Maximum Likelihood Estimation of Continuous-time Diffusion Models for Exchange Rates
0
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
Five diffusion models are estimated using three different foreign exchange rates to find an appropriate model for each. Daily spot exchange rates expressed as the prices of 1 euro, 1 British pound and 100 Japanese yen in US dollars, respectively denoted by USD/EUR, USD/GBP, and USD/100JPY, are used. The maximum likelihood estimation method is implemented after deriving an approximate log-transition density function (log-TDF) of the diffusion processes because the true log-TDF is unknown. Of the five models, the most general model is the best fit for the USD/GBP, and USD/100JPY exchange rates, but it is not the case for the case of USD/EUR. Although we could not find any evidence of the mean-reverting property for the USD/EUR exchange rate, the USD/GBP, and USD/ 100JPY exchange rates show the mean-reversion behavior. Interestingly, the volatility function of the USD/EUR exchange rate is increasing in the exchange rate while the volatility functions of the USD/GBP and USD/100Yen exchange rates have a U-shape. Our results reveal that more care has to be taken when determining a diffusion model for the exchange rate. The results also imply that we may have to use a more general diffusion model than those proposed in the literature when developing economic theories for the behavior of the exchange rate and pricing foreign currency options or derivatives.
Abstract
Five diffusion models are estimated using three different foreign exchange rates to find an appropriate model for each. Daily spot exchange rates expressed as the prices of 1 euro, 1 British pound and 100 Japanese yen in US dollars, respectively denoted by USD/EUR, USD/GBP, and USD/100JPY, are used. The maximum likelihood estimation method is implemented after deriving an approximate log-transition density function (log-TDF) of the diffusion processes because the true log-TDF is unknown. Of the five models, the most general model is the best fit for the USD/GBP, and USD/100JPY exchange rates, but it is not the case for the case of USD/EUR. Although we could not find any evidence of the mean-reverting property for the USD/EUR exchange rate, the USD/GBP, and USD/ 100JPY exchange rates show the mean-reversion behavior. Interestingly, the volatility function of the USD/EUR exchange rate is increasing in the exchange rate while the volatility functions of the USD/GBP and USD/100Yen exchange rates have a U-shape. Our results reveal that more care has to be taken when determining a diffusion model for the exchange rate. The results also imply that we may have to use a more general diffusion model than those proposed in the literature when developing economic theories for the behavior of the exchange rate and pricing foreign currency options or derivatives.
참고문헌 (36)
- Aït-Sahalia, Y., "Testing Continuous-Time Models of the Spot Interest Rate", Review of Financial Studies, 9(2), pp.385~426 (1996)
- Aït-Sahalia, Y., "Transition Densities for Interest Rate and Other Nonlinear Diffusions", Journal of Finance, 54(4), pp.1361~1395 (1999)
- Aït-Sahalia, Y., "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approximation Approach", Econometrica, 70(1), pp.223~262 (2002)
- Aït-Sahalia, Y., "Closed-Form Likelihood Expansions for Multivariate Diffusions", Annals of Statistics, 36(2), pp.906~937 (2008)
- Aït-Sahalia, Y., "Maximum Likelihood Estimation of Stochastic Volatility Models", Journal of Financial Economics, 83(2), pp.413~452 (2007)
- Aït-Sahalia, Y., "Estimating Affine Multifactor Term Structure Models Using Closed-Form Likelihood Expansions", Journal of Financial Economics, 98(1), pp.113~144 (2010)
- Akaike, H., "Second International Symposium on Information Theory", pp.267~281 (1973), Akademiai Kiado
- Akgiray, V., "Mixed Diffusion-Jump Process Modeling of Exchange Rate Movements", Review of Economics and Statistics, 70(4), pp.631~637 (1988)
- Bakshi, G., "A Refinement to Aït-Sahaliaís. 2000. Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach", Journal of Business, 78(5), pp.2037~2052 (2005)
- Bakshi, G., "Estimation of Continuous-time Models with an Application to Equity Volatility", Journal of Financial Economics, 82(1), pp.227~249 (2006)
- Baldeaux, J., "Pricing Currency Derivatives under the Benchmark Approach", Journal of Banking and Finance, 53, pp.34~48 (2015)
- Bates, D. S., "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options", Review of Financial Studies, 9(1), pp.69~107 (1996)
- Black, F., "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, 81(3), pp.637~654 (1973)
- Bollerslev, T., "Estimating Stochastic Volatility Diffusion Using Conditional Moments of Integrated Volatility", Journal of Econometrics, 109(1), pp.33~65 (2002)
- Chan, K., "An Empirical Comparison of Alternative Models of the Short-Term Interest Rate", Journal of Finance, 47(3), pp.1209~1227 (1992)
- Chang, J., "On the Approximate Maximum Likelihood Estimation For Diffusion Processes", Annals of Statistics, 39(6), pp.2820~2851 (2011)
- Seungmoon Choi, "Regime-Switching Univariate Diffusion Models of the Short-Term Interest Rate", Studies in Nonlinear Dynamics & Econometrics, 13(1) (2009), Walter de Gruyter GmbH[DOI]
- Choi, S., "Closed-Form Likelihood Expansions for Multivariate Time-Inhomogeneous Diffusions", Journal of Econometrics, 174(2), pp.45~65 (2013)
- Choi, S., "Explicit Form of Approximate Transition Probability Density Functions of Diffusion Processes", Journal of Econometrics, 187(1), pp.57~73 (2015)
- Approximate Transition Probability Density Function of a Multivariate Time-inhomogeneous Jump Diffusion Process in a Closed-Form Expression, "Choi, S.", University of Seoul, ~ (2019)
- Cox, J. C., "A Theory of the Term Structure of Interest Rates", Econometrica, 53(2), pp.385~407 (1985)
- Egorov, A. V., "Maximum Likelihood Estimation of Time Inhomogeneous Diffusions", Journal of Econometrics, 114(1), pp.107~139 (2003)
- Froot, K. A., "Exchange-Rate Dynamics under Stochastic Regime Shifts: A Unified Approach", Journal of International Economics, 31(3-4), pp.203~229 (1991)
- Froot, K. A., "Stochastic Process Switching: Some Simple Solutions", Econometrica, 59(1), pp.241~250 (1991)
- Garman, M. B., "Foreign Currency Option Values", Journal of International Money and Finance, 2(3), pp.231~237 (1983)
- Goutte, S., "ContinuousTime Regime-Switching Model Applied to Foreign Exchange Rate", Mathematical Finance Letters, 8 (2013)
- Grabbe, J. O., "The Pricing of Call and Put Options on Foreign Exchange", Journal of International Money and Finance, 2(3), pp.239~253 (1983)
- Krugman, P. R., "Target Zones and Exchange Rate Dynamics", Quarterly Journal of Economics, 106(3), pp.669~682 (1991)
- Larsen, K. S., "Diffusion Models for Exchange Rates in a Target Zone", Mathematical Finance, 17(2), pp.285~306 (2007)
- Li, M., "A Damped Diffusion Framework for Financial Modelingand Closed-Form Maximum Likelihood Estimation", Journal of Economic Dynamics and Control, 34(2), pp.132~157 (2010)
- Merton, R. C., "Theory of Rational Option Pricing", Bell Journal of Economics and Management Science, 4(1), pp.141~183 (1973)
- Schwarz, G., "Estimating the Dimension of a Model", Annals of Statistics, 6(2), pp.461~464 (1978)
- Swishchuk, A., "Currency Derivatives Pricing for Markov-Modulated Merton Jump-Diffusion Spot Forex Rate", Journal of Mathematical Finance, 4(4), pp.265~278 (2014)
- Tucker, A. L., "A Study of Diffusion Processes for Foreign Exchange Rates", Journal of International Money and Finance, 6(4), pp.465~478 (1987)
- Vasicek, O., "An Equilibrium Characterization of the Term Structure", Journal of Financial Economics, 5(2), pp.177~188 (1977)
- Yu, J., "Closed-Form Likelihood Approximation and Estimation of Jump-Diffusions with an Applicaiton to the Realignment Risk of the Chinese Yuan", Journal of Econometrics, 141(2), pp.1245~1280 (2007)