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A jump to default extended CEV model: an application of Bessel processes

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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.

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Authors and Affiliations

  1. Bloomberg LP and NYU Courant Institute, 251 Mercer Street, New York, NY, 10012, USA

    Peter Carr

  2. Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA

    Vadim Linetsky

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  1. Peter Carr
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  2. Vadim Linetsky
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Correspondence to Vadim Linetsky.

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Carr, P., Linetsky, V. A jump to default extended CEV model: an application of Bessel processes. Finance Stoch 10, 303–330 (2006). https://doi.org/10.1007/s00780-006-0012-6

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