Finance and Stochastics

, Volume 10, Issue 3, pp 303–330 | Cite as

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

  • Peter Carr
  • Vadim Linetsky


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.


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

Mathematics Subject Classification (2000)

60J35 60J60 60J65 60G70 

JEL Classification

G12 G13 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersen L., Buffum D. (2004): Calibration and implementation of convertible bond models. J. Comput. Financ. 7, 1–34Google Scholar
  2. 2.
    Andreasen, J.: Dynamite dynamics. In: Gregory, J. (ed.) Credit Derivatives: The Definitive Guide. Risk Books, pp. 371–384 (2003)Google Scholar
  3. 3.
    Aquilina J., Rogers L.C.G. (2004): The squared Ornstein-Uhlenbeck market. Math. Financ. 14(4): 487–513CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Atlan M., Leblanc B. (2005): Hybrid equity-credit modelling. Risk Mag. 18(8): 61–66Google Scholar
  5. 5.
    Bekaert G., Wu G. (2000): Asymmetric volatilities and risk in equity markets. Rev. Financ. Stud. 13, 1–42CrossRefGoogle Scholar
  6. 6.
    Benton D., Krishnamoorthy K. (2003): Computing discrete mixtures of continuous distributions: Noncentral chi-square, noncentral t and the distribution of the square of the sample multiple correlation coefficient. Comput. Stat. Data Anal. 43, 249–267MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bielecki T., Rutkowski M. (2002): Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin Heidelberg New YorkGoogle Scholar
  8. 8.
    Black, F.: Studies of stock price volatility changes. In: Proceedings of the 1976 American Statistical Association, Business and Economics Statistics Section, pp. 177–181. American Statistical Association, Alexandria, (1976)Google Scholar
  9. 9.
    Bloch D. (2005): Jumps as components in the pricing of credit and equity products. Risk 18(2): 67–73MathSciNetGoogle Scholar
  10. 10.
    Borodin A., Salminen P. (2002): Handbook of Brownian Motion, 2nd edn. Birkhäuser, BostonzbMATHGoogle Scholar
  11. 11.
    Campbell J., Hentschel L. (1992): No news is good news: An asymmetric model of changing volatility in stock returns. J. Financ. Econ. 31, 281–318CrossRefGoogle Scholar
  12. 12.
    Campbell J., Kyle A. (1993): Smart money, noise trading and stock price behavior. Rev. Econ. Stud. 60, 1–34CrossRefzbMATHGoogle Scholar
  13. 13.
    Campbell J., Taksler G. (2003): Equity volatility and corporate bond yields. J. Financ. 58, 2321–2349CrossRefGoogle Scholar
  14. 14.
    Campi, L., Polbennikov, S., Sbuelz, A.: Assessing credit with equity: a CEV model with jump to default. In: Proceedings of "AMASES Meetings, XXIX Edition”, Palermo. abstract=675061 (2005)Google Scholar
  15. 15.
    Carr P., Javaheri A. (2005): The forward PDE for European options on stocks with fixed fractional jumps. Int. J. Theor. Appl. Financ. 8, 239–253CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Carr, P., Wu, L.: Stock options and credit default swaps, a joint framework for valuation and estimation. NYU working paper, (2005)Google Scholar
  17. 17.
    Chen L., Filipović D. (2005): A simple model for credit migration and spread curves. Financ. Stoch. 9, 211–231CrossRefzbMATHGoogle Scholar
  18. 18.
    Christie A. (1982): The stochastic behavior of common stock variances. J. Financ. Econ. 10, 407–432CrossRefGoogle Scholar
  19. 19.
    Consigli, G.: Credit default swaps and equity volatility: theoretical modeling and market evidence. Departement of Applied Mathematics, University Ca’Foscari, Venice (2004)Google Scholar
  20. 20.
    Cox, J.: Notes on option pricing I: Constant elasticity of variance diffusions. Working Paper, Stanford University (reprinted in J. Portf. Manage., 1996, 22, 15–17) (1975)Google Scholar
  21. 21.
    Cremers, M., Driessen, J., Maenhout, P., Weinbaum, D.: Individual stock-option prices and credit spreads (December 2004). Yale ICF working paper No. 04-14,EFA 2004 Maastricht meetings Paper No. 5147, (2004)Google Scholar
  22. 22.
    Das, S., Sundaram, R.: A simple model for pricing securities with equity, interest rate, and default risk. NYU working paper,∼srdas/ (2003)Google Scholar
  23. 23.
    Davydov D., Linetsky V. (2001): The valuation and hedging of barrier and lookback options under the CEV process. Manage. Sci. 47, 949–965CrossRefGoogle Scholar
  24. 24.
    Davydov D., Linetsky V. (2003): Pricing options on scalar diffusions: an eigenfunction expansion approach. Oper. Res. 51, 185–209CrossRefMathSciNetGoogle Scholar
  25. 25.
    Delbaen F., Shirakawa H. (2002): A note on option pricing for constant elasticity of variance model. Asia-Pacific Financ. Mark. 9, 85–99CrossRefzbMATHGoogle Scholar
  26. 26.
    Dennis P., Mayhew S. (2002): Risk-neutral skewness: Evidence from stock options. J. Financ. Quant. Anal. 37, 471–493CrossRefGoogle Scholar
  27. 27.
    Dennis P., Mayhew S., Stivers C. (2006): Stock returns, implied volatility innovations and the asymmetric volatility phenomenon. J. Financ. Quant. Anal. 41, 381–406CrossRefGoogle Scholar
  28. 28.
    Ding C.G. (1992): Computing the non-central χ2 distribution function. Appl. Stat. 41, 478–482CrossRefGoogle Scholar
  29. 29.
    Duffie D., Schroder M., Skiadas C. (1996): Recursive valuation of defaultable securities and the timing of resolution of uncertainty, Ann. Appl. Probab. 6, 1075–1096CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Duffie D., Singleton K. (1999): Modeling term structures of defaultable bonds, Rev. Financ. Stud. 12, 653–686CrossRefGoogle Scholar
  31. 31.
    Duffie D., Singleton K. (2003): Credit Risk. Princeton University Press, PrincetonGoogle Scholar
  32. 32.
    Dyrting S., (2004): Evaluating the noncentral chi-square distribution for the Cox-Ingersoll-Ross process. Comput. Econ. 24, 35–50CrossRefzbMATHGoogle Scholar
  33. 33.
    Elliott R.J., Jeanblanc M., Yor M. (2000): On models of default risk. Math. Financ. 10, 179–196CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Feller W. (1951): Two singular diffusion problems, Ann. Math. 54, 173–182CrossRefMathSciNetGoogle Scholar
  35. 35.
    Göing-Jaeschke A., Yor M. (2003): A survey and some generalizations of Bessel processes. Bernoulli 9, 313–349CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Grundke P., Riedel K. (2004): Pricing the risks of default: a note on Madan and Unal. Rev. Deriv. Res. 7, 169–173CrossRefzbMATHGoogle Scholar
  37. 37.
    Haugen R., Talmor E., Torous W. (1991): The effect of volatility changes on the level of stock prices and subsequent expected returns. J. Financ. 46, 985–1007CrossRefGoogle Scholar
  38. 38.
    Heath D., Platen E. (2002): Consistent pricing and hedging for a modified constant elasticity of variance model. Quant. Financ. 2, 459–467CrossRefMathSciNetGoogle Scholar
  39. 39.
    Hilscher, J.: Is the corporate bond market forward looking? Harvard University working paper (2005)Google Scholar
  40. 40.
    Jarrow R., Lando D., Turnbull S. (1997): A Markov model for the term structure of credit risk spreads. Rev. Financ. Stud. 10, 481–523CrossRefGoogle Scholar
  41. 41.
    Jarrow R., Turnbull S. (1995): Pricing derivatives on financial securities subject to credit risk. J. Financ. 50, 53–85CrossRefGoogle Scholar
  42. 42.
    Jeanblanc M., Yor M., Chesney M. (2006): Mathematical Methods for Financial Markets. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  43. 43.
    Kassam, A.: Options and volatility. Goldman Sachs Derivatives and Trading Research Report, January (2003)Google Scholar
  44. 44.
    Lando D. (2004): Credit Risk Modeling. Princeton University Press, PrincetonGoogle Scholar
  45. 45.
    Lehnigk, S.: The Generalized Feller Equation and Related Topics. Pitman Monographs and Surveys in Applied Mathematics, vol. 68, Longman (1993)Google Scholar
  46. 46.
    Linetsky V. (2004): Lookback options and diffusion hitting times: a spectral expansion approach. Financ. Stoch. 8, 373–398CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    Linetsky V. (2006): Pricing equity derivatives subject to bankruptcy. Math. Financ. 16, 255–282CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    Madan D., Unal H. (1998): Pricing the risk of default. Rev. Deriv. Res. 2, 121–160Google Scholar
  49. 49.
    Merton R. (1974): On the pricing of corporate debt: the risk structure of interest rates. J. Financ. 29, 449–470CrossRefGoogle Scholar
  50. 50.
    Merton R. (1976): Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144CrossRefzbMATHGoogle Scholar
  51. 51.
    Pitman J.W., Yor M. (1981): Bessel processes and infinitely divisible laws. In: Williams D (eds). Stochastic Integrals. Lecture Notes in Mathematics, vol. 851, Springer, Berlin Heidelberg New York, pp. 285–370CrossRefGoogle Scholar
  52. 52.
    Revuz D., Yor M. (1999): Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  53. 53.
    Schönbucher P.J. (2003): Credit Derivatives Pricing Models. Wiley, New YorkGoogle Scholar
  54. 54.
    Schroder M. (1989): Computing the constant elasticity of variance option pricing formula. J. Financ. 44, 211–219CrossRefGoogle Scholar
  55. 55.
    Yor M. (1980): Loi de l’indice du lacet brownien, et distribution de Hartman-Watson. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 53, 71–95CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    Zhu H., Zhang, Y., Zhou, H.: Explaining credit default swap spreads with the equity volatility and jump risks of individual firms. Bank for International Settlements working paper, (2005)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Bloomberg LP and NYU Courant InstituteNew YorkUSA
  2. 2.Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied SciencesNorthwestern UniversityEvanstonUSA

Personalised recommendations