  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.
