The Early Exercise Premium Representation for American Options on Multiply Assets

Tomasz Klimsiak,Andrzej Rozkosz

doi:10.1007/s00245-015-9293-5

Abstract

In the paper we consider the problem of valuation of American options written on dividend-paying assets whose price dynamics follow the classical multidimensional Black and Scholes model. We provide a general early exercise premium representation formula for options with payoff functions which are convex or satisfy mild regularity assumptions. Examples include index options, spread options, call on max options, put on min options, multiply strike options and power-product options. In the proof of the formula we exploit close connections between the optimal stopping problems associated with valuation of American options, obstacle problems and reflected backward stochastic differential equations.

Highlights

In the paper we study American options written on dividend-paying assets
We assume that the underlying assets dynamics follow the classical multidimensional Black and Scholes model
[20]; see [21] for nice exposition and additional references), in terms of variational inequalities (Jaillet et al [19]) and in terms of solutions of reflected BSDEs (El Karoui and Quenez [14]). These approaches provide complete characterization of the option value, the paper by Broadie and Detemple [7] shows that it is of interest to provide alternative representation, which expresses the value of an American option as the value of the corresponding European option plus the gain from early exercise

Summary

Introduction

In the paper we study American options written on dividend-paying assets. We assume that the underlying assets dynamics follow the classical multidimensional Black and Scholes model. [20]; see [21] for nice exposition and additional references), in terms of variational inequalities (Jaillet et al [19]) and in terms of solutions of reflected BSDEs (El Karoui and Quenez [14]) These approaches provide complete characterization of the option value In the important paper [7] and in Detemple et al [11] (see [10]) the early exercise premium formula was established for concrete classes of options on multiply assets. In the proof of the exercise premium formula we rely on some results on reflected BSDEs and their links with optimal stopping problems (see [14]) and with parabolic variational inequalities established in Bally et al [2]. The basic idea of the proof comes from our earlier paper [25] devoted to standard American call and put options on single asset

Preliminaries

Obstacle Problem for the Black and Scholes Equation

Convex Payoffs

The Early Exercise Premium Representation