Modelling The Volatility Of Currency Exchange Rate Using GARCH Model

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Pertanika J. Soc. Sci. & Hum. 10(2): 85-95 (2002)ISSN: 0128-7702 Universiti Putra Malaysia PressModelling the Volatility of Currency Exchange Rate Using GARCH ModelCHaO WEI CHONG, LOa SIN CHUN & MUHAMMAD IDREES AHMADFaculty of Economics & ManagementUniversiti Putra Malaysia43400 UPM, Serdang, SelangOTE-mail: [email protected]: Exchange rates, volatility, forecasting, GARCH, random walkABSTRAKKertas ini mengkaji model GARCH dan modifikasinya dalam menguasai kemeruapan kadarpertukaran mata wang. Parameter model tersebut dianggar dengan menggunakan kaedahkebolehjadian maksimum. Prestasi bagi penganggaran dalam sam pel didiagnosis denganmenggunakan beberapa statistik kebagusan penyuaian dan kejituan telahan satu langkah kedepan dan luar sampel dinilai dengan menggunakan min ralat kuasa dua. Keputusan kajianmenunjukkan kegigihan kemeruapan kadar pertukaran mata wang RM/Sterling. Keputusandaripada penganggaran dalam sampel menyokong kebergunaan model GARCH dan modelvariasi malar pula ditolak, sekurang-kurangnya dalam sampel. Statistik Q dan ujian pendarabLangrange (LM) mencadangkan penggunaan model GARCH yang beringatan panjangmenggantikan model ARCH yang beringatan pendek dan berperingkat lebih tinggi. ModelGARCH-M pegun berprestasi lebih tinggi daripada model GARCH lain yang digunakan dalamkajian ini, dalam telahan satu langkah ke depan dan luar sam pel. Apabila menggunakan modelpeIjalanan rawak sebagai tanda aras, semua model GARCH berprestasi lebih baik daripada modeltanda aras ini dalam meramal kemeruapan kadar pertukaran mata wang RM/Sterling.ABSTRAGrThis paper attempts to study GARCH models with their modifications, in capturing the volatilityof the exchange rates. The parameters of these models are estimated using the maximumlikelihood method. The performance of the within-sample estimation is diagnosed using severalgoodness-of-fit statistics and the accuracy of the out-of-sample and one-step-ahead forecasts isevaluated using mean square error. The results indicate that the volatility of the RM/Sterlingexchange rate is persistent. The within sample estimation results support the usefulness of theGARCH models and reject the constant variance model, at least within-sample. The Qstatistic andLM tests suggest that long memory GARCH models should be used instead of the short-termmemory and high order ARCH model. The stationary GARCH-M outperforms other GARCHmodels in out-of-sample and one-step-ahead forecasting. When using random walk model as thenaive benchmark, all GARCH models outperform this model in forecasting the volatility of theRM/Sterling exchange rates.INTRODUCTIONIssues related to foreign exchange rate havealways been the interest of researchers inmodern financial theory. Exchange rate, whichis the price of one currency in terms of anothercurrency, has a great impact on the volume offoreign trade and investment. Its volatility hasincreased during the last decade and is harmfulto economic welfare (Laopodis 1997). Theexchange rate fluctuated according to demandand supply of currencies. The exchange ratevolatility will reduce the volume of internationaltrade and the foreign investment.Modelling and forecasting the exchange ratevolatility is a crucial area for research, as it hasimplications for many issues in the arena offmance and economics. The foreign exchangevolatility is an important determinant for pricing

Chao Wei Chong. Lao Sin Chun & Muhammad Idrees Ahmadof currency derivative. Currency options andforward contracts constitute approximately halfof the U.S. 880bn per day global foreignexchange market (Isard 1995). In view of this,knowledge of currency volatility should assistone to formulate investment and hedgingstrategies.The implication of foreign exchange ratevolatility for hedging strategies is also a recentissue. These strategies are essential for anyinvestment in a foreign asset, which is acombination of an investment in theperformance of the foreign asset and aninvestment in the performance of the domesticcurrency relative to the foreign currency. Hence,investing in foreign markets that are exposed tothis foreign currency exchange rate risk shouldhedge for any source of risk that is notcompensated in terms of expected returns (Santiset al. 1998).Foreign exchange rate volatility may alsoimpact on global trade patterns that will affect acountry's balance of payments position and thusinfluence the government's national policymaking decisions. For instance, Malaysia fixedthe exchange rate at RM3.80/US in September,1998, due to the economic turmoil and currencycrisis in 1997. This turmoil has spread todeveloped countries such as USA, Hong Kong,Europe and other developing South Americancountries such as Brazil and Mexico. Due to thiscurrency crisis, various governments haveresorted to different national policies so as tomitigate the effect of this crisis.In international capital budgeting ofmultinational companies, the knowledge offoreign exchange volatility will help them inestimating the future cash Hows of projects andthus the viability of the projects.Consequently, forecasting the futuremovement and volatility of the foreign exchangerate is crucially important and of interest tomany diverse groups including marketparticipants and decision makers.Beginning with the seminal works ofMandelbrot (1963a, 1963b, 1967) and Fama(1965), many researchers have found that thestylized characteristics of the foreign currencyexchange returns are non-linear temporaldependence and the distribution of exchangerate returns are leptokurtic, such as Friedmanand Vandersteel (1982), Bollerslev (1987),Diebold (1988), Hsieh (1988, 1989a, 1989b),86Diebold anderlove (1989), Baillie andBollerslev (1989). Their studies have found thatlarge and small changes in returns are ' clustered'together over time, and that their distribution isbell-shaped, symmetric and fat-tailed.These features of data are normally thoughtto be captured by using the AutoregressiveConditional Heteroskedasticity (ARCH) modelintroduced by Engle (1982) and the GeneralisedARCH (GARCH) model developed by Bollerslev(1986), which is an extension of the ARCHmodel to allow for a more flexible lag structure.The use of ARCH/GARCH models and itsextensions and modifications in modeling andforecasting stock market volatility is now verycommon in finance and economics, such asFrench et at. (1987), Akgiray (1989), Lau et at.(1990), Pagan and Schwert (1990) , Day andLewis (1992), Kim and Kon (1994), Franses andVan Dijk (1996) and Choo et al. (1999).On the other hand, the ARCH model wasfirst applied in modeling the currency exchangerate by Hsieh only in 1988. In a study done byHsieh (1989a) to investigate whether dailychanges in five major foreign exchange ratescontain any nonlinearities, he found thatalthough the data contain no linear correlation,evidence indicates the presence of substantialnonlinearity in a multiplicative rather thanadditive form. He further concludes that ageneralized ARCH (GARCH) model can explaina large part of the nonlinearities for all fiveexchange rates.Since then, applications of these models tocurrency exchange rates have increasedtremendously, such as Hsieh (1989b), Bollerslev,T. (1990), Pesaran and Robinson (1993),Copeland et al. (1994), Takezawa (1995),Episcopos and Davies (1995), Brooks (1997),Hopper (1997), Cheung et al. (1997), Laopodis(1997), Lobo et at. (1998) and Duan et at. (1999).In many of the applications, it was foundthat a very high-order ARCH model is requiredto model the changing variance. The alternativeand more flexible lag structure is the GeneralisedARCH (GARCH) introduced by Bollerslev(1986). Bollerslev et at. (1992) indicated thatthe squared returns of not only exchange ratedata, but all speculative price series, typicallyexhibit autocorrelation in that large and smallerrors tend to cluster together in contiguoustime periods in what has come to be known asvolatility clustering. It is also proven that smallPertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 2002

Modelling the Volatility of Currency Exchange Rate Using GARCH Modellag such as GARCH(I,l) is sufficient to modelthe variance changing over long sample periods(French et at. 1987; Franses and Van Dijk 1996;Choo et al. 1999).Even though the GARCH model caneffectively remove the excess kurtosis in returns,it cannot cope with the skewness of thedistribution of returns, especially the financialtime series which are commonly skewed. Hence,the forecasts and forecast error variances from aGARCH model can be expected to be biased forskewed time series. Recently, a few modificationsto the GARCH model have been proposed, whichexplicitly take into account skewed distributions.One of the alternatives of non-linear modelsthat can cope with skewness is the ExponentialGARCH or EGARCH model introduced byNelson (1990). For stock indices, Nelson'sexponential GARCH is proven to be the bestmodel of the conditional heteroskedasticity.In 1987, Engle et aL developed the GARCH-Mto formulate the conditional mean as functionof the conditional variance as well as anautoregressive function of the past values of theunderlying variable. This GARCH in the mean(GARCH-M) model is the natural extension dueto the suggestion of the financial theory that anincrease in variance (risk proxy) will result in ahigher expected return.Choo et al. (1999) studies the performanceof GARCH models in forecasting the stockmarket volatility and they found that i) thehypotheses of constant variance models couldbe rejected since almost all the parameterestimates of the non-eonstant variance (GARCH)models are significant at the 5% level; ii) theEGARCH model has no restrictions andconstraints on the parameters; iii) the longmemory GARCH model is more suitable thanthe short-memory and high-order ARCH modelin modelling the heteroscedasticity of thefinancial time series; iv) the GARCH-M is best infitting the historical data whereas the EGARCHmodel is best in out-of-sample (one-step-ahead)forecasting; v) the IGARCH is the poorest modelin both aspects.Since Choo et al. (1999) have indicated thatthe GARCH-M model performs well in withinsample estimation and the EGARCH modelperforms best in out-of-sample forecasting, thecombination of both models, EGARCH-M shouldbe able to enhance the performance in bothaspects.In order to know the out-of-sampleforecasting performance of EGARCH-M, wecompare the performance of EGARCH-M andthe other modifications of the GARCH model tothe simple random walk forecasting scheme.The models are presented in the followingsection. The third section is the background ofcurrency exchange rate data and themethodology used in this study. All the resultswill be discussed in the fourth section. Theconclusion will be in the final section.MODELThe conditional distribution of the series ofdisturbances which follows the GARCH processcan be written aswhere 'lJl' 1 denotes all available informationat time t - 1. The conditional variance h, ispa'. i ht w .]f3it-jj-lHence, the GARCH regression model forthe series of rt can be written as ,(B)r, jJ. " withiP,(B) I- IB-K ,B sf, .p;:;:e, -N(O,I)h,w f a;f ; f f3A-ji . lj 1where B is the backward shift operatordefmed by JJyt yt - k. The parameter jJ. reflectsa constant term, which in practice is typicallyestimated to be close or equal to zero. Theorder of s is usually 0 or small, indicating thatthere are usually no opportunities to forecast r,from its own past. In other words, there is alwaysno auto-regressive process inr,.1) ARCHThe GARCH(p,q) model is reduced to theARCH(q) model when p 0 and at least one ofthe ARCH parameters must be nonzero(q 0).PertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 200287

Choo Wei Chong, Loo Sin Chun & Muhammad 1drees Ahmad2) Stationary GARCH, SG(P,q)If the parameters are constrained such thatq2: a.-1pi 2: f3 j)-1 1, they imply the weakly stationaryGARCH (SG(P,q» model since the mean,variance and autocovariance are finite andconstant over time.3) Unconstrained GARCH, UG(P,q)The parameter of w, a and f3) can beunconstrained, thus yielding the unconstrainedGARCH (UG(P,q» model.4) Non-negative GARCH, NG(P,q)If P ::. 0 , q 0 and w 0, a i ::. 0, f3j ::. 0, yieldsthe non-negative GARCH (NG(P,q» model.5) Integrated GARCH, IG(P,q)Sometimes, the multistep forecasts of the variancedo not approach the unconditional variancewhen the model is integrated in variance; that isq2: ai 7) GARCH-in-Mean, G(p,q)-MThe GARCH-in-Mean, G(P,q)-M model has theadded regressor that is the conditional standarddeviationr, , 2: f3 j !l (j.Jh: ,.Jh:e,where h, follows the GARCH process.8) Stationary GARCH-in-Mean, SG(p,q)-MThis model has the added regressor that is theconditional standard deviationr, , pi-IThe coefficient of the second term in g(Z,)is set to be 1 (y 1) in this formulation. Notethat E/ ZI (2/n)I/2 if Z, - N(O,I).!l (j.Jh: ,.Jh:e,1. The unconditional variance forj-Ithe IGARCH model does not exist. However, itis interesting that the integrated GARCH orIGARCH (IG(p,q» model can be stronglystationary even though it is not weakly stationary(Nelson 1990a, b).where h, follows the stationary GARCH,SG(P,q) process.9) Unconstrained GARCH-in-Mean, UG(P,q)-MThis model has the added regressor that is theconditional standard deviation6) Exponential GARCH, EG(P,q)The exponential GARCH or EGARCH (EG(P,q»model was proposed by Nelson (1991). Nelsonand Cao (1992) argue that the nonnegativityconstraints in the linear GARCH model are toorestrictive. The GARCH model imposes thenonnegative constraints on the parameters, a iand f3, while there is no restriction on these).parameters III the EGARCH model. In theEGARCH model, the conditional variance, h" isan asymmetric function of lagged disturbances,r, , (j.Jh: .Jh:e,!l ,where h, follows the unconstrained GARCH,UG(P,q) process.10) Non-negative GARCH-in-Mean, NG(P,q)-MThis model has the added regressor that is theconditional standard deviation i :r, , ,where h, follows the non-negative GARCH,NG(P,q) process.whereg(Z,)Z,88(j.Jh: .Jh:e,!l OZ, Y[/ Z, / - E / Z, / ] ,/.Jh:11) Integrated GARCH-in-Mean, IG(P,q)-MThis model has the added regressor that is theconditional standard deviationPertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 2002

Modelling the Volatility of Currency Exchange Rate Using GARCH Modelr,fJ, (j.Jh: , .Jh:e, , where h, follows the integrated GARCH,IG(P,q) process.12) Exponential GARCH-in-Mean, EG(P,q)-MThis model has the added regressor that is theconditional standard deviationr,fJ, (j.Jh: , , .Jh:e,where h, follows the exponential GARCH,EG(P,q) process.Since a small lag of the GARCH model issufficient to model the long-memory process ofchanging variance (French et al. 1987; Fransesand Van Dijk 1996; Choo et al. 1999), theperformance of GARCH models in forecastingRM-Sterling exchange rate volatility is evaluatedby using SG(l,l), UG(l,l), G(l,l) IG(l,l),EG(l,l), G(l,l)-M, SG(l,l)-M, UG(l,l)-M,NG(l,l)-M, IG(l,l)-M, and EG(l,l)-M.DATA AND METHODOLOGYIn this study, simple rate of returns is employedto model the currency exchange rate volatility ofRM-Sterling. Consider a foreign exchange rateE" its rate of return Tl' is constructed asr,E -E '.-1 . The exchange rate t denotes dailyEt-lexchange rate observations.The foreign exchange rate used in this studyis focused on the Malaysian Ringgit (RM) to thePound Sterling. This exchange rate is chosenbecause in addition to the US dollar, the PoundSterling is also one of the major currenciestraded in the foreign exchange markets.Traditionally and historically, the UK has alwaysbeen one of the important trading partners ofMalaysia. The data was collected from 2 January1990 to 13 March 1997, from 1810 observations.The daily dosing exchange rates were used asthe daily observations. The first 1760 observationsare used for parameters estimation and the last50 observations reserved for forecastingevaluation.Fig. 1 shows nearly 1810 daily observer crossrates of the Malaysian Ringgit to the PoundSterling, covering the seven years from 2January1990 to 13 March 1997. Some characteristics ofthe rate of returns, r, are given in Table 1. Themeans and variances'are quite small. The excesskurtosis indicates the necessity of fat-taileddistribution to describe these variables. Theskewness of -0.200 indicates that the distributionof rate of returns for RM-Sterling is negativelyskewed.The family of GARCH models is estimatedusing the maximum likelihood method. Thismethod enables the rate of return and varianceprocesses being estimated jointly. The loglikelihood function is computed from theproduct of all conditional densities of thepre iction errors.l 1[nL-In( 2Jt ) ,-12 2]1n( h, ) - -L.h,where , r. - fJ, and h, is the conditionalvariance. When the GARCH(P,q)-M model isr, - fJ, - (j.Jh: . When there are noregressors (trend or constant, fJ, the residuals ,estimated, . are denoted as rIortr -tr -fh. The likelihood(j "\jft ,function is maximized via the dual quasi-Newtonand trust region algorithm. The starting valuesfor the regression parameters fJ, are obtainedfrom the OLS estimates. When there areautoregressive parameters in the model, theinitial values are obtained from the Yule-WalkerTABLE 1Summary statistics of currency exchange rate data on rate of returns from 2 January 1990 to13 March 1997Currency ExchangeRateRMjSterlingMeann1809Variancex 10-5Skewnessx 10-5ExcessKurtosis-3.1834.076-0.2002.370Source of data: The Federal Reserve, the Central Bank of the United StatesPertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 200289

Choo Wei Chong. Loo Sin Chun & Muhammad Idrees Ahmad5.5T-------------------, ::!\A 4.035l - . -.- -. .-I18392365274547456729638911820- -.-.- - -., -., .109310021:!75118414571366163915481730Time in trading day unitsFig. 1: RM/Sterling, daily from 2 January 1990 to 13 March 1997estimates. The starting value IE - 6 is used forthe GARCH process parameters. The variancecovariance matrix is computed using the Hessianmatrix. The dual quasi-Newton methodapproximated the Hessian matrix while the quasiNewton method gets an approximation of theinverse of Hessian. The trust region methoduses the Hessian matrix obtained using numericaldifferentiation. This algorithm is numericallystable, though computation is expensive.In order to test for the independence of theindices series, the portmanteau test statistic basedon squared residual is used (McLeod and Li1983). This Q statistic is used to test the nonlinear effects, such as GARCH effects, present inthe residuals. The GARCH (P,q) process can beconsidered as an ARMA (max(p,q),P) process.Therefore, the Q statistic calculated from thesquared residuals can be used to identify theorder of the GARCH process. The Lagrangemultiplier test for ARCH disturbances is proposedby Engle (1982). The test statistic is asymptoticallyequivalent to the test used by Breusch and Pagan(1979) .The LM and Q statistics are computed fromthe OLS residuals assuming that disturbance iswhite noise. The Q and LM statistics have anapproximate») distribution under the whitenoise null hypoilieses.Various goodness-of-fit statistics are used tocompare the six models in this study. Thediagnostics are the mean of square error (MSE),the loglikelihood (Log L), Schwarz's Bayesianinformation criterion (SBC) by Schwarz (1978)and Akaike's information criterion (AlC) (Judgeet at. 1985).(Xi90The 'true volatility' is measured to evaluatethe performance of the six GARCH models inforecasting the volatility in stock returns. As inthe studies by Pagan et at. (1990) and Day et at.(1992), the volatility is measured byv, (r, - r)2where r is the average return. The measureof the one-step-ahead forecast error iswhere h' 1 is generated using the h, equationsof the GARCH models being studied. Theestimated parameters of the GARCH modelssuch as w, a, f3, () and {) are substituted duringthe generation of h' I' In order to show theperformance of GARCH models over a naive nochange forecast, the forecast errors of therandom walk (RW) are calculated as follows:This is a very important naive benchmark inthe comparison of the forecasts from the GARCHmodels (Brooks 1997).RESULTS AND DISCUSSIONParameter EstimationsThe parameter estimates for eleven variations ofGARCH models of the rate of returns series arepresented in Table 2 (a) and Table 2 (b). Thesewithin-sample estimation results enable us toknow the possible usefulness of the GARCHPertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 2002

Modelling the Volatility of Currency Exchange Rate Using GARCH Modelmodels in modeling the currency exchange rateseries.It can be seen from Table 2(a) that exceptfor /1-, all the parameter estimates of the RM/Sterling (w, a and f3) are significant at 5% level.However, in Table 2(b), all the two additionalparameter estimates ({) and J) of the EGARCHand all the GARCH models with means are notsignificant. It appears that for the within-sampleestimations, all the family GARCH modelsperform well in modeling the exchange rate ofRM/Sterling.In general, it can be concluded that almostall a and f3 (ARCH and GARCH terms) of theRM/Sterling series examined are significant.Hence, the constant variance model can berejected, at least for the within-sample estimation.For the linear GARCH models such as SG(I,I),the sum of a and f3 is close to unity. Theproperties of I of IG(I,I) also hold for theseries.Diagnostics CheckingThe basic ARCH (q) model is a short memoryprocess in that only the most recent q squaredresiduals are used to estimate the changingvariance. The results for Q statistic and LagrangeMultiplier (LM) test are shown in Table 3. Thesecan help to determine the order of the ARCHprocess in modeling the RM/Sterling series.The tests are significant at less then 1 %level though order 12. These indicate that theheteroscedasticity terms of the daily RM/Sterlingexchange rate series needed to be modeled by aTABLE 2(a)Estimation results of rate of returns for the currency exchange rateParameter estimatesCurrencyExchange RateRM/Sterlingt .125-D.125-D.125-D.125-D. 104-D.056-1.305-1.308-1.306-1.306-1.229-D.622et Ratio-D.047-D.518-D.093

Choo Wei Chong. Loo Sin Chun & Muhammad Idrees AhmadTABLE 3Diagnostics for currency exchange rate using Q statistic and Lagrange Multiplier testDiagnosticsCurrencyExchange Raterm/poundQ(12)Prob Q(12)LM(l2)Prob LM(12)273.4470.0001147.3730.0001very high order of ARCH model. These resultssupport the use of GARCR model, which allowslong memory processes to estimate the currentvariance of the daily RM/Sterling series insteadof the ARCR model.models in the SBC and AlC test while in theMSE and Log L test, all the GARCR in meanmodels perform well to model the daily exchangerate compared to their ordinary GARCR modelcounterparts.Goodness of Fit TestsThe result of the goodness-of-fit statistics for theRM/Sterling series is presented in Table 4. Table5 shows the rankings of various GARCR models.From Table 5, the ranking of the MSE valueindicates that all the family of GARCH in meanmodels outperform the GARCH models with aslight value of 0.000001. The Log L valueshowever, suggest EG(l,l)-M to be the best modelfor modeling the volatility of RM/Sterling,followed by UG(l,I)-M, NG(l,l)-M andG(l,l)-M. The SBC values in contrast, rankedindifferently SG(I,l), UG(l,l) and G(l,l) tobe the best model followed by IG(l,l). The AlCvalues on the other hand, proposed UG(l,l)and NG(l,l) to be the best two models, followedby SG(l,l).From the goodness-of-fit test, it appears thatfor within-sample estimations, almost all theGARCR models outperform the GARCR in meanOne Step Ahead ForecastingThe good performance in the parameterestimation and goodness-of-fit statistics do notguarantee the good performance in forecasting(Choo et at. 1999). The performance of theGARCH models is evaluated through the onestep-ahead forecasting. 50 one-step-aheadforecasts are generated and the mean squareerror (MSE) is calculated to evaluate theforecasting performance. The results of theforecasting for the GARCH models and therandom walk model are shown in Table 6. Therankings of the models based on the performanceof the one-step-ahead forecasting are presentedin Table 7.In Table 7, the ranking results of MSEsuggest that SG(l,l)-M is the best model forone-step-ahead forecasts, followed by SG(l,l)and G(l,l)-M. It is also noted that, SG(l,l)-M,UG(l,l)-M and NG(l,l)-M clearly outperform

Modelling the Volatility of Currency Exchange Rate Using GARCH ModelTABLE 5Rankings of the models averaged across the currency exchange based on the performance ofvarious goodness-of-fit MEG(l,l)-MMSELog 644119TABLE 6Out-of-sample forecasting performance of various GARCH models and randomwalk models for the volatility of the currency exchange ratesMSE (x1Q-9) of one-step-ahead forecast (forecast period 73.6253.1506.849TABLE 7Rankings of the models averaged across the currency exchange ratesbased on the performance of one-step-ahead l)-MRWMSE of one-step-ahead forecast for RM/pound276108314511912PertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 200293

Choo Wei Chong, Loo Sin Chun & Muhammad Idrees Ahmadtheir ordinary GARCH models counterparts whileEG(I,I) and IG(I,I), in contrast, outperformtheir with mean GARCH counterparts.In general, almost all the GARCH in meanmodels outperform the ordinary GARCH modelswith the exception of EG(I,I) and IG(l,I).However, the family of GARCH models is clearlybeing proposed instead of their naive benchmark,the random walk model.CONCLUSIONUsing seven years of daily observed RM/Sterlingexchange rate, the performance of GARCHmodels, including the family of GARCH in meanmodels to explain the commonly observedcharacteristics of the unconditional distributionof daily rate of returns series, were examined.The results indicate that the hypotheses ofconstant variance model could be rejected, atleast within-sample, since almost all the parameterestimates of the ARCH and GARCH models aresignificant at 5% level.The Q statistics and the Lagrange Multipliertest reveal that the use of the long memoryGARCH model is preferable to the short memoryand high-order ARCH model.The results from various goodness-of-fitstatistics are not consistent for RM/Sterlingexchange rates. It appears that the SBC and AlCtest proposed GARCH models to be the best forwithin-sample modeling while the MSE and LogL test, suggest the GARCH in mean models tobe best to model the heteroscedasticity of dailyexchange rates.The forecasting results show that SG(l,l)-Mis the best model for forecasting purpose,followed by SG(I,I) and G(I,I)-M. Almost allthe GARCH in mean models outperform theordinary GARCH models. On the other hand,the family of GARCH models has clearly shownthat they perform better than the naivebenchmark, the random walk model.ACKNOWLEDGEMENTSWe would like to thank the anonymous refereesand reviewers of this paper who have providedus with many useful comments and suggestions.This research was supported by the short termresearch grant funded by the Ministry of Science,Technology and the Environment, Malaysia,through the Faculty of Economics andManagement, Universiti Putra Malaysia.94REFERENCESAKGIRAY, V. 1989. Conditional heteroskedasticity intime series of stock returns: Evidence andforecasts. Journal of Business 62: 55-80.BAILLIE, R. T. and T. BOLLERSLEV. 1989. The messagein daily exchange rates: A conditional variancetele. Journal of Business and Economic Statistics7: 297-305.BOLLERSLEV, T. 1986. Generalized autoregressiveconditional heteroskedasticity. Journal ofEconometrics 31: 307-327.BOLLERSLEV, T. 1987. A conditional heteroskedastictime series model for speculative prices andrates of return. Review ofEconomics and Statistics69: 542-547.BOLLERSLEV, T. 1990. Modelling the coherence inshort-run nominal exchange rates: Amultivariate generalized ARCH model. TheReview of Economics and Statistics: 498-504.BOLLERSLEV, T., R. Y. CHOU and K. F. KRONER. 1992.Arch modelling in finance. A review of thetheory and empirical evidence. Journal ofEconometrics 52: 5-59.BROOKS, C. and S. P. BURKE. 1998. Forecastingexchange rate volatility using conditionalvariance models selected by informationcriteria. Economics Letters 61: 273-278.CAPORALE, G. M. and N. PITTIS. 1996. Modellingsterling-deutschmark exchange rate: Non-lineardependence and thick tails. Economic Modelling13: 1-14.CHEN, A. S. 1997. Forecasting the S & P 500 indexvolatility. International Review of Economics &Finance 6: 391-404.CHEUNG, Y. W. and C. Y. P. WONG. 1997. Theperformance of trading rules on four Asiancurrency exchange rates. Multinational FinanceJournal 1(1): 1-22.CHoo, W. C., M. I. AHMAD and M. Y. ABDULLAH.1999. Performance of GARCH models inforecasting stock market volatility. Journal ofForecasting 18: 333-343.COPElAND, L. S. and P. WANG. 1994. Estimating dailyseasonality in foreign exchange rate changes.Journal of Forecasting 13(6): 519-528.DAY, T. E. and C. M. LEWIS. 1992. Stock marketvolatility and information content of stockindex options. Journal of Econometrics 52: 267287.PertanikaJ. Soc. Sci. & Hum. Vol. 10 No.2 2002

Modelling the Volatility of Currency Exchange Rate Using GARCH ModelDIEBOLD, F. X. and M. NERLOVE. 1989. The dynamicsof exchange rate volatility : A multivariatelatent variable factor ARCH model. Journal ofApplied Econometrics 4(1): 1-21.HSIEH, D. A. 1989a. Modeling hetereskedasticity indaily foreign exchange rates. Journal ofBusinessand Economic Statistics 7: 306-317.DIEBOLD, F. X. 1988. Empirical Modeling of ExchangeRat

currencyrelative to the foreign currency. Hence, investing in foreign markets that are exposed to this foreign currency exchange rate risk should hedge for any source of risk that is not compensated in termsofexpected returns (Santis et al. 1998). Foreign exchange rate volatility may also impact on global trade patterns thatwill affect a

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On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.

1.2.8 Volatility in terms of delta 11 1.2.9 Volatility and delta for a given strike 11 1.2.10 Greeks in terms of deltas 12 1.3 Volatility 15 1.3.1 Historic volatility 15 1.3.2 Historic correlation 18 1.3.3 Volatility smile 19 1.3.4 At-the-money volatility interpolation 25 1.3.5 Volatility

Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được

Transaction currency The currency in which a transaction originates. Accounting currency The primary currency in which a legal entity maintains its financial records. The accounting currency is stored in the Ledger table. Reporting currency The reporting currency of the ledger. The reporting currency is stored in the Ledger table. It is optional.

exchange rate volatility in the majority of the sample countries. On the other hand, there are also research that show evidence against the relationship between currency substitution and exchange rate volatility. Petrovic et al. (2016) empiri-cally investigated the e ect of currency substitution on exchange rate depreciation volatility

quanto factor from the domestic currency into the quanto currency. We let r d r f ˆ ; (12) be the adjusted drift, where r d and r f denote the risk free rates of the domestic and foreign underlying currency pair respectively, 1 the volatility of this currency pair, 2 the volatility of the currency pair DOM-QUANTO and ˆ .

Good Volatility, Bad Volatility and Option Pricing . by Bruno Feunou and Cédric Okou . 2 Bank of Canada Staff Working Paper 2017-52 . December 2017 . Good Volatility, Bad Volatility and Option Pricing by Bruno Feunou 1 and Cédric Okou 2 1 Financial Markets Department

the currency of the bond's denomination as the local currency and the chosen currency of the portfolio or index as the base currency. The return of this security in the base currency on day t can be computed using the following inputs. and 1 are the market values in local currency at the close of day t and t-1 respectively.

Currency Harvest was 15.22%. The Global currency harvest returned 12.07% and the G10 Currency harvest returned 8.02%. The Sharpe ratios for the Balanced Currency Harvest, Global Currency Harvest and G10 Currency Harvest over the period were 1.60, 1.23 and 1.07 respectively. (The returns are in USD terms

a sterling/Euro currency account held with us.) Each payment up to the currency equivalent of 100 Free Each payment over the currency equivalent of 100 6 Sterling and foreign currency payments payable to an account held with us in a different currency from the payment Each payment up to 100 or the currency equivalent of 100 Free

Currency Exchange Setup. Adding Currency Code and Exchange Rates2-1. Editing Currency Exchange Rate2-2. Removing Currency Exchange Rate2-2. Commission Charge Code and Account2-2. Setting Up Commission Charge Code2-4. Setting up System Account2-4. Performing Currency Exchange Transaction. Buy Sell Currency Exchange3-1. Exchange Charge to Account3-2

from an underlying asset which in foreign currency options is the exchange rate. There are two types of foreign currency options, a call currency option and a put currency option. A call option on a particular currency gives the holder the right but not an obligation to buy that currency at a predetermined exchange rate at a particular date and .

In this dissertation we take up the problem of pricing a European style FX quanto option under stochastic volatility. An FX quanto option has as its underlying an exchange rate with a domestic and foreign currency. The payofi at maturity is converted into a third currency. This third currency is called the quanto currency.

dominant (currency) countries that will be the focus of our analysis. Furthermore, this volatility is driven by uctuations in the value of the country's currency relative to the dominant currency, re-gardless of the country of origin of the imported goods. Consequently, when a country's currency

Volatility Strategies How to profit from interest rate volatility . Source: Ardea Investment Management, Bloomberg. 5 These dynamics of abnormally low market pricing of interest rate volatility and compressed volatility risk premia used to be rare but are now becoming more common. Just as risk premia have shrunk in other

ratio portfolios with volatility derivatives. Intuitively, one expects that a portfolio strategy mixing a well-diversified equity benchmark and a suitably designed long exposure to volatility through trading in volatility index futures and/or volatility index options can be enginee

pricing certain kinds of exotic and structured products. keywords: volatility of volatility, variance derivatives, exotic options, structured products. 0.1 Introduction It is intuitively clear that for exotic products that are strongly dependent on the dynamics of the volatility surface pro

One of the fastest, most efficient trading platforms in the world, offering complex trading and price improvement auctions with industry-leading risk control features. Powered by Volatility, reimagined. 30-day volatility The SPIKES Volatility Index (index symbol: SPIKE) is a measure of the expected 30-day volatility in the SPDR S&P 500 ETF .

may be taken on an instrument with only one manual. Consequently, in Grades 1–3, some notes may be transposed or omitted, provided the result is musically satisfactory. Elements of the exam All ABRSM graded Organ exams comprise the following elements: three Pieces; Scales, arpeggios and exercises; Sight-reading (with an additional Transposition exercise in Grades 6–8); and Aural tests .