Math­em­at­ic­al Fin­ance 2: Con­tinu­ous Time Fin­ance

Course content

Over the last few decades, financial analysts have started using more and more sophisticated mathematical concepts and models to describe the behaviour of markets and to derive computing methods. This course provides an introduction to stochastic calculus and shows how it is applied to the pricing and hedging of financial contracts such as equity options and interest rate derivatives. The goal is to make students confident with the probabilistic techniques required to understand the most widely used financial models. The course is mainly suitable for students who would like to become quantitative analysts, asset managers, traders, risk managers, structurers, or who are simply generally interested in financial markets and want to gain the technical skills needed to understand and model their behaviour.

The course is split into 4 sections:

1. Discrete Time Finance: the Binomial Model

• Introduction to measure theoretic probability
• Derivative pricing and hedging in the generalized market model
• Risk-neutral measure
• Self-financing strategies
• Absence of arbitrage    

2. Introduction to Stochastic Processes and Stochastic Calculus

• Continuous time limits
• Brownian motion
• Stochastic integrals
• Ito’s lemma
• Change of probability and the risk neutral measure
• Simulation

3. Pricing and Hedging in Continuous Time

• Arbitrage pricing and hedging theory
• Martingale approach to arbitrage theory
• Black-Scholes-Merton equation
• Feynman-Kac Theorem
• Hedging in the BSM model
• The implied volatility surface

4. Advanced material (time permitting)

• Stochastic volatility models
• Change of numeraire
• Currency derivatives
• Term structure models