Skip to main content

Implicit American Monte Carlo methods for nonlinear functional of future portfolio value


We introduce a new approximation algorithm for nonlinear functional of future portfolio value. We call it as “Implicit method” in contrast with calling existing method as “Explicit method”. In this paper, we show both “Implicit” and “Explicit” algorithms converge to true value using Stochastic mesh which is an typical example of American Monte Carlo methods. And we show the efficiency of “Implicit method” through the theoretical convergence order and numerical simulation.

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

Fig. 1
Fig. 2


  1. 1.

    Broadie, M., Glasserman, P., et al.: Astochastic mesh method for pricing high-dimensional American options. J. Comput. Finance. 7(4), 35–72 (2004)

    Article  Google Scholar 

  2. 2.

    Doersek, P., Teichmann, J., Velušček, D.: Cubature methods for stochastic (partial) differential equations in weighted spaces, vol. 1, pp. 634–663. Springer, Berlin (2013)

  3. 3.

    Duffie, D., Huang, M.: Swap rates and credit quality, vol. 51, pp. 921–949. Wiley, New York (1996)

  4. 4.

    Fujii, M., Takahashi, A.: Derivative pricing under asymmetric and imperfect collateralization and CVA, vol. 13, pp. 749–768. Taylor & Francis, Boca Raton (2013)

  5. 5.

    Glasserman, P.: Monte Carlo methods in financial engineering, vol. 53. Springer Science & Business Media, Berlin (2013)

    MATH  Google Scholar 

  6. 6.

    Green, A.: XVA: Credit, Funding and Capital Valuation Adjustments. Wiley, New York (2015)

    Book  Google Scholar 

  7. 7.

    Gregory, J.: The XVA challenge: Counterparty credit risk, funding, collateral, and capital. Wiley, New York (2015)

    Book  Google Scholar 

  8. 8.

    Kusuoka, S.: Malliavin calculus revisited. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo (2003)

    MATH  Google Scholar 

  9. 9.

    Kusuoka, S., Morimoto, Y.: Stochastic mesh methods for hörmander type diffusion processes. In: Advances in Mathematical Economics, vol. 18, pp. 61–99. Springer, Tokyo (2014)

  10. 10.

    Kusuoka, S., Morimoto, Y.: Least square regression methods for Bermudan derivatives and systems of functions. In: Advances in Mathematical Economics, vol. 19, pp. 57–89. Springer, Tokyo (2015)

  11. 11.

    Shigeo, K., Stroock, D.: Applications of the malliavin calculus. II. Faculty of Science, The University of Tokyo, Tokyo (1985)

  12. 12.

    Longstaff, F.A., Schwartz, E.S.: Valuing American options by simulation: a simple least-squares approach. Rev. Financial Stud. 14(1), 113–147 (2001)

    Article  Google Scholar 

  13. 13.

    Shigekawa, I.: Stochastic analysis, vol. 224. American Mathematical Soc, Providence (2004)

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Yusuke Morimoto.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Morimoto, Y. Implicit American Monte Carlo methods for nonlinear functional of future portfolio value. Japan J. Indust. Appl. Math. 34, 635–674 (2017).

Download citation


  • Computational finance
  • Option pricing
  • Malliavin calculus
  • Stochastic mesh method
  • XVA

Mathematics Subject Classification

  • 65C05
  • 60G40

JEL Classification

  • C63
  • G12