Browse > Article
http://dx.doi.org/10.11644/KIEP.EAER.2020.24.1.372

Maximum Likelihood Estimation of Continuous-time Diffusion Models for Exchange Rates  

Choi, Seungmoon (University of Seoul)
Lee, Jaebum (University of Seoul)
Publication Information
East Asian Economic Review / v.24, no.1, 2020 , pp. 61-87 More about this Journal
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.
Keywords
Foreign Exchange Rate; Diffusion Model; Maximum Likelihood Estimation; US Dollar; Euro; British Pound; Japanese Yen;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Ait-Sahalia, Y. 1996. "Testing Continuous-Time Models of the Spot Interest Rate," Review of Financial Studies, vol. 9, no. 2, pp. 385-426.   DOI
2 Ait-Sahalia, Y. 1999. "Transition Densities for Interest Rate and Other Nonlinear Diffusions," Journal of Finance, vol. 54, no. 4, pp. 1361-1395.   DOI
3 Ait-Sahalia, Y. 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approximation Approach," Econometrica, vol. 70, no. 1, pp. 223-262.   DOI
4 Ait-Sahalia, Y. 2008. "Closed-Form Likelihood Expansions for Multivariate Diffusions," Annals of Statistics, vol. 36, no. 2, pp. 906-937.   DOI
5 Ait-Sahalia, Y. and R. Kimmel. 2007. "Maximum Likelihood Estimation of Stochastic Volatility Models," Journal of Financial Economics, vol. 83, no. 2, pp. 413-452.   DOI
6 Ait-Sahalia, Y. and R. Kimmel. 2010. "Estimating Affine Multifactor Term Structure Models Using Closed-Form Likelihood Expansions," Journal of Financial Economics, vol. 98, no. 1, pp. 113-144.   DOI
7 Akaike, H. 1973. "Information Theory and an Extension of the Maximum Likelihood Principle, In Petrov, B. N. and F. Csaki. (eds.) Second International Symposium on Information Theory. Budapest: Akademiai Kiado. pp. 267-281.
8 Akgiray, V. and G. G. Booth. 1988. "Mixed Diffusion-Jump Process Modeling of Exchange Rate Movements," Review of Economics and Statistics, vol. 70, no. 4, pp. 631-637.   DOI
9 Bakshi, G. and N. Ju. 2005. "A Refinement to Ait-Sahaliais. 2000. Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach," Journal of Business, vol. 78, no. 5, pp. 2037-2052.   DOI
10 Bakshi, G., Ju, N. and H. Ou-Yang. 2006. "Estimation of Continuous-time Models with an Application to Equity Volatility," Journal of Financial Economics, vol. 82, no. 1, pp. 227-249.   DOI
11 Bollerslev, T. and H. Zhou. 2002. "Estimating Stochastic Volatility Diffusion Using Conditional Moments of Integrated Volatility," Journal of Econometrics, vol. 109, no. 1, pp. 33-65.   DOI
12 Baldeaux, J., Grasselli, M. and E. Platen. 2015. "Pricing Currency Derivatives under the Benchmark Approach," Journal of Banking and Finance, vol. 53, pp. 34-48.   DOI
13 Bates, D. S. 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, vol. 9, no. 1, pp. 69-107.   DOI
14 Black, F. and M. Scholes. 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, vol. 81, no. 3, pp. 637-654.   DOI
15 Chan, K., Karolyi, G. A., Longstaff, F. and A. Sanders. 1992. "An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, vol. 47, no. 3, pp. 1209-1227.   DOI
16 Chang, J. and S. X. Chen. 2011. "On the Approximate Maximum Likelihood Estimation For Diffusion Processes," Annals of Statistics, vol. 39, no. 6, pp. 2820-2851.   DOI
17 Choi, S. 2009. "Regime-Switching Univariate Diffusion Models of the Short-Term Interest Rate," Studies in Nonlinear Dynamics & Econometrics, vol. 13, no. 1, Article 4. (accessed January 21, 2020)
18 Choi, S. 2013. "Closed-Form Likelihood Expansions for Multivariate Time-Inhomogeneous Diffusions," Journal of Econometrics, vol. 174, no. 2, pp. 45-65.   DOI
19 Choi, S. 2015. "Explicit Form of Approximate Transition Probability Density Functions of Diffusion Processes," Journal of Econometrics, vol. 187, no. 1, pp. 57-73.   DOI
20 Choi, S. 2019. Approximate Transition Probability Density Function of a Multivariate Time-inhomogeneous Jump Diffusion Process in a Closed-Form Expression, Working paper, University of Seoul.
21 Cox, J. C., Ingersoll, J. E. and S. A. Ross. 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, vol. 53, no. 2, pp. 385-407.   DOI
22 Egorov, A. V., Li, H. and Y. Xu. 2003. "Maximum Likelihood Estimation of Time Inhomogeneous Diffusions," Journal of Econometrics, vol. 114, no.1, pp. 107-139.   DOI
23 Froot, K. A. and M. Obstfeld. 1991a. "Exchange-Rate Dynamics under Stochastic Regime Shifts: A Unified Approach," Journal of International Economics, vol. 31, no. 3-4, pp. 203-229.   DOI
24 Garman, M. B. and S. Kohlhagen.1983. "Foreign Currency Option Values," Journal of International Money and Finance, vol. 2, no. 3, pp. 231-237.   DOI
25 Goutte, S. and B. Zou. 2013. "ContinuousTime Regime-Switching Model Applied to Foreign Exchange Rate," Mathematical Finance Letters, Article ID 8.
26 Grabbe, J. O. 1983. "The Pricing of Call and Put Options on Foreign Exchange," Journal of International Money and Finance, vol. 2, no. 3, pp. 239-253.   DOI
27 Krugman, P. R. 1991. "Target Zones and Exchange Rate Dynamics," Quarterly Journal of Economics, vol. 106, no. 3, pp. 669-682.   DOI
28 Larsen, K. S. and M. Sorensen. 2007. "Diffusion Models for Exchange Rates in a Target Zone," Mathematical Finance, vol. 17, no. 2, pp. 285-306.   DOI
29 Li, M. 2010. "A Damped Diffusion Framework for Financial Modelingand Closed-Form Maximum Likelihood Estimation," Journal of Economic Dynamics and Control, vol. 34, no. 2, pp. 132-157.   DOI
30 Merton, R. C. 1973. "Theory of Rational Option Pricing," Bell Journal of Economics and Management Science, vol. 4, no. 1, pp. 141-183.   DOI
31 Schwarz, G. 1978. "Estimating the Dimension of a Model," Annals of Statistics, vol. 6, no. 2, pp. 461-464.   DOI
32 Froot, K. A. and M. Obstfeld. 1991b. "Stochastic Process Switching: Some Simple Solutions," Econometrica, vol. 59, no. 1, pp. 241-250.   DOI
33 Swishchuk, A., Tertychnyi, M. and W. Hoang. 2014. "Currency Derivatives Pricing for Markov-Modulated Merton Jump-Diffusion Spot Forex Rate," Journal of Mathematical Finance, vol. 4, no. 4, pp. 265-278.   DOI
34 Tucker, A. L. and E. Scott. 1987. "A Study of Diffusion Processes for Foreign Exchange Rates," Journal of International Money and Finance, vol. 6, no. 4, pp. 465-478.   DOI
35 Vasicek, O. 1977. "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, vol. 5, no. 2, pp. 177-188.   DOI
36 Yu, J. 2007. "Closed-Form Likelihood Approximation and Estimation of Jump-Diffusions with an Applicaiton to the Realignment Risk of the Chinese Yuan," Journal of Econometrics, vol. 141, no. 2, pp. 1245-1280.   DOI