ation behaviour in the shorter sample, with the correlation
−0.15
Ident.
Ident.2
1m.Real Rate
Jan-98
−6
Jan-97
−0.10
Jan-96
−4
Jan-95
−0.05
Jan-94
−2
Jan-93
0.00
Jan-92
0
Jan-91
0.05
Jan-90
2
Jan-89
0.10
Jan-88
4
Jan-87
0.15
3m.Real Rate
Non-ident.
121
6
Jan-86
Ident. and real rate (%)
Modelling the Term Structure of Interest Rates
Non-ident.
Ident.
1m. Real Rate
3m. Real Rate
Non-ident.
Non-ident. and ident.2
−0.6
Sep-98
−6
Sep-96
−0.4
Sep-94
−4
Sep-92
−0.2
Sep-90
−2
Sep-88
0.0
Sep-86
0
Sep-84
0.2
Sep-82
2
Sep-80
0.4
Sep-78
4
Sep-76
0.6
Sep-74
6
Sep-72
Ident. and real rate (%)
(a)
Ident.2
(b)
Figure 3.16 Time-series evolution: (a) ﬁrst factor, 1986–1998; (b) ﬁrst factor, 1972–1998;(c) second factor, 1986–1998; (d) second factor, 1972–1998; (e) inﬂation factor, 1986–1998;(f) inﬂation factor, 1972–1998
coefﬁcient achieving ﬁgures above 0.98 in both samples. As the model in (3.56) includesa speciﬁc factor, the inﬂation, the correlation between the second factor and that variabledecreases to around 0.5 in the shorter sample and remains close to 0.7 in the larger one.In Figure 3.16e and f, the behaviour of the speciﬁc inﬂation factor vis-à-vis the observedinﬂation is presented. This factor in both samples is highly correlated to the inﬂation,
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Applied Quantitative Methods for Trading and Investment432
%
10−1−2
Non-ident.
Ident.
Ident.2
Jan-98
Jan-97
Jan-96
Jan-95
Jan-94
Jan-93
Jan-92
Jan-91
Jan-90
Jan-89
Jan-88
Jan-87
−4
Jan-86
−3
Yearly CPI
(c)6
4
%
2
0
−2
Non-ident.
Ident.
Ident.2
Sep-98
Sep-96
Sep-94
Sep-92
Sep-90
Sep-88
Sep-86
Sep-84
Sep-82
Sep-80
Sep-78
Sep-76
Sep-74
−6
Sep-72
−4
Yearly CPI
(d)
Figure 3.16 (continued )
which implies that the correlation between the second factor and the inﬂation rate becomeslower.One of the most striking results obtained is that the second factor appears to be aleading indicator of inﬂation, according to Figure 3.16c and d. This is in line with thepioneer ﬁndings of Fama (1975) regarding the forecasting ability of the term structure
Modelling the Term Structure of Interest Rates
1236
100 00080 000
5
60 000
Factor
20 000
Inflation (%)
4
40 000
3
02
−20 000−40 000
1
−60 0000
Factor
Jan-98
Jan-97
Jan-96
Jan-95
Jan-94
Jan-93
Jan-92
Jan-91
Jan-90
Jan-89
Jan-88
Jan-87
−100 000
Jan-86
−80 000
−1
Inflation(e)
20
8
15
76
10
Factor
403−5
Inflation (%)
5
5
2
Factor
Sep-98
Sep-96
Sep-94
Sep-92
Sep-90
Sep-88
Sep-86
Sep-84
Sep-82
−1
Sep-80
−20
Sep-78
0Sep-76
−15Sep-74
1
Sep-72
−10
Inflation(f)
Figure 3.16
(continued )
on future inﬂation. It is also consistent with the idea that inﬂation expectations drivelong-term interest rates, given that the second factor is relatively more important for thedynamics of interest rates at the long end of the curve, as previously stated.If z2t is a leading indicator for πt , then the highest (positive) correlation should occurbetween lead values of πt and z2t . Table 3.6 shows cross-correlation between πt and z2t
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Applied Quantitative Methods for Trading and Investment
in the larger sample. The shaded ﬁgures are the correlation coefﬁcients between z2t andlags of πt . The ﬁrst ﬁgure in each row is k and the next corresponds to the correlationbetween πt−k and z2t (negative k means lead). The next to the right is the correlationbetween πt−k−1 and z2t , etc.According to Table 3.6, the highest correlation (in bold) is at the fourth lead of inﬂation.Additionally, that correlation increases with leads of inﬂation up to four and steadilydeclines for lags of inﬂation.Table 3.7 presents the Granger causality test and Figure 3.17 the impulse-response functions. The results strongly support the conjecture that z2,t has leading indicator propertiesfor inﬂation. At the 5% level of conﬁdence one can reject that z2,t does not Granger causeinﬂation. This is conﬁrmed by the impulse-response analysis.A positive shock to the innovation process of z2t is followed by a statistically signiﬁcantincrease in inﬂation, as illustrated by the ﬁrst panel in Figure 3.17, where the conﬁdenceinterval of the impulse-response function does not include zero. However a positive shockto the innovation process of inﬂation does not seem to be followed by a statisticallysigniﬁcant increase in z2t , according to the last panel in Figure 3.17, where the conﬁdenceinterval includes zero. These results suggest that an innovation to the inﬂation processdoes not contain “news” for the process of expectation formation. This is in line withthe forward-looking interpretation of shocks to z2t as reﬂecting “news” about the futurecourse of inﬂation.Table 3.6 Cross-correlations of series πt and z2t (monthly data from 1972:09 to 1998:12)k−25:−19:−13:−7:−1:5:11:17:23:29:
0.39747590.56390120.68293680.74320240.73682740.63606910.49602540.33116990.1527293−0.0167756
0.42749280.58822780.69796260.74533450.73110080.61595400.47065180.30117140.1226024−0.0421135
0.45706310.61061810.71198330.74570820.71447270.59494720.44279140.27207730.0927202−0.0673895
0.48613070.63102820.72358750.74581290.69545110.57260030.41569290.24258880.0636053−0.0911283
0.51294880.65013700.73249130.74437890.67616490.54789670.38706900.21268040.0362558−0.1139021
Note: Correlation between πt−k and z2t (i.e., negative k means lead).
Table 3.7 Granger causality testsInﬂation does not Granger cause z2tVariableF -StatisticSigniﬁcance5.80.03az2t does not Granger cause inﬂationVariableF -StatisticSigniﬁcance
πt
z2ta
6.12
Means rejection at 5% level.
0.03a
0.53940650.66708200.73854230.74115560.65614130.52291930.35905920.18299180.0085952−0.1356261
Modelling the Term Structure of Interest RatesRESP OF PISIZE =
0.40
125RESP OF PI TOSIZE =
0.5
0.350.4
0.300.25
0.3
0.200.2
0.150.10
0.1
0.050.00
0.02
4
6
RESP OF Z2SIZE =
0.30
2
8 10 12 14 16 18 20
0.25
4
6
8 10 12 14 16 18 20RESP OF Z2 TOSIZE =
0.640.56
0.20
0.48
0.150.10
0.40
0.05
0.32
0.00
0.24
−0.05
0.16
−0.10−0.15
0.082
4
6
8 10 12 14 16 18 20
Figure 3.17
2
4
6
8 10 12 14 16 18 20
Impulse-response functions and two standard error bands
Overall, considering the results for the factor loadings and the relationship betweenthe latent factors and the economic variables, the short-term interest rates are mostlydriven by the real interest rates. In parallel, the inﬂation expectations exert the mostimportant inﬂuence on the long-term rates, as is usually assumed. Similar results concerning the information content of the German term structure, namely in the longerterms, regarding future changes in inﬂation rate were obtained in several previous papers,namely Schich (1996), Gerlach (1995) and Mishkin (1991), using different samples andtesting procedures.5353The Mishkin (1991) and Jorion and Mishkin (1991) results on Germany are contradictory as, accordingto Mishkin (1991), the short end of the term structure does not contain information on future inﬂation for allOECD 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 inﬂation is low in the USA, Germany andSwitzerland.Fama (1990) and Mishkin (1990a,b) present identical conclusions concerning the information content ofUS term structure regarding future inﬂation and state that the US dollar short rates have information contentregarding future real interest rates and the longer rates contain information on inﬂation expectations. Mishkin(1990b) also concludes that for several countries the information on inﬂation expectations is weaker than forthe United States. Mehra (1997) presents evidence of a cointegration relation between the nominal yield on a10-year Treasury bond and the actual US inﬂation rate. Koedijk and Kool (1995), Mishkin (1991), and Jorionand Mishkin (1991) supply some evidence on the information content of the term structure concerning inﬂationrate in several countries.
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Fleming and Remolona (1998) also found that macroeconomic announcements of theCPI and the PPI affect mostly the long end of the term structure of interest rates, usinghigh frequency data. Nevertheless, it is also important to have in mind that, accordingto Schich (1999), the information content of the term structure on future inﬂation istime-varying and depends on the country considered.The relationship between the factors and the referred variables is consistent with thetime-series properties of the factors and those variables. Actually, as observed in Claridaet al. (1998), the I (1) hypothesis is rejected in Dickey–Fuller tests for the German inﬂation rate and short-term interest rate, while, as previously stated, the factors are stationary.
3.8 CONCLUSIONSThe identiﬁcation of the factors that determine the time-series and cross-section behaviourof the term structure of interest rates is one of the most challenging research topics inﬁnance. In this chapter, it was shown that a two-factor constant volatility model describesquite 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 theterm premium structure can be calculated and short-term interest rate expectations derivedfrom the adjusted forward rate curve. The estimates obtained for the term premium curveare not inconsistent with the ﬁgures usually conjectured. Nevertheless, poorer results areobtained if the second factor is directly linked to the inﬂation rate, given that restrictionson the behaviour of that factor are imposed, generating less plausible shapes and ﬁguresfor the term premium curve.We identiﬁed within the sample two periods of poorer model performance, both relatedto world-wide gyrations in bond markets (Spring 1994 and 1998), which were characterised by sharp changes in long-term interest rates while short-term rates remainedstable.As to the evolution of bond yields in Germany during 1998, it seems that it is morethe (low) level of inﬂation expectations as compared to the level of real interest rates thatunderlies the dynamics of the yield curve during that year. However, there still remainssubstantial volatility in long bond yields to be explained. This could be related to thespillover effects of international bond market developments on the German bond marketin 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 realinterest rate, while a second factor is linked to inﬂation expectations. This conclusion ismuch in accordance with the empirical literature on the subject and is a relevant resultfor modelling the yield curve using information on macroeconomic variables.Therefore, modelling the yield curve behaviour, namely for VaR purposes, seems to bereasonably approached by simulations of (ex-post) real interest rates and lagged inﬂationrate. In addition, the results obtained suggest that a central bank has a decisive roleconcerning the bond market moves, given that it inﬂuences both the short and the longends of the yield curve, respectively by inﬂuencing the real interest rate and the inﬂationexpectations. Accordingly, the second factor may also be used as an indicator of monetarypolicy credibility.
REFERENCESAı̈t-Sahalia, Y. (1996), “Testing Continuous-Time Models of the Spot Interest Rate”, Review ofFinancial Studies, 9, 427–470.
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Babbs, S. H. and K. B. Nowman (1998), “An Application of Generalized Vasicek Term StructureModels 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 Forward Rates”, mimeo.Backus, D., S. Foresi and C. Telmer (1998), “Discrete-Time Models of Bond Pricing”, NBERworking paper no. 6736.Balduzzi, P., S. R. Das, S. Foresi and R. Sundaram (1996), “A Simple Approach to Three FactorAfﬁne 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 ofAtlanta, Economic Review, Fourth Quarter.Bolder, D. J. (2001), “Afﬁne 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 andInvestment Opportunities”, Journal of Financial Economics, 7, 265–296.Buhler, W., M. Uhrig-Homburg, U. Walter and T. Weber (1999), “An Empirical Comparison ofForward-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. Luı́s (1996), “The Term Structure of Interest Rates: a Comparison of Alternative Estimation Methods with an Application to Portugal”, Banco de Portugal, working paperno. 17/96, October 1996.Cassola, N. and J. B. Luı́s (2003), “A Two-Factor Model of the German Term Structure of InterestRates”, Applied Financial Economics, forthcoming.Chen, R. and L. Scott (1993a), “Maximum Likelihood Estimations for a Multi-Factor EquilibriumModel 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 Inﬂation: Motivation and Strategy, NBER Studies inBusiness Cycles, Vol. 30.Clarida, R., J. Gali and M. Gertler (1998), “Monetary Policy Rules in Practice – Some InternationalEvidence”, 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 AssetPrices”, Econometrica, 53, 363–384.Deacon, M. and A. Derry (1994), “Deriving Estimates of Inﬂation Expectations from the Prices ofUK Government Bonds”, Bank of England, working paper 23.De Jong, F. (1997), “Time-Series and Cross-section Information in Afﬁne Term Structure Models”,Center for Economic Research.Doan, T. A. (1995), RATS 4.0 User’s Manual, Estima, Evanston, IL.Dufﬁe, 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 Inﬂation”, American EconomicReview, 65, 269–282.Fama, E. F. (1990), “Term Structure Forecasts of Interest Rates, Inﬂation and Real Returns”, Journal 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 Inﬂation Expectations and RiskPremiums from Internationally Integrated Financial Markets”, Bank of Canada, working paper99-6.
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Gerlach, S. (1995), “The Information Content of the Term Structure: Evidence for Germany”, BISworking paper no. 29, September.Geyer, A. L. J. and S. Pichler (1996), “A State-Space Approach to Estimate and Test MultifactorCox–Ingersoll–Ross Models of the Term Structure”, mimeo.Gong, F. F. and E. M. Remolona (1997a), “A Three-factor Econometric Model of the US TermStructure”, FRBNY Staff Reports, 19, January.Gong, F. F. and E. M. Remolona (1997b), “Inﬂation Risk in the U.S. Yield Curve: The Usefulnessof Indexed Bonds”, Federal Reserve Bank, New York, June.Gong, F. F. and E. M. Remolona (1997c), “Two Factors Along the Yield Curve”, The ManchesterSchool 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 LongHorizons”, Journal of Financial Economics, 29, 59–80.Koedijk, K. G. and C. J. M. Kool (1995), “Future Inﬂation and the Information in InternationalTerm Structures”, Empirical Economics, 20, 217–242.Litterman, R. and J. Scheinkman (1991), “Common Factors Affecting Bond Returns”, Journal ofFixed Income, 1, June, 49–53.Luı́s, J. B. (2001), Essays on Extracting Information from Financial Asset Prices, PhD thesis, University of York.Mehra, Y. P. (1997), “The Bond Rate and Actual Future Inﬂation”, Federal Reserve Bank of Richmond, working paper 97-3, March.Mishkin, F. (1990a), “What Does the Term Structure Tell Us About Future Inﬂation”, Journal ofMonetary Economics, 25, 77–95.Mishkin, F. (1990b), “The Information in the Longer Maturity Term Structure About Future Inﬂation”, Quarterly Journal of Economics, 55, 815–828.Mishkin, F. (1991), “A Multi-country Study of the Information in the Shorter Maturity Term Structure About Future Inﬂation”, Journal of International Money and Finance, 10, 2–22.Nelson, C. R. and A. F. Siegel (1987), “Parsimonious Modelling of Yield Curves”, Journal ofBusiness, 60, 4.Remolona, E., M. R. Wickens and F. F. Gong (1998), “What was the Market’s View of U.K.Monetary Policy? Estimating Inﬂation Risk and Expected Inﬂation with Indexed Bonds”, FRBNYStaff 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 Speciﬁcations of the German Term Structure and its InformationContent Regarding Inﬂation”, Deutsche Bundesbank, D.P. 8/96.Schich, S. T. (1999), “What the Yield Curves Say About Inﬂation: 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 FinancialEconomics, 5, 177–188.Zin, S. (1997), “Discussion of Evans and Marshall”, Carnegie-Rochester Conference on PublicPolicy, November.
4
Forecasting and Trading Currency Volatility:An Application of Recurrent NeuralRegression and Model Combination∗CHRISTIAN L. DUNIS AND XUEHUAN HUANG
ABSTRACTIn this chapter, we examine the use of nonparametric Neural Network Regression (NNR)and Recurrent Neural Network (RNN) regression models for forecasting and trading currency volatility, with an application to the GBP/USD and USD/JPY exchange rates. Boththe results of the NNR and RNN models are benchmarked against the simpler GARCHalternative and implied volatility. Two simple model combinations are also analysed.The intuitively appealing idea of developing a nonlinear nonparametric approach toforecast FX volatility, identify mispriced options and subsequently develop a tradingstrategy based upon this process is implemented for the ﬁrst time on a comprehensivebasis. Using daily data from December 1993 through April 1999, we develop alternativeFX volatility forecasting models. These models are then tested out-of-sample over theperiod April 1999–May 2000, not only in terms of forecasting accuracy, but also in termsof trading 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 yet, somewhat surprisingly,a model combination which has the best overall performance in terms of forecastingaccuracy 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 inefﬁcient and/or the pricing formulae applied by marketparticipants were inadequate.
4.1 INTRODUCTIONExchange rate volatility has been a constant feature of the International Monetary Systemever since the breakdown of the Bretton Woods system of ﬁxed parities in 1971–73. Notsurprisingly, in the wake of the growing use of derivatives in other ﬁnancial 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. 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
Edited by C.L. Dunis, J. Laws and P. Naı̈m
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following the extension of the seminal work of Black–Scholes (1973) to foreign exchangeby Garman–Kohlhagen (1983), currency options have become an ever more popular wayto 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 thebest volatility prediction has become ever more crucial. True, the only unknown variablein the Garman–Kohlhagen pricing formula is precisely the future foreign exchange ratevolatility during the life of the option. With an “accurate” volatility estimate and knowingthe other variables (strike level, current level of the exchange rate, interest rates on bothcurrencies and maturity of the option), it is possible to derive the theoretical arbitrage-freeprice of the option. Just because there will never be such thing as a unanimous agreementon the future volatility estimate, market participants with a better view/forecast of theevolution of volatility will have an edge over their competitors.In a rational market, the equilibrium price of an option will be affected by changesin volatility. The higher the volatility perceived by market participants, the higher theoption’s price. Higher volatility implies a greater possible dispersion of the foreignexchange rate at expiry: all other things being equal, the option holder has logically anasset with a greater chance of a more proﬁtable exercise. In practice, those investors/marketparticipants who can reliably predict volatility should be able to control better the ﬁnancial risks associated with their option positions and, at the same time, proﬁt from theirsuperior 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 describethe volatility on hourly data, West and Cho (1995) analysed the predictive ability ofGARCH, AR and nonparametric models on weekly data, Jorion (1995)