Pricing formulae for derivatives in insurance using the Malliavin calculus *

Caroline Hillairet (1) , Ying Jiao (2) , Anthony Réveillac (3, 4)

1 ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique
2 SAF - Laboratoire de Sciences Actuarielle et Financière
3 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse
4 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Domaines: Finance quantitative [q-fin.CP]; Probabilités [math.PR]; Economies et finances

Caroline Hillairet, Ying Jiao, Anthony Réveillac. Pricing formulae for derivatives in insurance using the Malliavin calculus *. Probability, Uncertainty and Quantitative Risk, 2018, 3 (7), pp.1-19. ⟨hal-01561987⟩