Pricing American Options Using Transition Probabilities: A Dynamical Systems Approach

Rocio Elizondo¹, Pablo Padilla², Mogens Bladt³

¹Directorate of Economic Studies, DGIE, Banco de México, Mexico City, Mexico.
²Department of Mathematics and Mechanics, IIMAS, UNAM, Mexico City, Mexico.
³Department of Probability and Statistics, IIMAS, UNAM, Mexico City, Mexico.

DOI: 10.4236/ojs.2015.56056                

Abstract

We give a new way to price American options by using Samuelson’s formula. We first obtain the option price corresponding to a European option at time t, weighing it by the probability that the underlying asset takes the value S at time t. We then use Samuelson’s formula with this factor which is given by the solution of the Fokker-Planck (Kolmogorov) equation for the transition probability density. The main advantage of this approach is that we can systematically introduce the effect of macroeconomic factors. If a macroeconomic framework is given by a dynamical system in the form of a set of ordinary differential equations we only have to solve a partial differential equation for the transition probability density. In this context, we verify, for the sake of consistency, that this formula coincides with the Black-Scholes model and compare several numerical implementations.

Keywords

American Options, Fokker-Planck, Black-Scholes, Samuelson, Probability Density Function

Conflicts of Interest

The authors declare no conflicts of interest.

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