The Binomial Tree Method and Explicit Difference Schemes for American Options with Time Dependent Coefficients

TL;DR

This paper develops and analyzes binomial tree methods and explicit difference schemes for American options with time-dependent coefficients, addressing issues of monotonicity, convergence, and symmetry in pricing models.

Contribution

It introduces a time interval partition method that maintains binomial tree dynamics for American options with time-dependent coefficients and proves properties like monotonicity and convergence.

Findings

BTM and EDS prices are monotonic in time under certain conditions.

Convergence of EDS to viscosity solutions is established.

Price functions exhibit linear homogeneity and call-put symmetry.

Abstract

Binomial tree methods (BTM) and explicit difference schemes (EDS) for the variational inequality model of American options with time dependent coefficients are studied. When volatility is time dependent, it is not reasonable to assume that the dynamics of the underlying asset's price forms a binomial tree if a partition of time interval with equal parts is used. A time interval partition method that allows binomial tree dynamics of the underlying asset's price is provided. Conditions under which the prices of American option by BTM and EDS have the monotonic property on time variable are found. Using convergence of EDS for variational inequality model of American options to viscosity solution the decreasing property of the price of American put options and increasing property of the optimal exercise boundary on time variable are proved. First, put options are considered. Then the linear homogeneity and call-put symmetry of the price functions in the BTM and the EDS for the variational inequality model of American options with time dependent coefficients are studied and using them call options are studied.