The Valuation of American Options with Stochastic Stopping Time Constraints

Abstract

This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.

Key Words:

• American options
• optimal stopping under constraints
• Feller process
• out-performance options
• management options
• Monte Carlo simulation

Acknowledgements

We thank Walter Farkas for helpful comments and suggestions. Markus Leippold acknowledges the financial support of the Swiss National Science Foundation (NCCR FINRISK).

Notes

¹ Meyer coined the term ‘class D’ in honour of J. L. Doob. For this and other anecdotes, see the historic review by Jarrow and Protter (2004).

² We also tested for larger values of m, but did not find much of an improvement.