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Unbiased simulation of Asian options

Bruno Bouchard, Xiaolu Tan

TL;DR

The paper develops an unbiased Monte Carlo framework for pricing Asian options modeled by a path-dependent SDE with $I_t = \\int_0^t X_s \, dA_s$, extending prior unbiased strategies to non-Markovian dynamics. It constructs a representation formula for $V_0 = \mathbb{E}[ g(X_T,I_T) ]$ using a random time grid and Malliavin-type weights derived from Dupire's path-dependent PDE theory, proving unbiasedness and establishing integrability under structural regularity conditions. It provides concrete sufficiency criteria for square integrability, including special cases like constant volatility, and derives explicit upper bounds on estimator moments. Numerical experiments on Asian options under the Bachelier and a local volatility model illustrate accurate unbiased estimates and favorable variance relative to Euler discretization, validating the method for path-dependent derivatives.

Abstract

We provide an extension of the unbiased simulation method for SDEs developed in Henry-Labordere et al. [Ann Appl Probab. 27:6 (2017) 1-37] to a class of path-dependent dynamics, pertaining for Asian options. In our setting, both the payoff and the SDE's coefficients depend on the (weighted) average of the process or, more precisely, on the integral of the solution to the SDE against a continuous function with bounded variations. In particular, this applies to the numerical resolution of the class of path-dependent PDEs whose regularity, in the sens of Dupire, is studied in Bouchard and Tan [Ann. I.H.P., to appear].

Unbiased simulation of Asian options

TL;DR

The paper develops an unbiased Monte Carlo framework for pricing Asian options modeled by a path-dependent SDE with , extending prior unbiased strategies to non-Markovian dynamics. It constructs a representation formula for using a random time grid and Malliavin-type weights derived from Dupire's path-dependent PDE theory, proving unbiasedness and establishing integrability under structural regularity conditions. It provides concrete sufficiency criteria for square integrability, including special cases like constant volatility, and derives explicit upper bounds on estimator moments. Numerical experiments on Asian options under the Bachelier and a local volatility model illustrate accurate unbiased estimates and favorable variance relative to Euler discretization, validating the method for path-dependent derivatives.

Abstract

We provide an extension of the unbiased simulation method for SDEs developed in Henry-Labordere et al. [Ann Appl Probab. 27:6 (2017) 1-37] to a class of path-dependent dynamics, pertaining for Asian options. In our setting, both the payoff and the SDE's coefficients depend on the (weighted) average of the process or, more precisely, on the integral of the solution to the SDE against a continuous function with bounded variations. In particular, this applies to the numerical resolution of the class of path-dependent PDEs whose regularity, in the sens of Dupire, is studied in Bouchard and Tan [Ann. I.H.P., to appear].

Paper Structure

This paper contains 14 sections, 8 theorems, 133 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Unbiased estimator for path-dependent SDEs
  3. The representation formula
  4. An intuitive overview.
  5. The representation formula.
  6. Proof of Theorem \ref{['theo:renorm_general_vol']}
  7. Integrability under structural conditions on the coefficients
  8. An upper bound of the estimators
  9. A sufficient condition for square integrability
  10. The case with constant volatility coefficient function
  11. Numerical examples
  12. The Bachelier model
  13. A local volatility model
  14. Technical lemmas

Key Result

Theorem 2.3

Let Assumption assum:smooth hold true. Suppose in addition that $\psi_n$ (resp. $\widehat{\psi}_n$, $\widetilde{\psi}_n$) is integrable for each $n \ge 1$, then Further assume that the sequence $(\psi_n)_{n \ge 1}$ (resp. $(\widehat{\psi}_n)_{n \ge 1}$, $(\widetilde{\psi}_n)_{n \ge 1}$) is uniformly integrable, then the random variable $\psi$ (resp. $\widehat{\psi}$, $\widetilde{\psi}$) is also i

Theorems & Definitions (17)

  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.2
  • Proposition 3.3
  • Remark 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 7 more