The Black-Scholes Model

Abstract

The option pricing model developed by Black and Scholes (1973), formalized and extended in the same year by Merton (1973a), enjoys great popularity. It is computationally simple and, like all arbitrage-based pricing models, does not require the knowledge of an investor’s risk preferences. Option valuation within the Black-Scholes framework is based on the already familiar concept of perfect replication of contingent claims. More specifically, we will show that an investor can replicate an option’s return stream by continuously rebalancing a self-financing portfolio involving stocks and risk-free bonds. For instance, a replicating portfolio for a call option involves, at any date t before the option’s expiry date, a long position in stock, and a short position in risk-free bonds. By definition, the wealth at time t of a replicating portfolio equals the arbitrage price of an option. Our main goal is to derive closed-form expressions for both the option’s price and the replicating strategy in the Black-Scholes setting. To do this in a formal way, we need first to construct an arbitrage-free market model with continuous trading. This can be done relatively easily if we take for granted certain results from the theory of stochastic processes, more precisely, from the Itô stochastic calculus. The theory of Itô stochastic integration is presented in several monographs; to cite a few, Elliott (1982), Karatzas and Shreve (1988), Protter (1990), Revuz and Yor (1991), and Durrett (1996).