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A Jump to Default Extended CEV Model: An application of Bessel processes

by Peter Carr of Bloomberg & NYU Courant Institute, and
Vadim Linetsky of Northwestern University

September 2006

Abstract: We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and equity volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a constant elasticity of variance (CEV) specification for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our stock price process to a standard Bessel process with killing. This reduction permits the development of completely explicit closed form solutions for risk-neutral survival probabilities, CDS spreads, corporate bond values, and European-style equity options. Furthermore, our valuation model is sufficiently flexible so that it can be calibrated to exactly match arbitrarily given term structures of CDS spreads, interest rates, dividend yields, and at-the-money implied volatilities.

JEL Classification: G12, G13.

AMS Classification: 60J35, 60J60, 60J65, 60G70.

Keywords: Default, credit spread, corporate bonds, equity derivatives, credit derivatives, implied volatility skew, CEV model, Bessel processes.

Published in: Finance and Stochastics, Vol. 10, No 3, (September 2006), pp. 303-330.

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