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An FBSDE Approach to American Option Pricing with an Interacting Particle Method

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Abstract

In the paper, we propose a new calculation scheme for American options in the framework of a forward backward stochastic differential equation (FBSDE). The well-known decomposition of an American option price with that of a European option of the same maturity and the remaining early exercise premium can be cast into the form of a decoupled non-linear FBSDE. We numerically solve the FBSDE by applying an interacting particle method recently proposed by Fujii and Takahashi (2012c), which allows one to perform a Monte Carlo simulation in a fully forward-looking manner. We perform the fourth-order analysis for the Black–Scholes (BS) model and the third-order analysis for the Heston model. The comparison to those obtained from existing tree algorithms shows the effectiveness of the particle method.

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Notes

  1. It is closely related to the research with a long history on the branching Markov process and a certain class of semi-linear PDEs. For instance, see Fujita (1966), Ikeda et al. (1965, 1966, 1968, 1966) and Nagasawa and Sirao (1969).

  2. A related but different approach was recently applied to evaluate CVA by Henry-Labordère (2012).

  3. Notice that there is no need to use unnecessarily small variance for the approximation of delta function. Intuitively speaking, the delta function within the expectation operation extracts the density where its argument vanishes. Thus, as long as the density functions of the underlyings do not change significantly within a given range, one can use it as a variance of the normal density function as an approximation of the corresponding delta function.

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Authors and Affiliations

  1. Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Masaaki Fujii, Seisho Sato & Akihiko Takahashi

Authors
  1. Masaaki Fujii
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  2. Seisho Sato
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  3. Akihiko Takahashi
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Correspondence to Seisho Sato.

Additional information

This research is partially supported by CARF (Center for Advanced Research in Finance), the global COE program “The research and training center for new development in mathematics,” as well as JSPS KAKENHI Grant Number 23500364. All the contents expressed in this research are solely those of the authors and do not represent any views or opinions of any institutions. The authors are not responsible or liable in any manner for any losses and/or damages caused by the use of any contents in this research.

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Fujii, M., Sato, S. & Takahashi, A. An FBSDE Approach to American Option Pricing with an Interacting Particle Method. Asia-Pac Financ Markets 22, 239–260 (2015). https://doi.org/10.1007/s10690-014-9195-6

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