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Smart expansion and fast calibration for jump diffusions

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Abstract

Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump processes. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black–Scholes volatilities for call options is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very rapid.

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References

  1. Albanese, C., Campolieti, G., Carr, P., Lipton, A.: Black–Scholes goes hypergeometric. Risk Mag. 14(12), 99–103 (2001)

    Google Scholar 

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

    Article  Google Scholar 

  3. Andersen, L., Andreasen, J.: Volatile volatilities. Risk Mag. 15(12), 163–168 (2002)

    Google Scholar 

  4. Antonelli, F., Scarlatti, S.: Pricing options under stochastic volatility: a power series approach. Finance Stoch. 13, 269–303 (2009)

    Article  MathSciNet  Google Scholar 

  5. Benhamou, E., Gobet, E., Miri, M.: Closed forms for European options in a local volatility model. Int. J. Theor. Appl. Finance (2008, to appear). Available at http://hal.archives-ouvertes.fr/hal-00325939/fr/

  6. Benhamou, E., Gobet, E., Miri, M.: Time-dependent Heston model. SSRN preprint (2009). http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1367955

  7. Benhamou, E., Miri, M.: Predictor corrector methods applied to PIDE and Monte Carlo simulations. Working Paper, Pricing Partners (2006). Available on request.

  8. Scholes, M., Black, F.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)

    Article  Google Scholar 

  9. Bouchard, B., Elie, R.: Discrete-time approximation of decoupled forward-backward SDE with jumps. Stoch. Process. Appl. 118, 53–75 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cass, T.: Smooth densities for solutions to stochastic differential equations with jumps. Stoch. Process. Appl. 119, 1416–1435 (2008)

    Article  Google Scholar 

  11. Cont, R., Tankov, P.: Non-parametric calibration of jump diffusion option pricing models. J. Comput. Finance 7(3), 1–49 (2003)

    Google Scholar 

  12. Cont, R., Voltchkova, E.: A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43, 1596–1626 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dupire, B.: Pricing with a smile. Risk Mag. 7(1), 18–20 (1994)

    Google Scholar 

  14. Fouque, J.P., Papanicolaou, G., Sircar, R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  15. Fournié, E., Lasry, J.M., Lebuchoux, J., Lions, P.L., Touzi, N.: Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3, 391–412 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fujiwara, T., Kunita, H.: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25, 71–106 (1985)

    MATH  MathSciNet  Google Scholar 

  17. Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. Wiley Finance, New York (2006)

    Google Scholar 

  18. Gobet, E.: Revisiting the Greeks for European and American options. In: Akahori, J., Ogawa, S., Watanabe, S. (eds.) Proceedings of the International Symposium on Stochastic Processes and Mathematical Finance, Ritsumeikan University, Kusatsu, Japan, pp. 53–71. World Scientific, Singapore (2004).

    Google Scholar 

  19. Gobet, E., Munos, R.: Sensitivity analysis using Itô–Malliavin calculus and martingales. Application to stochastic control problem. SIAM J. Control Optim. 43, 1676–1713 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. Wilmott Mag., 84–108 (2002)

  21. Hagan, P.S., Woodward, D.E.: Equivalent Black volatilities. Appl. Math. Finance 6, 147–157 (1999)

    Article  MATH  Google Scholar 

  22. Henry Labordère, P.: A general asymptotic implied volatility for stochastic volatility models (2005). http://arxiv.org/abs/cond-mat/0504317

  23. Lewis, A.: Option Valuation under Stochastic Volatility. Finance Press, New York (2000)

    MATH  Google Scholar 

  24. Matache, A.M., von Petersdorff, T., Schwab, C.: Fast deterministic pricing of options on Lévy driven assets. Math. Modell. Numer. Anal. 38, 37–71 (2004)

    Article  MATH  Google Scholar 

  25. Matytsin, A.: Perturbative analysis of volatility smiles. Working paper (2000). http://www.math.columbia.edu/~smirnov/matytsin2000.pdf

  26. Merton, R.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1–2), 125–144 (1976)

    Article  MATH  Google Scholar 

  27. Miri, M.: Stochastic expansions and closed pricing formulas of European options (tentative title). PhD thesis, Université de Grenoble (2009, in preparation). http://www-ljk.imag.fr/MATHFI/

  28. Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  29. Piterbarg, V.V.: A multi-currency model with FX volatility skew. SSRN working paper (2005). http://papers.ssrn.com/sol3/papers.cfm?abstract_id=685084

  30. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  31. Rubinstein, M.: Implied binomial trees. J. Finance 49, 771–818 (1994)

    Article  Google Scholar 

  32. Rudin, W.: Real and Complex Analysis. McGraw-Hill, Toronto (1966)

    MATH  Google Scholar 

  33. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  34. Siopacha, M., Teichmann, J.: Weak and strong Taylor methods for numerical solutions of stochastic differential equations (2007). http://arxiv.org/abs/0704.0745

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Authors and Affiliations

  1. Pricing Partners, 204 Rue de Crimée, 75019, Paris, France

    E. Benhamou & M. Miri

  2. Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, 38041, Grenoble cedex 9, France

    E. Gobet & M. Miri

Authors
  1. E. Benhamou
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  2. E. Gobet
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  3. M. Miri
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Correspondence to E. Benhamou.

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Benhamou, E., Gobet, E. & Miri, M. Smart expansion and fast calibration for jump diffusions. Finance Stoch 13, 563–589 (2009). https://doi.org/10.1007/s00780-009-0102-3

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