ABSTRACT
Show how differential equations model continuously varying asset prices. Introduce properties of Brownian motion: normal distribution, independent increments, variance and the martingale property. Random walk properties mimic and illustrate Brownian motion. Derive the Black-Scholes formula as an expectation of the lognormal distribution. Implement the formulas in software. Apply Calculus to obtain formulas for partial derivatives of Black-Scholes prices with respect to model parameters. Explore numerical methods for differentiation, interpolation, and approximation in cases where formulas are not available.
