# Derivative pricing and numerical solving of PDE for finance

## Presentation

Program:

• Hypotheses, derivation and interpretation of Black and Scholes p.d.e. model for pricing European Vanilla options.
• Extensions of Black and Scholes p.d.e. model to non-vanilla options (Barrier options, Asiatic options …)
• Finite difference method applied to Black and Scholes p.d.e. model: usual schemes, consistency, stability, explicit versus implicit schemes
• Stochastic interpretation of Black and Scholes equation, Feynmann-Kac formula.
• Monte-Carlo method for solving parabolic p.d.e. Basic methods for variance reduction.
• Labwork devoted to the numerical solution of Black-Scholes pde models by several methods.

## Objectives

At the end of this module, the student should know the main p.d.e.-based models for derivative pricing and as well the numerical methods (finite difference, Monte-Carlo) that are used to solve these models.

The student will be able to:

• derive a p.d.e model for pricing a derivative,
• solve this p.d.e. model by using either a finite difference method or a Monte-Carlo method,
• program each of this method in MATLAB or VBA.

## Needed prerequisite

Basis of stochastic calculus (Brownian motion, Ito integral, Ito formula)

Basis of Financial mathematics

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...