British Call Option on Stocks under Stochastic Interest Rate

Abstract

The closed form expression for the price of the British put and call options have long
been established where both interest rate and volatility are assumed to be constant. In reality,
these assumptions do not fully reflect the variable nature of the financial markets. In this paper, we
derived a closed form expression for the arbitrage-free price of the British call option by assuming
stochastic interest rate which follows the Cox-Ingersoll-Ross model and constant volatility as

where the first term is the arbitrage-free price of the European call option under stochastic interest
rate and the second term is the early-exercise premmium. We have also shown that the price
function of the British call option satisfies the partial differential equation given by

Moreover, we have shown that the contract drift satisfies μc < rt+ρσ1σ2√rtλ(0, t+u) for u ∈ [0, τ ] and t ∈ [0, T].