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Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching. (English) Zbl 1147.60320

Summary: 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 \(\theta\)-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.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
91B28 Finance etc. (MSC2000)

Software:

Mathematica
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References:

[1] W. J. Anderson, Continuous-Time Markov Chains, Springer Series in Statistics: Probability and Its Applications, Springer, New York, 1991. · Zbl 0731.60067
[2] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637-659, 1973. · Zbl 1092.91524 · doi:10.1086/260062
[3] M. Broadie and Kaya, “Exact simulation of stochastic volatility and other affine jump diffusion processes,” Working paper, 2003. · Zbl 1167.91363
[4] 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. · Zbl 1107.91325 · doi:10.1142/S0219024902001523
[5] 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. · Zbl 1274.91447 · doi:10.2307/1911242
[6] A. David, “Fluctuating confidence in stock markets: Implications for returns and volatility,” Journal of Financial and Quantitative Analysis, vol. 32, pp. 427-462, 1997. · doi:10.2307/2331232
[7] J. B. Detemple, “Further results on asset pricing with incomplete information,” Journal of Economic Dynamics & Control, vol. 15, no. 3, pp. 425-453, 1991. · Zbl 0732.90017 · doi:10.1016/0165-1889(91)90001-H
[8] \uI. \BI. Gīhman and A. V. Skorohod, Stochastic Differential Equations, vol. 72 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, New York, 1972. · Zbl 0242.60003
[9] P. Glasserman, Monte Carlo Methods in Financial Engineering, vol. 53 of Applications of Mathematics (New York), Springer, New York, 2004. · Zbl 1038.91045
[10] 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. · Zbl 0988.60038 · doi:10.1239/jap/996986756
[11] X. Guo, “Information and option pricings,” Quantitative Finance, vol. 1, no. 1, pp. 38-44, 2001. · doi:10.1088/1469-7688/1/1/302
[12] 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. · Zbl 1026.91048 · doi:10.1239/jap/1005091029
[13] 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. · Zbl 1061.90082 · doi:10.1137/S0036139903426083
[14] 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. · Zbl 1384.35131 · doi:10.1093/rfs/6.2.327
[15] 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.
[16] J. Hull and A. White, “The pricing of options on assets with stochastic volatilities,” Journal of Finance, vol. 42, pp. 281-300, 1987. · Zbl 1126.91369 · doi:10.2307/2328253
[17] A. Jobert and L. C. G. Rogers, “Option pricing with Markov-modulated dynamics,” preprint, 2004. · Zbl 1158.91380
[18] A. L. Lewis, Option Valuation under Stochastic Volatility. With Mathematica Code, Finance Press, California, 2000. · Zbl 0937.91060
[19] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood, Chichester, 1997. · Zbl 0892.60057
[20] R. C. Merton, “Theory of rational option pricing,” Bell Journal of Economics and Management Science, vol. 4, pp. 141-183, 1973. · Zbl 1257.91043 · doi:10.2307/3003143
[21] 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. · Zbl 0695.60055
[22] 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. · Zbl 1076.91021
[23] 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. · doi:10.1093/rfs/12.5.975
[24] 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. · Zbl 1117.91366 · doi:10.1142/S0219024905003323
[25] 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. · Zbl 1366.91148 · doi:10.1109/TAC.2004.824479
[26] 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. · Zbl 1044.65007 · doi:10.1016/j.matcom.2003.09.001
[27] Q. Zhang, “Stock trading: an optimal selling rule,” SIAM Journal on Control and Optimization, vol. 40, no. 1, pp. 64-87, 2001. · Zbl 0990.91014 · doi:10.1137/S0363012999356325
[28] 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. · Zbl 1136.93466 · doi:10.1109/9.811209
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