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