By using specific examples--mostly from computational finance--this paper shows that Mathematica is capable of transforming Feynman-type integrals on the pathspace into powerful numerical procedures for solving general partial differential equations (PDEs) of parabolic type, with boundary conditions prescribed at some fixed or free time boundary. Compared to the widely used finite difference methods, such procedures are more universal, are considerably easier to implement, and do not require any manipulation of the PDE. An economist or a financial engineer who works with very general models for asset values, including the so-called stochastic volatility models, will find that this method is just as easy to work with as the classical Black-Scholes model, in which asset values can follow only a geometric Brownian motion process.
2. Some Background: Solutions to Second-order Parabolic Partial Differential Equations as Path Integrals
3. Path Integrals as Approximation Procedures
4. The Black-Scholes Formula versus the Feynman-Kac Representation of the Solution of the Black-Scholes PDE
5. Approximation of the Fundamental Solution by Way of a Discrete Dynamic Programming Procedure
6. Dynamic Programming with Mathematica: The Case of American and European Stock Options
7. Concluding Remarks
About the Author
Andrew Lyasoff is associate professor and director of the M.A. degree program in Mathematical Finance at Boston University. His research interests are mostly in the area of Stochastic Calculus and Mathematical Finance. Over the last few years he has been using Mathematica extensively to develop new numerical methods for PDEs and new instructional tools for graduate courses in Operations Research and Mathematical Finance. In addition, he is developing new theoretical models that reflect long-range dependencies in market data.
Department of Mathematics and Statistics