Pricing cumulative loss derivatives under additive models via Malliavin calculus

Abstract

We show that integration by parts formulas based on Malliavin-Skorohod calculus techniques for additive processes help us to compute quantities like ${\E}(L_T h(L_T))$ for different suitable functions $h$ and different models for the cumulative loss process $L_T$. These quantities are important in Insurance and Finance. For example they appear in computing expected shortfall risk measures or stop-loss contracts. The formulas given in the present paper, obtained by simple proofs, generalize the formulas given in a recent paper by Hillairet, Jiao and Réveillac using Malliavin calculus techniques for the standard Poisson process, a particular case of additive process.