Option Prices with Stochastic Interest Rates - Black/Scholes and Ho/Lee Unified

Abstract

The option pricing model by Black and Scholes (1973) and the term structure model by Ho and Lee (1986) are among the most influential models of capital market theory. While Black/Scholes consider stock option prices under the assumption of a constant deterministic interest rate, Ho and Lee were the first to model the term structure of interest rates as a stochastic object where the initial term structure concides with the empirically observed one. Whereas the original Ho/Lee-paper used a binomial setting, Heath/Jarrow/Morton (1990) could describe the limit behaviour of that model which implies normally distributed interest rates. The present paper will show that a properly enriched Black/Scholes-model and an extended Ho/Lee-model are natural companions such that an option pricing model results which is compatible with Ho/Lee term structures. The method we use is stochastic discounting. We assume the economy's asset prices to be governed by a lognormally distributed stochastic discount factor which implies a term structure compatible to the limit case of the Ho/Lee model but is much more general in that it is also compatible with, e.g., the Vasicek-model of the term structure. Assuming that the stock price at maturity is lognormally distributed implies that the stock price must be log-normally distributed at any point in time and, under an additional assunmption, it follows a geometric Brownian motion as it is assumed in the classical Black/Scholes-world but, contrary to their model the volatility is no longer time-invariant. However, shifting from modelling the stock price itself to modelling the forward price of the stock proves to be the natural way to link the Black-Scholes model to stochastic interest rates. Nevertheless, it turns out that the stock price process and the process of the (term structure of) interest rate must not be specified independently of each other.Given the combined model it is an easy task to compute prices for European style derivatives as, e.g., call options on such a stock or on default-free bonds as well. The resulting option pricing formula (for the stock price) is a natural extension of the Black-Scholes-formula.