# American Institute of Mathematical Sciences

January  2018, 3: 7 doi: 10.1186/s41546-018-0028-9

## Pricing formulae for derivatives in insurance using Malliavin calculus

 1. ENSAE Universite Paris Saclay, CREST, 5 avenue Henry Le Chatelier, 91120 Palaiseau, France; 2. Université Claude Bernard-Lyon 1, Institut de Science Financière et d'Assurances, 69007 Lyon France; 3. INSA de Toulouse, IMT UMR CNRS 5219, Université de Toulouse, 135 avenue de Rangueil 31077 Toulouse Cedex 4 France

Received  July 14, 2017 Revised  April 06, 2018 Published  June 2018

Fund Project: The authors acknowledge anonymous referees and the Associate Editor for comments and suggestions that have allowed us to improve the paper. The authors acknowledge Projet PEPS égalité (part of the European project INTEGER-WP4) "Approximation de Stein:approche par calcul de Malliavin et applications à la gestion des risques financiers" for financial support.

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 Malliavin calculus. Similar to the celebrated Black-Scholes formula, we aim to express 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.
Citation: Caroline Hillairet, Ying Jiao, Anthony Réveillac. Pricing formulae for derivatives in insurance using Malliavin calculus. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 7-. doi: 10.1186/s41546-018-0028-9
##### References:
 [1] Albers, W:Stop-loss premiums under dependence. Insur. Math. Econ. 24(3), 173-185 (1999) [2] Albrecher, H, Boxma, O:A ruin model with dependence between claim sizes and claim intervals. Insur. Math. Econ. 35(2), 245-254 (2004) [3] Albrecher, H, Constantinescu, C, Loisel, S:Explicit ruin formulas for models with dependence among risks. Insur. Math. Econ. 48(2), 265-270 (2011) [4] Bakshi, G, Madan, D, Zhang, F:Understanding the role of recovery in default risk models:empirical comparisons and implied recovery rates (2006). FDIC Center for Financial Research Working Paper No. 2006-06. https://doi.org/10.2139/ssrn.285940 [5] Borodin A, Salminen P:Handbook of Brownian motion-facts and formulae, 2nd edn. Probability and its Applications. Birkhäuser Verlag, Basel (2002) [6] Boudreault, M, Cossette, H, Landriault, D, Marceau, E:On a risk model with dependence between interclaim arrivals and claim sizes. Scandinavian Actuarial Journal. 2006(5), 265-285 (2006) [7] de Lourdes Centeno, M:de Lourdes Centeno. Dependent risks and excess of loss reinsurance. Insur. Math. Econ. 37(2), 229-238 (2005) [8] Denuit, M, Dhaene, J, Ribas, C:Does positive dependence between individual risks increase stop-loss premiums? Insur. Math. Econ. 28(3), 305-308 (2001) [9] Hans, F, Schied, A:Stochastic finance. Walter de Gruyter & Co. An introduction in discrete time, Berlin, extended edition (2011) [10] Gerber, HU:On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums. Insur. Math. Econ. 1(1), 13-18 (1982) [11] Panjer, HH:Recursive evaluation of a family of compound distributions. ASTIN Bull. J. IAA. 12(1), 22-26 (1981) [12] Picard, J:Formules de dualité sur léspace de Poisson. Ann. Inst. H. Poincaré Probab. Statist. 32(4), 509-548 (1996a) [13] Picard, J:On the existence of smooth densities for jump processes. Probab. Theory Related Fields. 105(4), 481-511 (1996b) [14] Privault, N:Stochastic analysis in discrete and continuous settings with normal martingales, volume 1982 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2009)

show all references

##### References:
 [1] Albers, W:Stop-loss premiums under dependence. Insur. Math. Econ. 24(3), 173-185 (1999) [2] Albrecher, H, Boxma, O:A ruin model with dependence between claim sizes and claim intervals. Insur. Math. Econ. 35(2), 245-254 (2004) [3] Albrecher, H, Constantinescu, C, Loisel, S:Explicit ruin formulas for models with dependence among risks. Insur. Math. Econ. 48(2), 265-270 (2011) [4] Bakshi, G, Madan, D, Zhang, F:Understanding the role of recovery in default risk models:empirical comparisons and implied recovery rates (2006). FDIC Center for Financial Research Working Paper No. 2006-06. https://doi.org/10.2139/ssrn.285940 [5] Borodin A, Salminen P:Handbook of Brownian motion-facts and formulae, 2nd edn. Probability and its Applications. Birkhäuser Verlag, Basel (2002) [6] Boudreault, M, Cossette, H, Landriault, D, Marceau, E:On a risk model with dependence between interclaim arrivals and claim sizes. Scandinavian Actuarial Journal. 2006(5), 265-285 (2006) [7] de Lourdes Centeno, M:de Lourdes Centeno. Dependent risks and excess of loss reinsurance. Insur. Math. Econ. 37(2), 229-238 (2005) [8] Denuit, M, Dhaene, J, Ribas, C:Does positive dependence between individual risks increase stop-loss premiums? Insur. Math. Econ. 28(3), 305-308 (2001) [9] Hans, F, Schied, A:Stochastic finance. Walter de Gruyter & Co. An introduction in discrete time, Berlin, extended edition (2011) [10] Gerber, HU:On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums. Insur. Math. Econ. 1(1), 13-18 (1982) [11] Panjer, HH:Recursive evaluation of a family of compound distributions. ASTIN Bull. J. IAA. 12(1), 22-26 (1981) [12] Picard, J:Formules de dualité sur léspace de Poisson. Ann. Inst. H. Poincaré Probab. Statist. 32(4), 509-548 (1996a) [13] Picard, J:On the existence of smooth densities for jump processes. Probab. Theory Related Fields. 105(4), 481-511 (1996b) [14] Privault, N:Stochastic analysis in discrete and continuous settings with normal martingales, volume 1982 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2009)
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