Pre-requisites: MA 224 (Real Analysis) & MA 371 (Financial Engineering I)

Stochastic processes, filtrations, conditional expectations, martingales and stopping times, Brownian motion and its properties; Ita-integral and its extension to wider classes of integrands, isometry and martingale properties of the integral; Ita-calculus, Ita-formula and its application in calculating stochastic integrals; Stochastic differential equations, existence and uniqueness of solutions; Risk-neutral measure, Girsanov's theorem for change of measure, martingale representation theorems, representation of Brownian martingales, Feynman-Kac formula; Stock prices as geometric Brownian motions, Black-Scholes option pricing, delta hedging, derivation of the Black-Scholes differential equation, the Black-Scholes formula and simple extensions of the model; Application of Girsanov's theorem to Black-Scholes dynamics, self-financing strategies and model completeness, risk neutral measures, the fundamental theorem of asset pricing; The Black-Scholes model, the Black-Scholes option pricing formula and the market price of risk. Continuous time optimal stopping and pricing of American options.

Texts:

• S. Shreve, Stochastic Calculus for Finance, Vol.2, Springer, 2004.
• R. A. Dana and M. Jeanblanc, Financial Markets in Continuous Time, Springer, 2001.
• N. H. Bingham and R. Kiesel, Risk Neutral Valuation, 2nd Edition, Springer, 2004.
• M. Baxter and A. Rennie, Financial Calculus, Cambridge University Press, 1996.
• J. M. Steele, Stochastic Calculus and Financial Applications, Springer, 2001.