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

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),

[1]

Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079

[2]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481

[3]

Tomasz R. Bielecki, Igor Cialenco, Marek Rutkowski. Arbitrage-free pricing of derivatives in nonlinear market models. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 2-. doi: 10.1186/s41546-018-0027-x

[4]

Linyi Qian, Wei Wang, Rongming Wang. Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 411-429. doi: 10.3934/jimo.2013.9.411

[5]

G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 485-501. doi: 10.3934/dcdss.2020027

[6]

Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019103

[7]

Kai Zhang, Xiaoqi Yang, Kok Lay Teo. A power penalty approach to american option pricing with jump diffusion processes. Journal of Industrial & Management Optimization, 2008, 4 (4) : 783-799. doi: 10.3934/jimo.2008.4.783

[8]

Jingzhen Liu, Ka-Fai Cedric Yiu, Tak Kuen Siu, Wai-Ki Ching. Optimal insurance in a changing economy. Mathematical Control & Related Fields, 2014, 4 (2) : 187-202. doi: 10.3934/mcrf.2014.4.187

[9]

Daniel Brinkman, Christian Ringhofer. A kinetic games framework for insurance plans. Kinetic & Related Models, 2017, 10 (1) : 93-116. doi: 10.3934/krm.2017004

[10]

Gusein Sh. Guseinov. Spectral method for deriving multivariate Poisson summation formulae. Communications on Pure & Applied Analysis, 2013, 12 (1) : 359-373. doi: 10.3934/cpaa.2013.12.359

[11]

Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435

[12]

Ping Huang, Ercai Chen, Chenwei Wang. Entropy formulae of conditional entropy in mean metrics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5129-5144. doi: 10.3934/dcds.2018226

[13]

Giséle Ruiz Goldstein, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. A generalized Cox-Ingersoll-Ross equation with growing initial conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1513-1528. doi: 10.3934/dcdss.2020085

[14]

Manuel V. C. Vieira. Derivatives of eigenvalues and Jordan frames. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 115-126. doi: 10.3934/naco.2016003

[15]

Adam Andersson, Felix Lindner. Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4271-4294. doi: 10.3934/dcdsb.2019081

[16]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[17]

Fabrizio Colombo, Graziano Gentili, Irene Sabadini and Daniele C. Struppa. A functional calculus in a noncommutative setting. Electronic Research Announcements, 2007, 14: 60-68. doi: 10.3934/era.2007.14.60

[18]

Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101

[19]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039

[20]

Jun-Jie Miao, Sara Munday. Derivatives of slippery Devil's staircases. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 353-365. doi: 10.3934/dcdss.2017017

 Impact Factor: 

Metrics

  • PDF downloads (2)
  • HTML views (3)
  • Cited by (0)

[Back to Top]