Stochastic Calculus for Finance

Introduction

The objective of the course is to provide the students with knowledge of the stochastic calculus that underlies the pricing and hedging of derivative instruments, including stochastic integrals and stochastic differential equations. The course is focused on the application of stochastic calculus methods in finance with both discrete-time and continuous-time stochastic models of financial markets, starting from the simple random walk and geometric Brownian motion all the way to models with jumps. The course is designed to deepen the students’ understanding of pricing techniques used in international derivatives markets.

Course content

• Review of probability theory, deterministic calculus, and discrete-time models
• Martingale processes, stochastic integrals, and stochastic differential equations (SDEs)
• Ito’s lemma
• Major models of SDEs, solving SDEs analytically and using simulations
• Fundamental theorems of asset pricing, the Girsanov theorem
• Equivalent martingale measure
• Black-Scholes model and partial differential equations
• Jumps