Chapter

Partial Differential Equations in Finance

Finite Mathematics for Financial Modeling

  1. Yves Achdou PhD1,
  2. Olivier Bokanowski PhD2,
  3. Tony Lelièvre PhD3

Published Online: 15 DEC 2012

DOI: 10.1002/9781118182635.efm0081

Encyclopedia of Financial Models

Encyclopedia of Financial Models

How to Cite

Achdou, Y., Bokanowski, O. and Lelièvre, T. 2012. Partial Differential Equations in Finance. Encyclopedia of Financial Models. .

Author Information

  1. 1

    Professor, Lab. Jacques-Louis Lions, University Paris-Diderot, Paris, France

  2. 2

    Associate Professor, Lab. Jacques-Louis Lions, University Paris-Diderot, Paris, France

  3. 3

    Professor, CERMICS, Ecole des Ponts ParisTech, Marne-la-Vallée, France

Publication History

  1. Published Online: 15 DEC 2012

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.

Keywords:

  • partial differential equations (PDEs) in finance;
  • Black and Scholes;
  • lookback options;
  • Asian options;
  • European options;
  • American options;
  • finite difference method;
  • finite element methods;
  • viscosity solution;
  • calibration;
  • local volatility;
  • barrier options;
  • Dupire's formula;
  • least squares;
  • Lecture Notes in Math