EDP in finance

Objective

This course is a short introduction to the partial differential equations (PDEs) related to option
pricing in mathematical finance and their approximation by finite difference type methods. An important
part of the course is devoted to numerical programming.
 

Planning

1. Link between expectations formula and PDEs (Feynman-Kac theorem), Black and Scholes PDE. Maximum principle. Examples.
2. Finite difference schemes: Euler Forward and Implicit Euler schemes, Crank-Nicolson scheme, stability, CFL condition, convergence analysis, numerical implementation.
3. American options, PDE inequality, Finite difference schemes.
4. Algorithms for solving linear or non-linear implicit schemes.
5. Project supervision.

References

• P.Wilmott, S. Howison, J. Dewynne, The mathematics of financial derivatives, Cambridge
  University Press, 1998. (An elementary introduction to PDE methods for finance)
• Y. Achdou, O. Pironneau, Computational methods for option pricing. Frontiers in applied
  mathematics, Siam, 2005. (A more advanced document with c++ solutions)
• H. Pham, Optimisation et contrôle stochastique appliqués à la finance, Springer-Verlag, 2007. (portfolio optimisation related PDEs)
• Y. Achdou, O. Bokanowski, T. Lelievre, PDE in finance, Encyclopedia of financial models,
  2012 (see http://www.math.jussieu.fr/ boka/enseignement/pdefinance.pdf)