An Introduction to Particle Methods with Financial Applications

  • René Carmona
  • Pierre Del Moral
  • Peng HuEmail author
  • Nadia Oudjane
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 12)


The aim of this article is to give a general introduction to the theory of interacting particle methods, and an overview of its applications to computational finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to the numerical solution of a variety of financial applications such as pricing complex path dependent European options, computing sensitivities, pricing American options or numerically solving partially observed control and estimation problems.


Advanced Monte Carlo Feynman-Kac Interacting particle system 


  1. 1.
    C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chain Monte Carlo methods. Journal Royal Statistical Society B, vol. 72, no. 3, pp. 269–342 (2010).Google Scholar
  2. 2.
    Y. Achdou and O. Pironneau. Computational methods for option pricing. SIAM, Frontiers in Applied Mathematics series, (2005).Google Scholar
  3. 3.
    R. Bahr and S. Hamori. Hidden Markov models : Applications to finance and economics. Advanced studies in theoretical and applied econometrics. vol. 40, Kluwer Academic Publishers (2004).Google Scholar
  4. 4.
    S. Ben Hamida and R. Cont. Recovering Volatility from Option Prices by Evolutionary Optimization. Journal of Computational Finance, Vol. 8, No. 4 (2005).Google Scholar
  5. 5.
    V. S. Borkar. Controlled diffusion processes. Probability Surveys, Vol. 2, pp. 213–244 (2005).Google Scholar
  6. 6.
    B. Bouchard and X. Warin. Monte-Carlo valuation of American options: facts and new algorithms to improve existing methods. Numerical Methods in finance, Springer (2011).Google Scholar
  7. 7.
    D. Brigo, T. Bielecki and F. Patras. Credit Risk Frontiers. Wiley–Bloomberg Press (2011).Google Scholar
  8. 8.
    M. Broadie and P. Glasserman. Estimating security prices using simulation. Management Science, 42, pp. 269–285 (1996).Google Scholar
  9. 9.
    M. Broadie and P. Glasserman. A Stochastic Mesh Method for Pricing High-Dimensional American Options. Journal of Computational Finance, vol. 7, pp. 35–72, (2004).Google Scholar
  10. 10.
    R. Carmona and S. Crépey. Importance Sampling and Interacting Particle Systems forthe Estimation of Markovian Credit Portfolios Loss Distribution. International Journal of Theoretical and Applied Finance, vol. 13, No. 4 (2010) 577–602.Google Scholar
  11. 11.
    R. Carmona, J.-P. Fouque and D. Vestal. Interacting Particle Systems for the Computation of Rare Credit Portfolio Losses. Finance and Stochastics, vol. 13, no. 4, 2009 pp. 613–633 (2009).Google Scholar
  12. 12.
    J. F. Carrière. Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression. Insurance : Mathematics and Economics, 19, 19–30 (1996).Google Scholar
  13. 13.
    R. Casarin. Simulation Methods for Nonlinear and Non-Gaussian Models in Finance. Premio SIE, Rivista Italiana degli Economisti, vol. 2, pp. 341–345 (2005).Google Scholar
  14. 14.
    R. Casarin and C. Trecroci. Business Cycle and Stock Market Volatility: A Particle Filter Approach, Cahier du CEREMADE N. 0610, University Paris Dauphine (2006).Google Scholar
  15. 15.
    N. Chen and P. Glasserman. Malliavin Greeks without Malliavin calculus. Stochastic Processes and their Applications 117, pp. 1689–1723 (2007).Google Scholar
  16. 16.
    E. Clément, D. Lamberton and P. Protter. An analysis of a least squares regression method for American option pricing. Finance and Stochastics, 6, 449–472, (2002).Google Scholar
  17. 17.
    R. Cont and P. Tankov. Non-parametric calibration of jump-diffusion option pricing models. Journal of computational finance, vol. 7, no. 3, pp. 1–49, (2004).Google Scholar
  18. 18.
    R. Cont and P. Tankov. Retrieving Levy processes from option prices: regularization of ill-posed inverse problem. SIAM Journal on Control and Optimization, vol. 45, no. 1, pp. 1–25 (2007).Google Scholar
  19. 19.
    A.L. Corcoran and R.L. Wainwright. A parallel island model genetic algorithm for the multiprocessor scheduling problem. Proceedings of the 1994 ACM/SIGAPP Symposium on Applied Computing, March 6–8, pp. 483–487, ACM Press (1994).Google Scholar
  20. 20.
    D. D. Creal. A survey of sequential Monte Carlo methods for economics and finance. To appear in Econometric Reviews (2011).Google Scholar
  21. 21.
    S. Crépey. Calibration of the local volatility in a trinomial tree using Tikhonov regularization. Inverse Problems, vol. 19, pp. 91–127 (2003)Google Scholar
  22. 22.
    D. Crisan, P. Del Moral and T. Lyons. Interacting Particle Systems Approximations of the Kushner Stratonovitch Equation. Advances in Applied Probability, vol.31, no. 3, 819–838 (1999).Google Scholar
  23. 23.
    Del Moral P. Non Linear Filtering: Interacting Particle Solution. Markov Processes and Related Fields, Volume 2 Number 4, 555–580 (1996).Google Scholar
  24. 24.
    P. Del Moral. Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, New York: Springer-Verlag (2004).Google Scholar
  25. 25.
    P. Del Moral, Measure Valued Processes and Interacting Particle Systems. Application to Nonlinear Filtering Problems. Annals of Applied Probabability, vol. 8, no. 2, pp. 1254–1278 (1998).Google Scholar
  26. 26.
    P. Del Moral and A. Doucet. Sequential Monte Carlo and Genetic particle models. Theory and Practice, in preparation, Chapman & Hall (2011).Google Scholar
  27. 27.
    P. Del Moral, A. Doucet and S. S. Singh. A Backward Particle Interpretation of Feynman-Kac Formulae HAL-INRIA RR-7019 (07-2009), M2AN, vol 44, no. 5, pp. 947–976 M2AN (sept. 2010).Google Scholar
  28. 28.
    P. Del Moral, J. Garnier. Genealogical Particle Analysis of Rare events. Annals of Applied Probability, vol. 15, no. 4, 2496–2534 (2005).Google Scholar
  29. 29.
    P. Del Moral and A. Guionnet. On the stability of interacting processes with applications to filtering and genetic algorithms, Annales de l’Institut Henri Poincaré, Vol. 37, No. 2, 155–194 (2001).Google Scholar
  30. 30.
    P. Del Moral and A. Jasra. SMC for option pricing. To appear in Stochastic Analysis and Applications Volume 29, Issue 2, pp. 292–316 (2011).Google Scholar
  31. 31.
    P. Del Moral and L. Miclo. Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering. Séminaire de Probabilités XXXIV, Ed. J. Azma and M. Emery and M. Ledoux and M. Yor, Lecture Notes in Mathematics, Springer-Verlag Berlin, Vol. 1729, 1–145 (2000).Google Scholar
  32. 32.
    P. Del Moral, P. Hu and N. Oudjane. Snell envelope with path dependent multiplicative optimality criteria HAL-INRIA, RR-7360 (2010).Google Scholar
  33. 33.
    P. Del Moral, P. Hu, N. Oudjane and Br. Rémillard. On the Robustness of the Snell envelope. HAL-INRIA, RR-7303, to appear in SIAM Journal on Financial Mathematics (2011).Google Scholar
  34. 34.
    P. Del Moral, P. Hu and L. Wu. On the concentration properties of Interacting particle processes. HAL-INRIA, RR-7677, to appear in Foundations and Trends in Machine Learning (2011).Google Scholar
  35. 35.
    P. Del Moral and Fr. Patras. Interacting path systems for credit risk. Credit Risk Frontiers. D. Brigo, T. Bielecki, F. Patras Eds. Wiley–Bloomberg Press, (2011), 649–674. Short announcement available as Interacting path systems for credit portfolios risk analysis. INRIA:RR-7196 (2010).Google Scholar
  36. 36.
    P. Del Moral, Br. Rémillard and S. Rubenthaler. Monte Carlo approximations of American options that preserve monotonicity and convexity. (2011).Google Scholar
  37. 37.
    P. Del Moral and E. Rio. Concentration Inequalities for Mean Field Particle Models. HAL-INRIA RR-6901, to appear in the Annals of Applied Probability (2011).Google Scholar
  38. 38.
    D. Egloff. Monte Carlo algorithms for optimal stopping and statistical learning. Annals of Applied Probability , 15, pp. 1–37 (2005).Google Scholar
  39. 39.
    E. Fournié, J. M. Lasry, J. Lebuchoux, P. L. Lions and N. Touzi. Applications of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics, vol. 3, pp. 391–412 (1999).Google Scholar
  40. 40.
    E. Fournié, J. M. Lasry, J. Lebuchoux and P. L. Lions. Applications of Malliavin calculus to Monte Carlo methods in finance. II. Finance and Stochastics, vol. 5, pp. 201–236 (2001).Google Scholar
  41. 41.
    M. C. Fu, D. B. Madan and T. Wang. Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods. Journal of Computational Finance, vol. 2, pp. 49–74, (1998).Google Scholar
  42. 42.
    V. Genon-Catalot, Th. Jeantheau and C. Laredo. Conditional Likelihood Estimators for Hidden Markov Models and Stochastic Volatility Models. Scandinavian Journal of Statistics, Vol 30: 297–316 (2003).Google Scholar
  43. 43.
    R. Lagnado and S. Osher. A technique for calibrating derivative security pricing models: numerical solution of the inverse problem. Journal of computational finance, vol. 1, pp. 13–25 (1997).Google Scholar
  44. 44.
    Fr. Le Gland and N. Oudjane. Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters. Annals Applied Probability, Vol. 14, no. 1, 144–187 (2004).Google Scholar
  45. 45.
    M. Giles and P. Glasserman, Smoking adjoints: fast Monte Carlo Greeks. Risk, pp. 92–96 (2006).Google Scholar
  46. 46.
    E. Gobet, J. P. Lemor and X. Warin. A regression-based Monte-Carlo method for backward stochastic differential equations. Annals Applied Probability, 15, pp. 2172–2202 (2005).Google Scholar
  47. 47.
    M. S. Johannes, N. G. Polson and J. R. Stroud. Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices. Review of Financial Studies, 22, Issue. 7, pp. 2759–2799 (2009).Google Scholar
  48. 48.
    I. Karatzas and S. Shreve. Methods of Mathematical Finance. Springer (1998).Google Scholar
  49. 49.
    A. Kohatsu-Higa and M. Montero. Malliavin Calculus in Finance. Handbook of Computational and Numerical Methods in Finance, Birkhauser, pp. 111–174 (2004).Google Scholar
  50. 50.
    J. M. Lasry and P. L. Lions. Contrôle stochastique avec informations partielles et applications à la Finance. Comptes Rendus de l’Académie des Sciences – Series I – Mathematics, vol. 328, issue 11, pp. 1003–1010 (1999).Google Scholar
  51. 51.
    G. Liu and L. J. Hong. Revisit of stochastic mesh method for pricing American options. Operations Research Letters. 37(6), 411–414 (2009).Google Scholar
  52. 52.
    F. A. Longstaff and E. S. Schwartz. Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies 14, pp. 113–147 (2001).Google Scholar
  53. 53.
    F. A. Matsen and J. Wakeley. Convergence to the Island-Model Coalescent Process in Populations With Restricted Migration. Genetics, vol. 172, pp. 701–708 (2006).Google Scholar
  54. 54.
    G. Pagès and B. Wilbertz. Optimal quantization methods for pricing American style options. Numerical Methods in finance, Springer (2011).Google Scholar
  55. 55.
    G. Pagès and H. Pham. Optimal quantization methods for nonlinear filtering with discrete-time observations. Bernoulli, vol.11, pp. 893–932 (2005).Google Scholar
  56. 56.
    G. Pagès, H. Pham and J. Printems. Optimal quantization methods and applications to numerical problems in finance. in Handbook of Computational and Numerical Methods in Finance. Ed. S. T. Rachev, Birkhuser, Boston, 2004, pp. 253–297 (2004).Google Scholar
  57. 57.
    G. Pagès. A space vector quantization method for numerical integration. Journal of Comput. Appl. Math., 89, pp. 1–38 (1997).Google Scholar
  58. 58.
    G. Pagès and J. Printems. Functional quantization for numerics with an application to option pricing. Monte Carlo Methods and Appl., 11(4), pp. 407–446 (2005).Google Scholar
  59. 59.
    H. Pham, W. Runggaldier and A. Sellami. Approximation by quantization of the filter process and applications to optimal stopping problems under partial observation. Monte Carlo Methods and Applications. Volume 11, Issue 1, pp. 57–81 (2005).Google Scholar
  60. 60.
    H. Pham and M. C. Quenez. Optimal portfolio in partially observed stochastic volatility models. Annals of Applied Probability, 11, pp. 210–238 (2001).Google Scholar
  61. 61.
    H. Pham, M. Corsi, and W. Runggaldier, Numerical Approximation by Quantization of Control Problems in Finance Under Partial Observations. Handbook of Numerical Analysis, vol. 15, pp. 325–360 (2009).Google Scholar
  62. 62.
    V. Rossi and J. P. Vila. Nonlinear filtering in discrete time : A particle convolution approach. Ann. I.SU.P., vol.50, no. 3, pp. 71–102 (2006).Google Scholar
  63. 63.
    W. Runggaldier and L. Stettner. Approximations of Discrete Time Partially Observed Control Problems Applied Mathematics Monographs CNR, Giardini Editori, Pisa (1994).Google Scholar
  64. 64.
    J. N. Tsitsiklis and B. Van Roy. Regression Methods for Pricing Complex American-Style Options. IEEE Transactions on Neural Networks, Vol. 12, No. 4 (special issue on computational finance), pp. 694–703 (2001).Google Scholar
  65. 65.
    R. Van Handel. Uniform time average consistency of Monte Carlo particle filters, Stoch. Proc. Appl., 119, pp. 3835–3861 (2009).Google Scholar
  66. 66.
    J. Vanneste. Estimating generalized Lyapunov exponents for products of random matrices. Phys. Rev. E, vol. 81, 036701 (2010).Google Scholar
  67. 67.
    D. Whitley, S. Rana and R.B. Heckendorn. The island Model Genetic algorithm: On separability, population size and convergence. CIT. Journal of computing and information technology. vol. 7, no. 1, pp. 33–47 (1999).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • René Carmona
    • 1
  • Pierre Del Moral
    • 2
    • 3
  • Peng Hu
    • 2
    Email author
  • Nadia Oudjane
    • 4
  1. 1.Department of Operations Research and Financial Engineering, Bendheim Center for FinancePrinceton UniversityPrincetonUSA
  2. 2.Bordeaux Mathematical Institute, INRIA Bordeaux-Sud Ouest CenterUniversit Bordeaux ITalence cedexFrance
  3. 3.Centre de Mathématiques AppliquéesÉcole Polytechnique CNRSPalaiseauFrance
  4. 4.EDF R&D, Université Paris 13 and FiME (Finance for Energy Market Research Centre (Dauphine, CREST, EDF R&D))ClamartFrance

Personalised recommendations