A PDE approach to jump - diffusions ∗

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-diffusions; (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. JEL classification: C02, C60, G12. AMS classification codes: 60G44, 60J25, 60J50, 91B28, 91B70.