Investment Theory and Risk Management, + Website

Black-Scholes-Merton Model

As Δt → 0, the number of nodes in our binomial lattice goes to infinity and we are essentially moving to a continuous time option model. The Black-Scholes model (independently derived by Robert Merton) is a closed form continuous time option pricing model. Its derivation is mathematically rigorous, but I think we can sketch its derivation using the tools we've derived thus far. The model is based on the assumptions that the underlying stock price is a geometric Brownian motion process and that the stock price S has a lognormal distribution. The intuition is this: there are three assets—a bond, a stock, and a derivative. These assets’ dynamics can be described by the following respective models, which we've already derived:

We recognize, first, the deterministic return on a riskless bond followed by geometric Brownian motion describing the stock price movement and, finally, Ito's lemma, which describes the dynamics underlying the derivative on the stock. The insight behind Black-Scholes is that we could form a riskless portfolio consisting of the stock and the derivative, thus equating the return on this portfolio to the return on the riskless bond. This ...