Synopsis for Numerical Methods 1: Finite difference methods

Michaelmas Term (8 Lectures)
Introduction: heat equation, explicit Euler time stepping, implementation, numerical ex- periments Analysis of dierence schemes: truncation error, stability, implicit and Crank- Nicolson schemes Sensitivities: Numerical dierentiation, stronger stability concepts Bound- aries: Asymptotic conditions, option payos as terminal conditions
Hilary Term (8 Lectures)
Path-dependent options: discretely and continuously sampled barrier and Asian options American options: explicit treatment, projected iterations, penalty methods Calibration: implied trees, ill-posedness and regularisation of inverse problems Multi-factor problems: analysis, computation and limitations

This course introduces numerical techniques which are fundamental to a large class of computational problems in nance. The rst half of the course (Michaelmas Term) covers nite dierence approximations to the solution of parabolic equations arising in nancial applications, their accuracy and stability, and related topics such as stable estimation of derivatives of the solution (Greeks).
The second part in Hilary Term deals with the numerical solution of obstacle problems (American options), and algorithms for exotic options, such as Asian options or multi-asset products. We briefly discuss the calibration of models to market data.

1. R Seydel, Tools for Computational Finance, Springer (2006).
2. D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Method, Wiley, 2000.