Special Semester on Stochastics with Emphasis on Finance
Linz, September 2008 - December 2008
Tutorial: Jean-Pierre Fouque

Title: Partial Differential Equations in Option Pricing.

Abstract: In the first part, we will review the partial differential equation approach to pricing financial derivatives. Starting with the classical constant volatility Black-Scholes model, we will review the link between risk neutral valuation and PDE through the Feynman-Kac formula. We will pay special attention to the various boundary conditions associated to contracts such as barrier options or American options. Asian options involving higher dimensions will also be discussed.
In the second part we will make the case for a need for varying volatility in order to capture the so-called smile/skew of implied volatilities. Stochastic volatility models and their associated PDEs will be introduced. The lack of closed-form solutions in general will lead us to some approximation techniques. In particular we will present regular and singular perturbation techniques relevant to stochastic volatility PDEs.

Presentation slides: Tutorial 1, Tutorial 2 , Tutorial 3

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