Euler-Maruyama approximations in mean-reverting
stochastic volatility  model under regime-switching

Mao, Xuerong; Truman, Aubrey; Yuan, Chenggui

doi:https://doi.org/10.1155/JAMSA/2006/80967

International Journal of Stochastic Analysis

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Volume 2006 | Article ID 080967 | https://doi.org/10.1155/JAMSA/2006/80967

Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching

Xuerong Mao,¹Aubrey Truman,²and Chenggui Yuan²

Received28 Dec 2005

Revised09 Feb 2006

Accepted09 Feb 2006

Published13 Jul 2006

Abstract

Stochastic differential equations (SDEs) under regime-switching have recently been developed to model various financial quantities. In general, SDEs under regime-switching have no explicit solutions, so numerical methods for approximations have become one of the powerful techniques in the valuation of financial quantities. In this paper, we will concentrate on the Euler-Maruyama (EM) scheme for the typical hybrid mean-reverting θ-process. To overcome the mathematical difficulties arising from the regime-switching as well as the non-Lipschitz coefficients, several new techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems.

References

W. J. Anderson, Continuous-Time Markov Chains, Springer Series in Statistics: Probability and Its Applications, Springer, New York, 1991.
View at: Zentralblatt MATH | MathSciNet
F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637–659, 1973.
View at: Publisher Site | Google Scholar
M. Broadie and Kaya, “Exact simulation of stochastic volatility and other affine jump diffusion processes,” Working paper, 2003.
View at: Google Scholar
J. Buffington and R. J. Elliott, “American options with regime switching,” International Journal of Theoretical and Applied Finance, vol. 5, no. 5, pp. 497–514, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
J. C. Cox, J. E. Ingersoll Jr., and S. A. Ross, “A theory of the term structure of interest rates,” Econometrica, vol. 53, no. 2, pp. 385–407, 1985.
View at: Google Scholar | MathSciNet
A. David, “Fluctuating confidence in stock markets: Implications for returns and volatility,” Journal of Financial and Quantitative Analysis, vol. 32, pp. 427–462, 1997.
View at: Google Scholar
J. B. Detemple, “Further results on asset pricing with incomplete information,” Journal of Economic Dynamics & Control, vol. 15, no. 3, pp. 425–453, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Ĭ. Ī. Gīhman and A. V. Skorohod, Stochastic Differential Equations, vol. 72 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, New York, 1972.
View at: Zentralblatt MATH | MathSciNet
P. Glasserman, Monte Carlo Methods in Financial Engineering, vol. 53 of Applications of Mathematics (New York), Springer, New York, 2004.
View at: Zentralblatt MATH | MathSciNet
X. Guo, “An explicit solution to an optimal stopping problem with regime switching,” Journal of Applied Probability, vol. 38, no. 2, pp. 464–481, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
X. Guo, “Information and option pricings,” Quantitative Finance, vol. 1, no. 1, pp. 38–44, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
X. Guo and L. Shepp, “Some optimal stopping problems with nontrivial boundaries for pricing exotic options,” Journal of Applied Probability, vol. 38, no. 3, pp. 647–658, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
X. Guo and Q. Zhang, “Closed-form solutions for perpetual American put options with regime switching,” SIAM Journal on Applied Mathematics, vol. 64, no. 6, pp. 2034–2049, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. L. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options,” Review of Financial Studies, vol. 6, no. 2, pp. 327–343, 1993.
View at: Publisher Site | Google Scholar
D. J. Higham and X. Mao, “Convergence of Monte Carlo simulations involving the mean-reverting square root process,” Journal of Computational Finance, vol. 8, pp. 35–61, 2005.
View at: Google Scholar
J. Hull and A. White, “The pricing of options on assets with stochastic volatilities,” Journal of Finance, vol. 42, pp. 281–300, 1987.
View at: Google Scholar
A. Jobert and L. C. G. Rogers, “Option pricing with Markov-modulated dynamics,” preprint, 2004.
View at: Google Scholar
A. L. Lewis, Option Valuation under Stochastic Volatility. With Mathematica Code, Finance Press, California, 2000.
View at: Zentralblatt MATH | MathSciNet
X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood, Chichester, 1997.
View at: Zentralblatt MATH | MathSciNet
R. C. Merton, “Theory of rational option pricing,” Bell Journal of Economics and Management Science, vol. 4, pp. 141–183, 1973.
View at: Google Scholar | MathSciNet
A. V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, vol. 78 of Translations of Mathematical Monographs, American Mathematical Society, Rhode Island, 1989.
View at: Zentralblatt MATH | MathSciNet
R. H. Stockbridge, “Portfolio optimization in markets having stochastic rates,” in Stochastic Theory and Control (Lawrence, KS, 2001), vol. 280 of Lecture Notes in Control and Information Sciencs, pp. 447–458, Springer, Berlin, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
P. Veronesi, “Stock market overreactions to bad news in good times: a rational expectations equilibrium model,” Review of Financial Studies, vol. 12, no. 5, pp. 975–1007, 1999.
View at: Publisher Site | Google Scholar
S. Wu and Y. Zeng, “A general equilibrium model of the term structure of interest rates under regime-switching risk,” International Journal of Theoretical and Applied Finance, vol. 8, no. 7, pp. 839–869, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
G. Yin and X. Y. Zhou, “Markowitz's mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits,” IEEE Transactions on Automatic Control, vol. 49, no. 3, pp. 349–360, 2004.
View at: Google Scholar | MathSciNet
C. Yuan and X. Mao, “Convergence of the Euler-Maruyama method for stochastic differential equations with Markovian switching,” Mathematics and Computers in Simulation, vol. 64, no. 2, pp. 223–235, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Q. Zhang, “Stock trading: an optimal selling rule,” SIAM Journal on Control and Optimization, vol. 40, no. 1, pp. 64–87, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Q. Zhang and G. Yin, “On nearly optimal controls of hybrid LQG problems,” IEEE Transactions on Automatic Control, vol. 44, no. 12, pp. 2271–2282, 1999.
View at: Google Scholar | MathSciNet

Copyright

Copyright © 2006 Xuerong Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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