A RBSDES APPROACH FOR THE NUMERICAL EVALUATION OF THE AMERICAN OPTION PRICING PROBLEM

R Di Franco

doi:10.12732/ijpam.v109i2.15

Abstract

The problem of pricing American type options is a typical example of a non lin- ear problem characterized by the absence of closed expressions for its evaluation. Therefore, during recent years, such an issue has been approached , both deterministically and randomly, from the algorithmic point of view, trying to derive suitable numerical approximations. In this paper, starting from the aforementioned solutions, we review some computational, stochastic inspired, methods, mainly based on the the existing link between the above recalled pric- ing task, and the theory of Reflected Backward Stochastic Differential Equations (BSDEs). In particular we show how suitable numerical schemes can be developed within the SBDEs framework by mean of quantization techniques as well as considering Monte Carlo methods.

Highlights

There is a vast literature about Backward Stochastic Differential Equations (BSDEs) and their applications
The BSDEs theory has been extensively used in stochastic control, see, e.g., [3] and references therein, as they appear as adjoint equations in the stochastic version of Pontryagin maximum principle, and in Mathematical Finance, since any pricing problem by replication can be written in terms of linear BSDEs, or non-linear BSDEs when portfolios constraints are taken into account, see [19, 11, 12, 13, 14]
The paper is organized as follows: in Section 2 we give some basic notions on BSDEs, as, e.g., existence and uniqueness of solutions and the comparison principle; in Section 3 we report almost the same basic notions, but in the Reflected Backward Stochastic Differential Equation (RBSDE) setting, in Section 4 we will show some possible applications of the theoretical techniques developed for BSDEs and RBSDEs in Finance, in particular in Subsection 4.1, resp. in Subsection 4.2, we present the link between RBSDEs and American option, resp. the link between quantization method and RBSDE, while the application of the Monte Carlo approach to the same problem will be treated in Subsection 4.3

Summary

Introduction

There is a vast literature about Backward Stochastic Differential Equations (BSDEs) and their applications. El Karoui, Kapoudjian, Pardoux, Peng and Quenez generalized these results to BSDEs with reflection, see [20], that is, to a setting with an additional continuous, increasing process added in the equation; the function of this additional process is to keep the solution above a certain prescribed lowerboundary process and to do so in a minimal fashion These authors make the crucial observation that the solution is the value function of an optimal stopping problem. Bossaerts [9] and Tilley [33], suggested that Monte Carlo methods could be used to price American style derivatives Such type of numerical approaches are outperformed by ad hoc solutions at least in some particular frameworks as those when once for all computations can be realized. In the present paper we shall exploit the approach mainly developed in [2, 34], to the links between the RBSDEs theory, Monte Carlo and quantization method, to price American options.

Backward Stochastic Differential Equations

Reflected Backward Stochastic Differential Equations

RBSDEs-Quantization Approach to American Option Pricing

Pricing American Option by Monte Carlo Method

Conclusion