The American put with finite-time maturity and stochastic interest rate

TL;DR

This paper analyzes the pricing and optimal exercise boundary of American put options with finite maturity in a stochastic interest rate environment, providing theoretical proofs and numerical methods applicable to models like CIR and Vasicek.

Contribution

It introduces a rigorous mathematical framework for American put options with stochastic interest rates, including existence, uniqueness, and numerical computation of the exercise boundary.

Findings

The option value is a $C^1$ function of initial parameters.

Derived an early exercise premium formula using Ito calculus.

Numerical computation of the exercise boundary for Vasicek model.

Abstract

In this paper we study pricing of American put options on the Black and Scholes market with a stochastic interest rate and finite-time maturity. We prove that the option value is a $C^1$ function of the initial time, interest rate and stock price. By means of Ito calculus we rigorously derive the option value's early exercise premium formula and the associated hedging portfolio. We prove the existence of an optimal exercise boundary splitting the state space into continuation and stopping region. The boundary has a parametrisation as a jointly continuous function of time and stock price, and it is the unique solution to an integral equation which we compute numerically. Our results hold for a large class of interest rate models including CIR and Vasicek models. We show a numerical study of the option price and the optimal exercise boundary for Vasicek model.