Basic ideas of hedging and pricing by arbitrage. Basic concepts from probability theory and stochastic processes, conditional expectation, martingales, random walk, Markov processes, Brownian motion. Stochastic integration, Itô’s integral, Itô’s formula. Stochastic differential equations. Risk-neutral pricing, Black-Scholes-Merton option pricing model, Girsanov’s Theorem, American derivative securities, term-structure models. Jump processes and their application to option pricing.

Texts:

• S.E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer, 2005.
• S.E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2005.
• J. M. Steele, Stochastic Calculus and Financial Applications, Springer, 2001.

References:

• T. Mikosh, Elementary Stochastic Calculus, with Finance in View, World Scientific, 1998.
• F.C. Klebaner, Introduction to Stochastic Calculus with Applications , World Scientific, 2005.