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American Option Valuation under Continuous-Time Markov Chains

Part of:Mathematical financeMarkov processesProbabilistic 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 securities65C40: Computational Markov chains
Secondary:60J27: Continuous-time Markov processes on discrete state spaces

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015, pp. 378 - 401
Copyright
© Applied Probability Trust 

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