Course description

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 simpler random walk and geometric Brownian motion all the way to models with jumps and models for bubbles.

Course content

  • Review of probability theory and discrete time models
  • Continuous time finance
  • Brownian motion
  • Continuous-time models for asset prices
  • Introduction to stochastic calculus
  • Ito’s lemma
  • Girsanov’s theorem
  • SDEs
  • Security valuation and advanced topics in stochastic calculus
  • No arbitrage
  • Black-Sholes model
  • PDEs
  • Multivariate Ito
  • Jumps

Learning outcome knowledge

By the end of the course the students are expected to know:

  • Definition of a Brownian motion
  • Arithmetic Brownian motion, geometric Brownian motion, Ornstein-Uhlenbeck process
  • Ito's lemma and Multivariate Ito's lemma
  • Girsanov's theorem
  • Evaluation of stochastic differential equations
  • No arbitrage models
  • Black - Sholes model
  • Models with jumps
  • Models with bubbles

Exam organisation

  • Written exam: 50%
  • Written exam: 50%