Advances in Applied Probability

American option valuation under continuous-time Markov chains

B. Eriksson and M. R. Pistorius

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 2 (2015), 378-401.

Dates
First available in Project Euclid: 25 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1435236980

Digital Object Identifier
doi:10.1239/aap/1435236980

Mathematical Reviews number (MathSciNet)
MR3360382

Zentralblatt MATH identifier
06458824

Subjects
Primary: 91G20: Derivative securities
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 65C40: Computational Markov chains

Keywords
Markov chain American option free-boundary problem optimal stopping Feller process numerical approximation

Citation

Eriksson, B.; Pistorius, M. R. American option valuation under continuous-time Markov chains. Adv. in Appl. Probab. 47 (2015), no. 2, 378--401. doi:10.1239/aap/1435236980. https://projecteuclid.org/euclid.aap/1435236980


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