Pricing formulae for derivatives in insurance using the Malliavin calculus

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.