Most realistic financial derivatives models are too complex to allow explicit analytic solutions. The computational techniques used to implement those models fall into two broad categories: finite difference methods for the solution of partial differential equations (PDEs) and Monte Carlo simulation. Accordingly, the course consists of two sections.
The first half of the course focuses on finite difference methods. The following topics are discussed; Parabolic PDEs, Black-Scholes PDE for European and American options; binomial and trinomial trees; explicit, implicit and Crank- Nicholson finite difference methods; far boundary conditions, convergence, stability, variance bias; early exercise and free boundary conditions; parabolic PDEs arising from fixed income derivatives; implied trees for exotic derivatives, adapted trees for interest rate derivatives.
The second half of the course focuses on Monte Carlo. The following topics are discussed; Random number generation and testing; evaluation of expected payoff by Monte Carlo simulation; variance reduction techniquesÃ¢â‚¬â€�antithetic variables, importance sampling, martingale control variables; stratification, low-discrepancy sequences and quasi-Monte Carlo methods; efficient evaluation of sensitivity measures; methods suitable for multifactor and term-structure dependent models.
Computational Methods of Financial Mathematics is taught by Marcel Blais, a professor at Worcester Polytechnic Institute.
Worcester Polytechnic Institute
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Copp, Jessica L., "Course Summary of Computational Methods of Financial Mathematics" (2009). Masters Theses (All Theses, All Years). 745.
finite difference methods, monte carlo, financial mathematics, computational methods