8229 - NUMERICAL METHODS IN FINANCE

This course provides essential tools to understand and solve the basic issues of computation in financial engineering. In many practical situations numerical methods are the only possible way to answer a question. For instance, in a Black-Scholes market there is no closed formula for pricing neither finite-maturity American put options nor finite-maturity American call options, if the underlying stock pays some dividend. We face the same problem also in the case of Asian options, whose payoff depends on the average value of the underlying asset. In this course we study the main numerical techniques used in finance, such as Monte Carlo simulation and the solution of the Black-Scholes partial differential equations via finite-difference schemes. Moreover, we analyze some extensions and alternatives to the Black and Scholes model, whose importance is crucial for the quantitative analysis of modern financial markets, when dealing with credit risk models and the evaluation of defaultable securities. Time is devoted to the computer implementation of the discussed numerical methods.

• Pricing and hedging of European and American-type derivative securities in discrete time: algorithms and computational issues.
• The Black-Scholes model with many underlying assets. Pricing and hedging derivatives.
• Finite difference schemes and the solution of partial differential equations for pricing European options.
• Monte Carlo simulation: confidence interval, convergence, bias and computational issues. Simulation of asset prices.
• Variance Reduction Techniques.
• Estimating the sensitivities.
• Extensions to the Black-Scholes model (time-varying coefficients, stochastic volatility, jump-diffusion processes).

Brief written exam and group work (numerical solution to an assigned problem). Group work can also be carried out individually.

• P. GLASSERMAN,  Monte Carlo Methods in Financial Engineering, Springer, 2003.