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.
Learning Outcomes:
It is intended that students will demonstrate:
- understanding of continuous-time stochastic processes and their role in modelling the evolution of random phenomena,
- understanding of the Wiener process,
- conceptual understanding of the stochastic Itô integral and Itô’s formula,
- conceptual understanding of the main results and basic applications of stochastic Ito calculus,
- understanding stochastic differential equations (SDE’s),
- understanding of equivalent measures and in particular Girsanov’s theorem.
- conceptual understanding of martingales in continuous time,
- understanding of the application of the theory of stochastic calculus to option pricing problems,
- understanding of the martingale representation theorem and its role in financial applications,
- conceptual understanding of the role of martingales in the theory of derivative pricing,
- conceptual understanding of the role of equivalent martingale measures in financial mathematics,
- conceptual understanding of SDEs in stochastic modelling and in particular in finance,
- understanding the concept of strategies in financial models,
by answering relevant exam questions.
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