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

In this paper, we study pricing of American put options on a nondividend-paying stock in the Black and Scholes market with a stochastic interest rate and finite-time maturity. We prove that the option value is a C1 function of the initial time, interest rate, and stock price. By means of Itô 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 parametrization 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.