The Black-Scholes- Merton Equation and Option Pricing
Darren E. Mason
Professor
Mathematics and Computer Science
Albion College
The Black-Scholes option pricing formula is a 1997 Nobel Prize winning result in economics
(& mathematical finance) that provides a framework for rational pricing of a large class of
stock options. In this talk we will discuss the basic idea of an option on an asset as well as
the problem of fair valuation of such a financial object. Then, assuming that the stock price
St follows a geometric Brownian motion, we will discuss a rough hedging argument that
results in the Black-Scholes partial differential equation as a necessary condition for risk-free
portfolio evolution. Using changes of coordinate systems and integrating factors, the Black-Scholes partial differential equation will be transformed into the classic heat (or diffusion)
equation, for which a standard integral solution form is known. Finally, we will use this
integral solution to derive the celebrated Black-Scholes option pricing formula. Some
limitations of this model will also be discussed.