Efficient option pricing under Feller models

Description

The Black and Scholes (BS) model, which relates derivative prices to current stock prices and quantifies risk through a constant volatility parameter, rests upon assumptions which turned out to be insufficient to capture modern market phenomena. An extension of the BS model is to make the volatility a function of the stock price, which leads to local volatility (LV) models.

A different approach is the inclusion of jumps into the spot price evolution. A big class of processes with finite and infinite jump activity are Lévy processes with Merton's jump diffusion model being one of the first representatives.

In this project option pricing under processes which combine these two approaches will be considered, that are general Feller processes with state space dependent jump measures. Option pricing under such models leads to parabolic partial integro-differential equations (PIDE) or inequalities and Fourier techniques are no longer available as they rely on the translation invariance of the process. Though there is some literature on theoretical aspects (i.e. existence and uniqueness) of Feller processes, computational aspects have hardly been treated.

The aim of the project is the development, analysis and implementation of wavelet based numerical schemes for pricing of European and American options under Feller processes in up to three dimensions. The construction of multidimensional jump measures and norm equivalences for the corresponding scale spaces will be treated.

References

• Schneider, R. and Reichmann, O. and Schwab, C., Wavelet solution of variable order pseudodifferential equations, Calcolo, 2010, 47, no.2, pp. 65-101
• Reichmann, O. and Schwab, C., Numerical analysis of additive, L\'evy and Feller processes with applications to option pricing, SAM Report 2010/06
  http://www.sam.math.ethz.ch/reports/2010/06
• Reichmann, O. and Schwab, C., Numerical analysis of additive, Lévy and Feller processes with applications to option pricing, Lévy Matters I, Lecture Notes in Mathematics, Springer, Vol. 2001 (2010).

Contacts

Prof. Christoph Schwab
Oleg Reichmann