David Ruiz Baños

Computing Deltas without Derivatives

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 payoff function. This allows numerically tractable computation of the Greeks even for discontinuous payoff functions. However, while the payoff function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example already excludes simple regime-switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to Itô diffusions with irregular drift coefficients, where we focus here on the computation of the delta, which is the option price sensitivity with respect to the initial value of the underlying. To this end, we first show existence, Malliavin differentiability and (Sobolev) differentiability in the initial condition for strong solutions of Itô 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 derivatives 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 European and path-dependent options. We conclude with a small simulation study of several regime-switching examples.

The Bismut–Elworthy–Li formula for mean-field stochastic differential equations

We generalise the so-called Bismut-Elworthy-Li formula to a class of stochastic differential equations whose coefficients might depend on the law of the solution. We give some examples of where this formula can be applied to in the context of finance and the computation of Greeks and provide with a simple but rather illustrative simulation experiment showing that the use of the Bismut-Elworthy-Li formula, also known as Malliavin method, is more efficient compared to the finite difference method.

Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation

Abstract In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the regularity of densities of such solutions. We also apply these results to construct a classical solution to the stochastic transport equation when the drift is Lipschitz.