This paper aims to derive and solve the Black-Scholes partial differential equation (PDE) used to price options. Options allow investors the possibility for great gain with small probability for a large loss. Thus, they can be very valuable if money is invested correctly. Brownian motion can be used to model the change in stock prices and is the jumping off point for the derivation of the Black-Scholes PDE. The derivation of the PDE and its solution using the method of Fourier transforms will be shown. An application of the pricing formula using actual stock values will also be given.
Engstrom, H. (2010). Financial Mathematics: Options, Arbitrage, and the Black-Scholes PDE (Undergraduate honors thesis, University of Redlands). Retrieved from http://inspire.redlands.edu/cas_honors/128