## Numerical Methods in Finance

Department:
MATH
Course Number:
6635
Hours - Lecture:
3
Hours - Lab:
0
Hours - Recitation:
0
Hours - Total Credit:
3
Typical Scheduling:
Every spring semester

This course contains the basic numerical and simulation techniques for the pricing of derivative securities.

Prerequisites:

MATH 2403 and MATH 3215 (or the equivalent), knowledge of computer programming, and MS QCF standing or some previous exposure to the topics of stocks, bonds and options.

Course Text:

Text at the level of The Mathematics of Financial Derivatives: A Student Introduction by P. Wilmott, S. Howison and J. Dewynne, published by Cambridge University Press

Topic Outline:
• Solution of a single non-linear equation and its applications to computing implied volatility and bond yield.
• The use of polynomials and piecewise polynomials to fit data and approximate functions by interpolation and least squares methods. Applications to the volatility smile and estimation of the discount curve.
• Simulation of Brownian Motion. Monte Carlo Simulation of Stochastic Differential Equations. Euler-Maruyama and Milstein Approximations, Low Discrepancy Sequences.
• Introduction to basic matrix factorizations: LU, Cholesky, Eigenvalue-Eigenvector, and SVD. Generation of correlated Brownian motion. Application to pricing of multifactor options and dimension reduction.
• Introduction to Numerical Integration and Differentiation: Richardson Extrapolation and Romberg integration. Application to numerical solution of ordinary differential equations with Euler, trapezoidal rule, and BDF2 methods of time stepping. Introduction to stability of time stepping methods.
• The heat equation and its solution, analytic properties and issues in its numerical solution.
• Numerical Solution of PDEs relevant to computational finance: the Black-Scholes equation for European options; solutions of the American option problem: boundary conditions implied by early exercise; numerical methods for the free boundary; bond pricing via solution of PDEs, if time permits.
• Comparisons among PDE. Monte Carlo, and basic tree methods for option pricing.