A Semigroup Expansion for Pricing Barrier Options

Takashi Kato,Akihiko Takahashi,Toshihiro Yamada

doi:10.1155/2014/268086

Abstract

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.

Highlights

Since the Merton’s seminal work ([15]) barrier options have been quite popular and important products in both academics and financial business for the last four decades
Let us note that the value of a continuously monitoring down-and-out barrier option is expressed as the following form under the so called risk-neutral probability measure: CBarrier(T, x)
Our approach is to firstly develop a general semi-group expansion scheme for the Cauchy-Dirichlet problem under multi-dimensional diffusion setting; as an application, we provide a new approximation formula under a certain class of stochastic volatility model

Summary

Introduction

Since the Merton’s seminal work ([15]) barrier options have been quite popular and important products in both academics and financial business for the last four decades. Fast and accurate computation of their prices and Greeks is highly desirable in the risk management, which is a tough task under the finance models commonly used in practice. It has been one of the central issues in the mathematical finance community. Let us note that the value of a continuously monitoring down-and-out barrier option is expressed as the following form under the so called risk-neutral probability measure: CBarrier(T , x) = E [ e− ∫T.

Objectives

Results

Conclusion