Abstract
This work develops a class of stochastic optimization algorithms for pricing American put options. The stock model is a regime-switching geometric Brownian motion. The switching process represents macro market states such as market trends, interest rates, etc. The solutions of pricing American options may be characterized by certain threshold values. Here, we show how one can use a stochastic approximation (SA) method to determine the optimal threshold levels. For option pricing in a finite horizon, a SA procedure is carried for a fixed time T. As T varies, the optimal threshold values obtained using stochastic approximation trace out a curve, called the threshold frontier. Convergence and rates of convergence are obtained using weak convergence methods and martingale averaging techniques. The proposed approach provides us with a viable computational approach, and has advantage in terms of the reduced computational complexity compared with the variational or quasi-variational inequality approach for optimal stopping.
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References
G. Barone-Adesi and R. Whaley, Efficient analytic approximation of American option values, J. Finance, 42 (1987), 301–320.
J. Buffington and R.J. Elliott, American options with regime switching, Internat. J. Theoretical Appl. Finance, 5 (2002), 497–514.
P. Billingsley, Convergence of Probability Measures, J. Wiley, New York, NY, 1968.
G.B. Di Masi, Y.M. Kabanov and W.J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory Probab. Appl., 39 (1994), 173–181.
J.P. Fouque, G. Papanicolaou, and K.R. Sircar, Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, 2000.
M.C. Fu, S.B. Laprise, D.B. Madan, Y. Su, and R. Wu, Pricing American options: A comparison of Monte Carlo simulation approaches, J. Comput. Finance, Vol. 4 (2001), 39–88.
M.C. Fu, R. Wu, G. Gürkan, and A. Y. Demir, A note on perturbation analysis estimators for American-style options, Probab. Eng. Informational Sci., 14 (2000), 385–392.
P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer-Verlag, New York, 2003.
X. Guo and Q. Zhang, Closed-form solutions for perpetual American put options with regime switching, SIAM J. Appl. Math., 64 (2004), 2034–2049.
Y.C. Ho and X.R. Cao, Perturbation Analysis of Discrete Event Dynamic Systems, Kluwer, Boston, MA, 1991.
J.D. Hamilton, A new approach to the economic analysis of nonstationary time series, Econometrica, 57 (1989), 357–384.
J.C. Hull and A. White, The pricing of options on assets with stochastic volatilities, J. Finance, 42 (1987), 281–300.
H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.
H.J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications, 2nd Ed., Springer-Verlag, New York, 2003.
F. A. Longstaff and E. S. Schwartz, Valuing American Options by Simulation: A simple least-squares approach, Rev. Financial Studies, 14 (2001), 113–147.
R.C. Merton, Lifetime portfolio selection under uncertainty: The continuous time case, Rev. Economics Statist., 51 (1969), 247–257.
D.D. Yao, Q. Zhang, and X. Zhou, A regime-switching model for European option pricing, in this volume (2006).
G. Yin, R.H. Liu, and Q. Zhang, Recursive algorithms for stock Liquidation: A stochastic optimization approach, SIAM J. Optim., 13 (2002), 240–263.
G. Yin, J.W. Wang, Q. Zhang, and Y.J. Liu, Stochastic optimization algorithms for pricing American put options under regime-switching models, to appear in J. Optim. Theory Appl. (2006).
Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim., 40, (2001), 64–87.
Q. Zhang and G. Yin, Nearly optimal asset allocation in hybrid stock-investment models, J. Optim. Theory Appl., 121 (2004), 197–222.
X.Y. Zhou and G. Yin, Markowitz mean-variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466–1482.
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Yin, G., Wang, J.W., Zhang, Q., Liu, Y.J., Liu, R.H. (2006). Pricing American Put Options Using Stochastic Optimization Methods. In: Yan, H., Yin, G., Zhang, Q. (eds) Stochastic Processes, Optimization, and Control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems. International Series in Operations Research & Management Science, vol 94. Springer, Boston, MA . https://doi.org/10.1007/0-387-33815-2_15
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DOI: https://doi.org/10.1007/0-387-33815-2_15
Publisher Name: Springer, Boston, MA
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