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American Option Valuation under Continuous-Time Markov Chains

Part of: Mathematical finance Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  22 February 2016

B. Eriksson  and
M. R. Pistorius
B. Eriksson*
Affiliation:
Imperial College London
M. R. Pistorius*
Affiliation:
Imperial College London
*
∗ Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
∗ Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
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Abstract

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This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.

Keywords

Markov chainAmerican optionfree-boundary problemoptimal stoppingFeller processnumerical approximation

MSC classification

Primary: 91G20: Derivative securities 65C40: Computational Markov chains
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 378 - 401
Copyright
© Applied Probability Trust 

References

Abergel, F. and Jedidi, A. (2011). A mathematical approach to order book modelling. In Econophysics of Order-Driven Markets, Springer, Milan, pp. 93–107.CrossRefGoogle Scholar
Ahn, J. and Song, M. (2007). Convergence of the trinomial tree method for pricing European/American options. Appl. Math. Comput. 189, 575–582.Google Scholar
Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Prob. 15, 2062–2080.Google Scholar
Boyarchenko, S. I. and Levendorskiī, S. Z. (2000). Option pricing for truncated Lévy processes. Internat. J. Theoret. Appl. Finance 3, 549–552.Google Scholar
Boyarchenko, S. and Levendorskiī, S. (2002). Perpetual American options under Lévy processes. SIAM J. Control Optimization 40, 1663–1696.Google Scholar
Boyarchenko, S. I. and Levendorskiī, S. Z. (2007). Irreversible Decisions under Uncertainty: Optimal Stopping Made Easy. Springer, Berlin.Google Scholar
Carr, P. (1998). Randomization and the American put. Rev. Financial. Stud. 11, 597–626.Google Scholar
Chung, K. L. and Walsh, J. B. (2005). Markov Processes, Brownian Motion, and Time Symmetry, 2nd edn. Springer, New York.Google Scholar
Cox, J. (1996). Notes on option pricing I: constant elasticity of variance diffusions. J. Portfolio Manag. 22, 15–17.Google Scholar
Cox, J. C., Ross, S. A. and Rubinstein, M. (1979). Option pricing: a simplified approach. J. Financial Econom. 7, 229–263.Google Scholar
Detemple, J. (2006). American-Style Derivatives. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Eriksson, B. (2013). On the valuation of barrier and American options in local volatility models with Jumps. , Imperial College London.Google Scholar
Jacka, S. D. (1991). Optimal stopping and the American put. Math. Finance 1, 1–14.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.Google Scholar
Kou, S. G. (2002). A Jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101.CrossRefGoogle Scholar
Kou, S. G. and Wang, H. (2004). Option pricing under a double exponential Jump diffusion model. Manag. Sci. 50, 1178–1192.Google Scholar
Kushner, H. J. (1997). Numerical methods for stochastic control problems in finance. In Mathematics of Derivative Securities (Publ. Newton Inst. 15), Cambridge University Press, pp. 504–527.Google Scholar
Kushner, H. J. and Dupuis, P. (2001). Numerical Methods for Stochastic Control Problems in Continuous Time, 2nd edn. Springer, New York.Google Scholar
Lamberton, D. (1998). Error estimates for the binomial approximation of American put options. Ann. Appl. Prob. 8, 206–233.Google Scholar
Lamberton, D. and Mikou, M.A. (2012). The smooth-fit property in an exponential Lévy model. J. Appl. Prob. 49, 137–149.Google Scholar
Lamberton, D. and Mikou, M. A. (2013). Exercise boundary of the American put near maturity in an exponential Lévy model. Finance Stoch. 17, 355–394.Google Scholar
McKean, H. P. Jr. (1965). Appendix: a free boundary problem for the heat equation arising from a problem in mathematical economics. Indust. Manag. Rev. 6, 32–39.Google Scholar
Mijatović, A. and Pistorius, M. (2013). Continuously monitored barrier options under Markov processes. Math. Finance 23, 1–38.Google Scholar
Peskir, G. (2005). On the American option problem. Math. Finance 15, 169–181.Google Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
Szimayer, A. and Maller, R. A. (2007). Finite approximation schemes for Lévy processes, and their application to optimal stopping problems. Stoch. Process. Appl. 117, 1422–1447.Google Scholar
Wong, H. Y. and Zhao, J. (2008). An artificial boundary method for American option pricing under the CEV model. SIAM J. Numerical Anal. 46, 2183–2209.Google Scholar