# Computing Deltas Without Derivatives

## Abstract

A well-known application of Malliavin calculus in Mathematical Finance is the probabilistic representation of option price sensitivities, the so-called Greeks, as expectation functionals that do not involve the derivative of the pay-off function. This allows for numerically tractable computation of the Greeks even for discontinuous pay-off functions. However, while the pay-off function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example excludes already simple regime switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to It^{o} diffusions with irregular drift coefficients, whereat we here focus on the computation of the Delta, which is the option price sensitivity with respect to the initial value of the underlying. To this purpose we first show existence, Malliavin differentiability, and
(Sobolev) differentiability in the initial condition of strong solutions of It^{o} diffusions with drift coefficients that can be decomposed into the sum of a bounded but merely measurable and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivative in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding Deltas for lookback and Asian type options. We conclude with a simulation
study of several regime-switching examples.

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## Bibtex

@ARTICLE{BanosEtAl2015b, author = {Ba{\~n}os, David Ruiz and Duedahl, Sindre and Meyer-Brandis, Thilo and Proske, Frank Norbert}, title = {Computing Deltas Without Derivatives}, journal = {Preprint}, year = {2015}, note = {Submitted}, abstract = {A well-known application of Malliavin calculus in Mathematical Finance is the probabilistic representation of option price sensitivities, the so-called Greeks, as expectation functionals that do not involve the derivative of the pay-off function. This allows for numerically tractable computation of the Greeks even for discontinuous pay-off functions. However, while the pay-off function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example excludes already simple regime switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to It^o diffusions with irregular drift coefficients, whereat we here focus on the computation of the Delta, which is the option price sensitivity with respect to the initial value of the underlying. To this purpose we first show existence, Malliavin differentiability, and (Sobolev) differentiability in the initial condition of strong solutions of It^o diffusions with drift coefficients that can be decomposed into the sum of a bounded but merely measurable and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivative in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding Deltas for lookback and Asian type options. We conclude with a simulation study of several regime-switching examples.}, url = {http://www.researchgate.net/publication/277813795_COMPUTING_DELTAS_WITHOUT_DERIVATIVES} }