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Handbook of Computational and Numerical Methods in Finance pp 111–174Cite as

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Malliavin Calculus in Finance

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Abstract

This article is an introduction to Malliavin Calculus for practitioners. We treat one specific application to the calculation of greeks in Finance. We consider also the kernel density method to compute greeks and an extension of the Vega index called the local vega index.

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References

  1. Bally V., Talay D.: The law of the Euler scheme for stochastic differential equations (I): convergence rate of the distribution function, Probab. Rel. Fields 104 (1996), 43–60.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bally V., Talay D.: The law of the Euler scheme for stochastic differential equations (II): convergence rate of the distribution function, Monte Carlo Methods Methods Appl 2 (1996), 93–128.

    MathSciNet  MATH  Google Scholar 

  3. Bermin H.-P., Kohatsu-Higa A., Montera M.: Local Vega index and variance reduction methods. Math. Finance 13 (2003), 85–97.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernis G., Gobet E., Kohatsu-Higa A.: Monte Carlo evaluation of Greeks for multidimensional barrier and lookback options. Math. Finance 13 (2003), 99–113.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bichteler K., Gravereaux J. and Jacod J.: Malliavin Calculus for Processes with Jumps. Gordon and Breach Science Publishers, 1987.

    Google Scholar 

  6. Bouchard B., Ekeland I., Touzi N.: On the Malliavin approach to Monte Carlo approximations to conditional expectations. To appear in Finance and Stochastics, 2004.

    Google Scholar 

  7. Broadie, M., Glasserman P.: Estimating security price derivatives using simulation. Management Science 42 (1996), 269–285.

    Article  MATH  Google Scholar 

  8. Cattiaux, P., Mesnager L.: Hypoelliptic non homogeneous diffusions. Probability Theory and Related Fields 123 (2002), 453–83.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cvitanic, J., Spivak G., Maximizing the probability of a perfect hedge. Annals of Applied Probability, 9(4) (1999), 1303–1328.

    Article  MathSciNet  MATH  Google Scholar 

  10. Dufresne, D.: Laguerre series for Asian and other options. Math. Finance 10 (2000), 407–28.

    Article  MathSciNet  MATH  Google Scholar 

  11. Föllmer H., Leukert P.: Quantile Hedging. Finance and Stochastics 3 (1999), 251–273.

    Article  MathSciNet  MATH  Google Scholar 

  12. Föllmer H., Leukert P.: Efficient hedging: Cost versus shortfall risk. Finance and Stochastics 4 (2000), 117–146.

    Article  MathSciNet  MATH  Google Scholar 

  13. Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L., Touzi, N.: An application of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics 3 (1999), 391–412.

    Article  MathSciNet  MATH  Google Scholar 

  14. Fournié E., Lasry J.M., Lebuchoux J., Lions P.L.: Applications of Malliavin calculus to Monte Carlo methods in finance II. Finance Stochast. 5 (2001), 201–236.

    Article  MATH  Google Scholar 

  15. Geman H., Yor M.: Bessel processes, Asian options and perpetuities. Math. Finance, 3 (1993), 349–375.

    Article  MATH  Google Scholar 

  16. Glynn P.W.: Likelihood ratio gradient estimation: an overview. In: Proceedings of the 1987 Winter Simulation Conference, A. Thesen, H. Grant and W.D. Kelton, eds, 366–375, 1987

    Google Scholar 

  17. Gobet, E.: LAMN property for elliptic diffusion: a Malliavin Calculus approach. Bernoulli 7 (2001), 899–912.

    Article  MathSciNet  MATH  Google Scholar 

  18. Gobet E., Kohatsu-Higa A.: Computation of Greeks for barrier and lookback options using Malliavin calculus. Electronic Communications in Probability 8 (2003), 51–62.

    Article  MathSciNet  MATH  Google Scholar 

  19. Gobet E., Munos R.: Sensitivity analysis using Itô-Malliavin Calculus and Martingales. Application to stochastic optimal control. Preprint (2002).

    Google Scholar 

  20. Hull J.C.: Options, Futures and Other Derivatives, Fourth Ed., Prentice-Hall, Upper Saddle River, NJ, 2000.

    Google Scholar 

  21. Ikeda N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1989.

    MATH  Google Scholar 

  22. Imkeller, P., Pontier, M., Weisz, F.: Free lunch and arbitrage possibilities in a financial market with an insider. Stochastic Proc. Appl. 92 (2001), 103–130.

    Article  MathSciNet  MATH  Google Scholar 

  23. Karatzas, I.: Lectures on the Mathematics of Finance. CRM Monographs 8, American Mathematical Society, 1996.

    Google Scholar 

  24. Karatzas, I., Shreve S.: Brvwnian Motion and Stochastic Calculus. Springer-Verlag, 1991.

    Google Scholar 

  25. Kohatsu-Higa, A.: High order Itô-Taylor approximations to heat kernels. Journal of Mathematics of Kyoto University 37 (1997), 129–151.

    MathSciNet  MATH  Google Scholar 

  26. Kohatsu-Higa, A., Pettersson R.: Variance reduction methods for simulation of densities on Wiener space. SIAM Journal of Numerical Analysis 40 (2002), 431–450.

    Article  MathSciNet  MATH  Google Scholar 

  27. Kulldorff, M.: Optimal control of favorable games with a time limit. SIAM Journal of Control & Optimization 31 (1993), 52–69.

    Article  MathSciNet  MATH  Google Scholar 

  28. Kunitomo, N., Takahashi A.: The asymptotic expansion approach to the valuation of interest rate contingent claims. Math. Finance 11 (2001), 117–151.

    Article  MathSciNet  MATH  Google Scholar 

  29. Kusuoka, S., Stroock, D.W.: Application of the Malliavin calculus I. In: Stochastic Analysis, Itô, K., ed. Proceedings of the Taniguchi International Symposium on Stochastic Analysis, Katata and Kyoto, 1982. Tokyo: Kinokuniya/North-Holland, 271–306, 1988.

    Google Scholar 

  30. L’Ecuyer P., Perron G.: On the convergence rates of IPA and FDC derivative estimators. Operations Research 42 (1994), 643–656.

    Article  MathSciNet  Google Scholar 

  31. Nualart D.: The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin (1995).

    MATH  Google Scholar 

  32. Nualart D., Vives J.: Continuité absolue de la loi du maximum d’un processus continu. C.R. Acad. sci. Paris 307 (1988), 349–354.

    MathSciNet  MATH  Google Scholar 

  33. Malliavin P.: Stochastic Analysis. Springer-Verlag, Berlin, Heidelberg, New York, 1997.

    MATH  Google Scholar 

  34. Oksendal B.: An Introduction to Malliavin Calculus with Applications to Economics. Working Paper 3, Norwegian School of Economics and Business Administration (1996).

    Google Scholar 

  35. Picard, J.: On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 (1996), 481–511.

    Article  MathSciNet  MATH  Google Scholar 

  36. Privault N.: Absolute continuity in infinite dimensions and anticipating stochastic calculus. Potential Analysis 8 (1998), 325–343.

    Article  MathSciNet  MATH  Google Scholar 

  37. Silverman W.: Density Estimation, Chapman Hall, London, 1986.

    MATH  Google Scholar 

  38. Üstünel, A.S.: An introduction to stochastic analysis on Wiener space. Springer-Verlag, Lecture Notes in Mathematics 1610, 1996.

    Google Scholar 

  39. Wilmott, P., Derivatives, Wiley, Chichester 1998.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Economics, Universitat Pompeu Fabra, Ramón Trias Fargas, 25-27, 08005, Barcelona, Spain

    Arturo Kohatsu-Higa

  2. Departament de Física Fonamental, Universitat de Barcelona, Diagonal 647, 08028, Barcelona, Spain

    Miquel Montero

Authors
  1. Arturo Kohatsu-Higa
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  2. Miquel Montero
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Editor information

Editors and Affiliations

  1. Department of Statistics and Applied Probability, University of California, Santa Barbara, CA, 93106, USA

    Svetlozar T. Rachev

  2. Department of Economics and Business Engineering, Universität Karlsruhe, D-76128, Karlsruhe, Germany

    Svetlozar T. Rachev

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© 2004 Springer Science+Business Media New York

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Kohatsu-Higa, A., Montero, M. (2004). Malliavin Calculus in Finance. In: Rachev, S.T. (eds) Handbook of Computational and Numerical Methods in Finance. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8180-7_4

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  • DOI: https://doi.org/10.1007/978-0-8176-8180-7_4

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6476-7

  • Online ISBN: 978-0-8176-8180-7

  • eBook Packages: Springer Book Archive

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