Partial Differential Equations in Finance

Abstract

Partial differential equations are useful in finance in various contexts, in particular for the pricing of European and American options, for stochastic portfolio optimization, and for calibration. They can be used for simple options as well as for more exotic ones, such as Asian or lookback options. They are particularly useful for nonlinear models. They allow for the numerical computations of several spot prices at the same time. Numerical aspects, discretization methods, algorithms, and analysis of the numerical schemes have been under constant development during the last three decades. Finite difference methods are the simplest and most basic approaches. Finite element methods allow the use of nonuniform meshes and refinement procedures can then be applied and improve accuracy near a region of interest. Deterministic approaches based on partial differential equation formulations can also be used for calibration of various volatility models (such as local, stochastic, or Levy-driven volatility models) and by making use of Dupire's formula. Current research directions include the development of discretization methods for high-dimensional problems.