A PDE Approach to Jump-Diffusions

37 Pages Posted: 28 Oct 2010  

Peter Carr

New York University (NYU) - Courant Institute of Mathematical Sciences

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 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 (NYU) - Courant Institute of Mathematical Sciences ( email )

251 Mercer Street
New York, NY 10012
United States

Laurent Cousot (Contact Author)

BNP Paribas ( email )

Paris
France

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