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

Peter Carr, Vadim Linetsky

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)303-330
Number of pages28
JournalFinance and Stochastics
Issue number3
StatePublished - Sep 2006


  • Bessel processes
  • CEV model
  • Corporate bonds
  • Credit derivatives
  • Credit spread
  • Default
  • Equity derivatives
  • Implied volatility skew

ASJC Scopus subject areas

  • Statistics and Probability
  • Finance
  • Statistics, Probability and Uncertainty


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