatility.Still, as noted by Dunis (1996), a good forecast may be a necessary but it is certainlynot a sufﬁcient condition for generating positive trading returns. Prediction accuracy is notthe ultimate goal in itself and should not be used as the main guiding selection criterionfor system traders. In the following section, we therefore use our volatility forecastingmodels to identify mispriced foreign exchange options and endeavour to develop proﬁtablecurrency volatility trading models.
4.6 FOREIGN EXCHANGE VOLATILITY TRADING MODELS4.6.1 Volatility trading strategiesKroner et al. (1995) point out that, since expectations of future volatility play such acritical role in the determination of option prices, better forecasts of volatility shouldlead to a more accurate pricing and should therefore help an option trader to identifyover- or underpriced options. Therefore a proﬁtable trading strategy can be establishedbased on the difference between the prevailing market implied volatility and the volatilityforecast. Accordingly, Dunis and Gavridis (1997) advocate to superimpose a volatilitytrading strategy on the volatility forecast.As mentioned previously, there is a narrow relationship between volatility and theoption price. An option embedding a high volatility gives the holder a greater chance ofa more proﬁtable 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 intaking no forward risk. Furthermore, as noted, amongst others, by Hull (1997), both theATMF 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. Althoughthis will entail paying two premia, the trader will proﬁt from a subsequent movementin volatility: if the foreign exchange market moves far enough either up or down, oneof the options will end deeply in-the-money and, when it is sold back to the writingcounterparty, the proﬁt will more than cover the cost of both premia. The other option willexpire worthless. Conversely, if both the call and put expire out-of-the-money followinga 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 thetwo option premia. This is a high-risk strategy if his market view is wrong as he mighttheoretically 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 modelsThe trading strategy adopted is based on the currency volatility trading model proposedby Dunis and Gavridis (1997). A long volatility position is initiated by buying the 1-month
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Applied Quantitative Methods for Trading and Investment
ATMF foreign exchange straddle if the 1-month volatility forecast is above the prevailing1-month implied volatility level by more than a certain threshold used as a conﬁrmationﬁlter or reliability indicator. Conversely, a short ATMF straddle position is initiated if the1-month volatility forecast is below the prevailing implied volatility level by more thanthe given threshold.To this effect, the ﬁrst stage of the currency volatility trading strategy is, based onthe threshold level as in Dunis (1996), to band the volatility predictions into ﬁve classes,namely, “large up move”, “small up move”, “no change”, “large down move” and “smalldown move” (Figure 4.4). The change threshold deﬁning the boundary between small andlarge movements was determined as a conﬁrmation ﬁlter. Different strategies with ﬁltersranging 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 ﬁlter rule,a position is only initiated when the 1-month volatility forecast is above or below theprevailing 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 straddlewhere Dt denotes the difference between the 1-month volatility forecast and the prevailing1-month implied volatility, and c represents the threshold (or ﬁlter).In terms of exit rules, our main test is to assume that the straddle is held until expiryand that no new positions can be initiated until the existing straddle has expired. As, dueto the drop in time value during the life of an option, this is clearly not an optimal tradingstrategy, we also consider the case of American options which can be exercised at anytime until expiry, and thus evaluate this second strategy assuming that positions are onlyheld for ﬁve trading days (as opposed to one month).18As in Dunis and Gavridis (1997), proﬁtability is determined by comparing the levelof implied volatility at the inception of the position with the prevailing 1-month realisedhistorical volatility at maturity.It is further weighted by the amount of the position taken, itself a function of thedifference between the 1-month volatility forecast and the prevailing 1-month impliedvolatility level on the day when the position is initiated: intuitively, it makes sense toassume that, if we have a “good” model, the larger |Dt |, the more conﬁdent we shouldbe about taking the suggested position and the higher the expected proﬁt. Calling G thisgearing of position, we thus have:19G = |Dt |/|c|
(4.8)
C = change threshold−CLarge down
C
0.0Small down
Small up
Large up
No change
Figure 4.4 Volatility forecasts classiﬁcation18For the “weekly” trading strategy, we also considered closing out European options before expiry by takingthe opposite position, unwinding positions at the prevailing implied volatility market rate after ﬁve tradingdays: this strategy was generally not proﬁtable.19Laws and Gidman (2000) adopt a similar strategy with a slightly different deﬁnition of the gearing.
Forecasting and Trading Currency Volatility
147
Proﬁtability is therefore deﬁned as a volatility net proﬁt (i.e. it is calculated in volatilitypoints or “vols” as they are called by options traders20 ). Losses are also deﬁned asa volatility loss, which implies two further assumptions: when short the straddle, nostop-loss strategy is actually implemented and the losing trade is closed out at the thenprevailing volatility level (it is thus reasonable to assume that we overestimate potentiallosses in a real world environment with proper risk management controls); when longthe straddle, we approximate true losses by the difference between the level of impliedvolatility 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 theposition (here again, we seem to overestimate potential losses). It is further assumed thatvolatility proﬁts 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 includedin our proﬁt and loss computations.4.6.3 Trading simulation resultsThe currency volatility trading strategy was applied from 31 December 1993 to 9 May2000. Tables 4.3 and 4.4 document our results for the GBP/USD and USD/JPY monthlytrading strategies both for the in-sample period from 31 December 1993 to 9 April 1999and the out-of-sample period from 12 April 1999 to 9 May 2000. The evaluation discussedbelow is focused on out-of-sample performance.For our trading simulations, four different thresholds ranging from 0.5 to 2.0 and twodifferent holding periods, i.e. monthly and weekly, have been retained. A higher thresholdlevel implies requiring a higher degree of reliability in the signals and obviously reducesthe overall number of trades.The proﬁtability criteria include the cumulative proﬁt and loss with and without gearing,the total number of trades and the percentage of proﬁtable trades. We also show the averagegearing of the positions for each strategy.Firstly, we compare the performance of the NNR/RNN models with the benchmarkGARCH (1,1) model. For the GBP/USD monthly volatility trading strategy in Table 4.3,the GARCH (1,1) model generally produces higher cumulative proﬁts not only in-samplebut also out-of-sample. NNR/RNN models seldom produce a higher percentage of profitable trades in-sample or out-of-sample, although the geared cumulative return of thestrategy based on the RNN (44-1-1) model is close to that produced with the benchmark model. With NNR/RNN models predicting more accurately directional change thanthe GARCH model, one would have intuitively expected them to show a better tradingperformance for the monthly volatility trading strategies.This expected result is in fact achieved by the USD/JPY monthly volatility tradingstrategy, as shown in Table 4.4: NNR/RNN models clearly produce a higher percentageof proﬁtable trades both in- and out-of-sample, with the best out-of-sample performancebeing that based on the RNN (44-5-1) model. On the contrary, the GARCH (1,1) modelbased strategies produce very poor trading results, often recording an overall negativecumulative proﬁt and loss ﬁgure.20In market jargon, “vol” refers to both implied volatility and the measurement of volatility in percent perannum (see, amongst others, Malz (1996)). Monetary returns could only be estimated by comparing the actualproﬁt/loss of a straddle once closed out or expired against the premium paid/received at inception, an almostimpossible task with OTC options.
148
Applied Quantitative Methods for Trading and InvestmentTable 4.3
1 Threshold = 0.5
GBP/USD monthly volatility trading strategy
Trading days = 21
Out-of-sampleIn-sampleSample(1330-1610)(1-1329)ObservationPeriod(31/12/1993-09/04/1999)(12/04/1999-09/05/2000)ModelsNNR(44-10-5-1) RNN(44-1-1) GARCH(1,1)COM1GRNNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1GRP/L without gearing58.39%54.47%81.75% 54.25% 81.40%5.77%5.22%12.90% 10.35% 11.98%P/L with gearing240.44%355.73%378.18% 182.38% 210.45%16.60%35.30%42.91% 16.41% 19.44%Total trades59596158611212111010Profitable trades67.80%70.00%77.05% 70.69% 83.61%50.00%58.33%72.73% 70.00% 80.00%Average gearing2.834.053.872.352.131.552.172.391.461.442 Threshold = 1.0
Trading days = 21
In-sampleSample(1-1329)ObservationPeriod(31/12/1993-09/04/1999)ModelsNNR(44-10-5-1) RNN(44-1-1) GARCH(1,1)COM161.48%57.52%82.61% 64.50%P/L without gearing134.25%190.80%P/L with gearing211.61% 116.99%515145Total trades5872.55%69.09%Profitable trades84.48% 80.00%1.682.212.081.53Average gearing3 Threshold = 1.5
Trading days = 21
In-sampleSampleObservation(1-1329)Period(31/12/1993-09/04/1999)ModelsNNR(44-10-5-1) RNN(44-1-1) GARCH(1,1)P/L without gearing53.