TY - JOUR

T1 - A Clark-Ocone type formula via Itô calculus and its application to finance

AU - Arai, Takuji

AU - Suzuki, Ryoichi

N1 - Publisher Copyright:
Copyright © 2019, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2019/6/16

Y1 - 2019/6/16

N2 - An explicit martingale representation for random variables described as a functional of a Lévy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Itô’s formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Lévy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Lévy processes.

AB - An explicit martingale representation for random variables described as a functional of a Lévy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Itô’s formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Lévy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Lévy processes.

KW - Digital options

KW - Local risk-minimization

KW - Lévy processes

KW - Malliavin calculus

KW - Martingale representation theorem

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M3 - Article

AN - SCOPUS:85093152186

JO - Mathematical Social Sciences

JF - Mathematical Social Sciences

SN - 0165-4896

ER -