Numerical Methods in Finance

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Every spring semester

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


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.