A PDE Approach to Jump-Diffusions

37 Pages Posted: 28 Oct 2010

See all articles by Peter Carr

Peter Carr

New York University Finance and Risk Engineering

Laurent Cousot

BNP Paribas

Date Written: October 4, 2010

Abstract

In this paper, we show that the calibration to an implied volatility surface and the pricing of contingent claims can be as simple in a jump-diffusion framework as in a diffusion one. Indeed, after defining the jump densities as those of diffusions sampled at independent and exponentially distributed random times, we show that the forward and backward Kolmogorov equations can be transformed into partial differential equations. It enables us to (i) derive Dupire-like equations (see Dupire (1994)) for coefficients characterizing these jump-di ffusions; (ii) describe sufficient conditions for the processes they induce to be calibrated martingales; and (iii) price path-independent claims using backward partial differential equations. This paper also contains an example of calibration to the S&P 500 market.

Keywords: martingale, jump-di ffusion, partial diff erential equation, calibration, option

JEL Classification: C02, C60, G12

Suggested Citation

Carr, Peter P. and Cousot, Laurent, A PDE Approach to Jump-Diffusions (October 4, 2010). Available at SSRN: https://ssrn.com/abstract=1698975 or http://dx.doi.org/10.2139/ssrn.1698975

Peter P. Carr

New York University Finance and Risk Engineering ( email )

6 MetroTech Center
Brooklyn, NY 11201
United States
9176217733 (Phone)

HOME PAGE: http://engineering.nyu.edu/people/peter-paul-carr

Laurent Cousot (Contact Author)

BNP Paribas ( email )

Paris
France

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