Functional Itô calculus, path-dependence and the computation of Greeks

Resumo

Dupire's functional Ito calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional It (o) over cap calculus framework. Namely, we consider the Lie bracket of the space and time functional derivatives, which we use to classify functionals accordingly to their degree of path-dependence. We then revisit the problem of efficient numerical computation of Greeks for path-dependent derivatives using integration by parts techniques. Special attention is paid to path-dependent functionals with zero Lie bracket, called locally weakly path dependent functionals in our classification. Hence, we derive the weighted-expectation formulas for their Greeks. In the more general case of fully path-dependent functionals, we show that, equipped with the functional Ito calculus, we are able to analyze the effect of the Lie bracket on the computation of Greeks. Moreover, we are also able to consider the more general dynamics of path-dependent volatility. These were not achieved using Malliavin calculus. (C) 2017 Elsevier B.V. All rights reserved.

Ficha do documento

Tipo

Artigo científico

Ano

2017

Instituição

Elsevier Science Bv

Fonte

Repositório da FGV

Idioma

Inglês

Acesso

Acesso restrito

DOI

10.1016/j.spa.2017.03.015

Identificador

oai:repositorio.fgv.br:10438/23826

Palavras-chave

Functional Itô calculusPath-dependenceMonte Carlo methodsMatemáticaGreeksMonte Carlo, Método de