8229 - NUMERICAL METHODS IN FINANCE

This course provides essential tools to understand and solve the basic issues of computation in financial engineering. For instance, you may think of exotic derivatives, or of underlying assets whose dynamics are described by time-dependent or stochastic volatility, jumps in the trajectories, or mean-reverting features. In such cases, numerical methods only allow us to obtain the price or the replicating strategy of a derivative.
In this course we discuss no-arbitrage and completeness in financial markets with several underlying assets, with constant (as the case of multi-dimensional Black-Scholes model) and time-dependent coefficients. We derive the partial differential equation to price European derivatives in this framework, and discuss finite-difference schemes to solve it. We analyze the foreign exchange market and valuate currency derivatives, accounting for the specific features of domestic/foreign market price of risk. We study Mertons jump diffusion model, the simplest model with jumps in asset paths, and one of the building blocks to describe defaultable securities. We discuss the dynamics of risky assets under both the historical and the risk-neutral measure, and price derivatives in this market.
We also deal with the valuation and replication of American and path-dependent derivatives in the event-tree framework. Moreover, we study Monte Carlo Methods, to generate sample paths of the financial assets, to price derivatives, and to estimate their greaks.We also analyze variance reduction techniques to improve the efficiency of the Monte Carlo estimate.
Sessions are devoted to the introduction to Visual Basic for Application, and to the computer implementation of the discussed numerical methods.

• Pricing and hedging of European, American, and path-dependent derivative securities in discrete time: algorithms and computational issues.
• The Black-Scholes model with many underlying assets. Pricing and hedging derivatives. Foreign exchange markets and currency derivatives.
• The Merton jump-diffusion model.
• Monte Carlo methods: confidence intervals, convergence, bias, computational issues. Simulation of asset prices (time-dependent coefficients, stochastic volatility).
• Variance reduction techniques.
• Estimating sensitivities.
• Finite-difference schemes for the Black-Scholes partial differential equation.

Final (short) written exam and an assignment. The assignment can be shared by a team of students.
The policy is the same for non-attending students.

• P. GLASSERMAN, Monte Carlo Methods in Financial Engineering, Springer, 2003
• Lecture Notes of the instructor.

 Suggested readings:

• P.T. KIMMEL, S. BULLEN, J. GREEN,  Excel 2003 VBA Programmer's Reference (Programmer to Programmer), Wrox 2004