Actuarial Finance

17Rigorous derivations of the Black-Scholes formula***

In Chapter 16 we provided a derivation of the Black-Scholes formula based on limiting arguments. Indeed, we have shown that when the time step in a binomial tree becomes increasingly small, the geometric random walk followed by the stock price gradually approaches that of a geometric Brownian motion. Although it provides a lot of the intuition and explains where many results come from, the approach lacks some rigor.

This chapter intends to fill these gaps by providing a more advanced treatment of the BSM model based upon advanced tools such as stochastic calculus (see Chapter 15), partial differential equations and changes of probability measures. Our main objective is to provide two rigorous derivations of the Black-Scholes formula using either partial differential equations or changes of probability measures. This chapter is therefore targeted at readers with a stronger mathematical background and who have read Chapter 15. This chapter is not mandatory to understand the upcoming chapters.

More specifically, the learning objectives are to:

• distinguish an ordinary differential equation (ODE) from a partial differential equation (PDE);
• understand the link between PDEs and diffusion processes as given by the Feynman-Kač formula;
• derive and solve the Black-Scholes PDE for simple payoffs;
• understand how to price and replicate simple derivatives with the Black-Scholes PDE;
• apply a change of probability measure to random variables ...