Pricing formulae for derivatives in insurance using the Malliavin calculus

TL;DR

This paper develops a valuation formula for insurance and financial derivatives based on a general loss process, utilizing Malliavin calculus to express expected cash flows in terms of fundamental building blocks.

Contribution

It introduces a novel Malliavin calculus-based approach to derive pricing formulas for contracts dependent on complex loss processes, extending classical models like Black-Scholes.

Findings

Provides explicit valuation formulas for various insurance contracts.

Demonstrates the use of Malliavin calculus in handling doubly stochastic Poisson processes.

Connects loss distribution functions with risk measures like Value at Risk and Expected Shortfall.

Abstract

In this paper we provide a valuation formula for different classes of actuarial and financial contracts which depend on a general loss process, by using the Malliavin calculus. In analogy with the celebrated Black-Scholes formula, we aim at expressing the expected cash flow in terms of a building block. The former is related to the loss process which is a cumulated sum indexed by a doubly stochastic Poisson process of claims allowed to be dependent on the intensity and the jump times of the counting process. For example, in the context of Stop-Loss contracts the building block is given by the distribution function of the terminal cumulated loss, taken at the Value at Risk when computing the Expected Shortfall risk measure.