Aims:
This course aims to provide a good and rigorous understanding of the mathematics used in derivative pricing and to enable students to understand where the assumptions in the models break down.
Summary:
- Continuous time processes: basic ideas, filtration, conditional expectation, stopping times.
- Continuous-time martingales, sub- and super-martingales, martingale inequalities, optional sampling.
- Wiener process and Wiener martingale, stochastic integral, Itô calculus and some applications.
- Multi-dimensional Wiener process, multi-dimensional Itô’s formula.
- Stochastic differential equations, Ornstein-Uhlenbeck processes, Black-Scholes SDE, Bessel processes and CIR equations.
- Change of measure, Girsanov’s theorem, equivalent martingale measures and arbitrage.
- Representation of martingales.
- The Black-Scholes model, self-financing strategies, pricing and hedging options, European and American options.
- Option pricing and partial differential equations; Kolmogorov equations.
- Further topics: dividends, reflection principle, exotic options, options involving more than one risky asset.
Detailed Information
Course Description: Link to Official Course Descriptor.
Pre-requisites: none.
Location: Edinburgh.
Reading list:
The students are referred to the following texts.
- Karatzas, I. & Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer.
- Baxter, M. & Rennie, A. (1996). Financial Calculus. CUP.
- Etheridge, A. (2002). A Course in Financial Calculus. CUP.
- Lamberton, D. & Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall.
SCQF Level: 11.
Credits: 15.
Other Information
Help: If you have any problems or questions regarding the course, you are encouraged to contact the course leader.
Canvas: further information and course materials are available on Canvas