Pricing cumulative loss derivatives under additive models via Malliavin calculus

Abstract

We show that the integration by parts formula based on Malliavin-Skorohod calculus techniques for additive processes helps us to compute quantities like $\mathbb{E}\left(L_{T} h\left(L_{T}\right)\right)$, or more generally $\mathbb{E}\left(H\left(L_{T}\right)\right)$, for different suitable functions $h$ or $H$ and different models for the cumulative loss process $L .$ These quantities are important in Insurance and Finance. For example they appear in computing expected shortfall risk measures or prices of stop-loss contracts. The formulas given in the present paper generalize the formulas given in a recent paper by Hillairet, Jiao and Réveillac (HJR). In the HJR paper, despite the use of advanced models, including the Cox process, the treatment of the formulas is based only on Malliavin calculus techniques for the standard Poisson process, a particular case of additive process. In the present paper, Malliavin calculus techniques for additive processes are used, more general results are obtained and proofs appears to be shorter.