Forward equations for option prices in semimartingale models


We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a—possibly discontinuous—semimartingale. This result generalizes Dupire’s forward equation to a large class of non-Markovian models with jumps.

This is a preview of subscription content, access via your institution.


  1. 1.

    Note, however, that the equation given in [9] does not seem to be correct: it involves the double tail of \(k(z)\,dz\) instead of the exponential double tail.


  1. 1.

    Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. Frontiers in Applied Mathematics. SIAM, Philadelphia (2005)

    Google Scholar 

  2. 2.

    Andersen, L., Andreasen, J.: Jump diffusion models: volatility smile fitting and numerical methods for pricing. Rev. Deriv. Res. 4, 231–262 (2000)

    Article  MATH  Google Scholar 

  3. 3.

    Avellaneda, M., Boyer-Olson, D., Busca, J., Friz, P.: Application of large deviation methods to the pricing of index options in finance. C. R. Math. Acad. Sci. Paris 336, 263–266 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions theory revisited. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, 567–585 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bentata, A., Cont, R.: Mimicking the marginal distributions of a semimartingale. Working paper (2012). arXiv:0910.3992v5 [math.PR]

  6. 6.

    Berestycki, H., Busca, J., Florent, I.: Asymptotics and calibration of local volatility models. Quant. Finance 2, 61–69 (2002)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 3, 637–654 (1973)

    Article  Google Scholar 

  8. 8.

    Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Finance 13, 345–382 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Carr, P., Geman, H., Madan, D.B., Yor, M.: From local volatility to local Lévy models. Quant. Finance 4, 581–588 (2004)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Carr, P., Hirsa, A.: Why be backward? Risk 16(1), 103–107 (2003)

    Google Scholar 

  11. 11.

    Cont, R., Minca, A.: Recovering portfolio default intensities implied by CDO tranches. Math. Finance 23, 94–121 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Cont, R., Savescu, I.: Forward equations for portfolio credit derivatives. In: Cont, R. (ed.) Frontiers in Quantitative Finance: Credit Risk and Volatility Modeling, pp. 269–288. Wiley, New York (2008), Chap. 11

    Google Scholar 

  13. 13.

    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. CRC Press, Boca Raton (2004)

    Google Scholar 

  14. 14.

    Cont, R., Voltchkova, E.: Integro-differential equations for option prices in exponential Lévy models. Finance Stoch. 9, 299–325 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Dupire, B.: Model art. Risk 6, 118–120 (1993)

    Google Scholar 

  16. 16.

    Dupire, B.: Pricing with a smile. Risk 7, 18–20 (1994)

    Google Scholar 

  17. 17.

    Dupire, B.: A unified theory of volatility. Working paper, Paribas (1996). Unpublished

  18. 18.

    Dupire, B.: Pricing and hedging with smiles. In: Dempster, M., Pliska, S. (eds.) Mathematics of Derivative Securities, pp. 103–111. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  19. 19.

    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin (2000)

    Google Scholar 

  20. 20.

    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986)

    Google Scholar 

  21. 21.

    Filipovic, D., Overbeck, L., Schmidt, T.: Dynamic CDO term structure modeling. Math. Finance 21, 53–71 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Garroni, M.G., Menaldi, J.L.: Second Order Elliptic Integro-Differential Problems. CRC Press, Boca Raton (2002)

    Google Scholar 

  23. 23.

    Giesecke, K.: Portfolio credit risk: top down vs. bottom up approaches. In: Cont, R. (ed.) Frontiers in Quantitative Finance: Credit Risk and Volatility Modeling, pp. 251–265. Wiley, New York (2008), Chap. 10

    Google Scholar 

  24. 24.

    Gyöngy, I.: Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Relat. Fields 71, 501–516 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    He, S.W., Wang, J.G., Yan, J.A.: Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe (Science Press), Beijing (1992)

    Google Scholar 

  26. 26.

    Hilber, N., Reich, N., Schwab, C., Winter, C.: Numerical methods for Lévy processes. Finance Stoch. 13, 471–500 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Jourdain, B.: Stochastic flows approach to Dupire’s formula. Finance Stoch. 11, 521–535 (2007)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Klebaner, F.: Option price when the stock is a semimartingale. Electron. Commun. Probab. 7, 79–83 (2002) (electronic)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Lopatin, A.V., Misirpashaev, T.: Two-dimensional Markovian model for dynamics of aggregate credit loss. In: Fouque, J.P., Fomby, T.B., Solna, K. (eds.) Advances in Econometrics, vol. 22, pp. 243–274. Emerald Group Publishing, Bingley (2008)

    Google Scholar 

  30. 30.

    Madan, D., Yor, M.: Making Markov martingales meet marginals. Bernoulli 8, 509–536 (2002)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Merton, R.: Theory of rational option pricing. Bell J. Econ. 4, 141–183 (1973)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Mikulevičius, R., Pragarauskas, H.: On the martingale problem associated with integro-differential operators. In: Grigelionis, B. (ed.) Probability Theory and Mathematical Statistics, vol. II, Vilnius, 1989. VSP, pp. 168–175. Mokslas, Vilnius (1990)

    Google Scholar 

  33. 33.

    Protter, P., Shimbo, K.: No arbitrage and general semimartingales. In: Ethier, S.N., et al. (eds.) Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz. Inst. Math. Stat. Collect., vol. 4, pp. 267–283. Institute of Mathematical Statistics, Beachwood (2008)

    Google Scholar 

  34. 34.

    Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005)

    Google Scholar 

  35. 35.

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)

    Google Scholar 

  36. 36.

    Schönbucher, P.: Portfolio losses and the term structure of loss transition rates: a new methodology for the pricing of portfolio credit derivatives. Working paper (2005).

  37. 37.

    Sidenius, J., Piterbarg, V., Andersen, L.: A new framework for dynamic credit portfolio loss modeling. Int. J. Theor. Appl. Finance 11, 163–197 (2008)

    MathSciNet  Article  MATH  Google Scholar 

Download references


We thank Damien Lamberton for numerous remarks, which helped improve the manuscript. Substantial assistance from an Associate Editor and the Editor is also gratefully acknowledged. Any remaining errors are our own.

Author information



Corresponding author

Correspondence to Amel Bentata.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bentata, A., Cont, R. Forward equations for option prices in semimartingale models. Finance Stoch 19, 617–651 (2015).

Download citation


  • Forward equation
  • Dupire equations
  • Jump process
  • Semimartingale
  • Tanaka–Meyer formula
  • Markovian projection
  • Call option
  • Option pricing

Mathematics Subject Classification

  • 60H30
  • 91G20
  • 35S10
  • 91G80

JEL Classification

  • C60
  • G13