TITLE:
Pricing American Options Using Transition Probabilities: A Dynamical Systems Approach
AUTHORS:
Rocio Elizondo, Pablo Padilla, Mogens Bladt
KEYWORDS:
American Options, Fokker-Planck, Black-Scholes, Samuelson, Probability Density Function
JOURNAL NAME:
Open Journal of Statistics,
Vol.5 No.6,
October
20,
2015
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.