Stochastic Functional Differential Equations and Sensitivity to Their Initial Path

Abstract

We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks connected to the choice of these models. In particular we focus on the impact of the initial condition on the evaluations. This problem is known as the analysis of sensitivity to the initial condition and, in the terminology of finance, it is referred to as the Delta. In this work the initial condition is represented by the relevant past history of the stochastic functional differential equation. This naturally leads to the redesign of the definition of Delta. We suggest to define it as a functional directional derivative, this is a natural choice. For this we study a representation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our formula is achieved by studying an appropriate relationship between Malliavin derivative and functional directional derivative. For this we introduce the technique of randomisation of the initial condition.

Sensitivity analysis in the infinite dimensional Heston model

“…We remark that there is a structural difference in the application perspectives between our paper and [1]. While we face an infinite dimensional problem all the way through, the authors in [1] consider a finite dimensional noise and path-dependent coefficients.…”

“…Thus we have to reinterpret the meaning of the Greeks. For the delta, a natural choice is to take inspiration from [1] and interpret it as a directional derivative. For the vega however, there is no natural generalization to our framework.…”

“…For this last one, we have to introduce the adequate concepts in the infinite dimensional setting. We resolve this by exploiting some ideas in the approach of [1], using the interplay of functional derivatives and the Malliavin calculus via a form of randomization. We remark that there is a structural difference in the application perspectives between our paper and [1].…”

“…We resolve this by exploiting some ideas in the approach of [1], using the interplay of functional derivatives and the Malliavin calculus via a form of randomization. We remark that there is a structural difference in the application perspectives between our paper and [1]. While we face an infinite dimensional problem all the way through, the authors in [1] consider a finite dimensional noise and path-dependent coefficients.…”