Hillairet, Caroline; Jiao, Ying; Réveillac, Anthony Pricing formulae for derivatives in insurance using Malliavin calculus. (English) Zbl 1435.62373 Probab. Uncertain. Quant. Risk 3, Paper No. 7, 19 p. (2018). Summary: 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. Cited in 3 Documents MSC: 62P05 Applications of statistics to actuarial sciences and financial mathematics 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 91G30 Interest rates, asset pricing, etc. (stochastic models) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) Keywords:Cox processes; pricing formulae; insurance derivatives; Malliavin calculus PDFBibTeX XMLCite \textit{C. Hillairet} et al., Probab. Uncertain. Quant. Risk 3, Paper No. 7, 19 p. (2018; Zbl 1435.62373) Full Text: DOI arXiv References: [1] Albers, W., Stop-loss premiums under dependence, Insur. Math. Econ, 24, 3, 173-185 (1999) · Zbl 0945.62108 [2] Albrecher, H.; Boxma, O., A ruin model with dependence between claim sizes and claim intervals, Insur. Math. Econ, 35, 2, 245-254 (2004) · Zbl 1079.91048 [3] Albrecher, H.; Constantinescu, C.; Loisel, S., Explicit ruin formulas for models with dependence among risks, Insur. Math. Econ, 48, 2, 265-270 (2011) · Zbl 1218.91065 [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] Handbook of Brownian motion—facts and formulae (2002), Basel: Birkhäuser Verlag, Basel · Zbl 1012.60003 [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) · Zbl 1145.91030 [7] De Lourdes Centeno, M., de Lourdes Centeno. Dependent risks and excess of loss reinsurance, Insur. Math. Econ, 37, 2, 229-238 (2005) · Zbl 1125.91062 [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) · Zbl 1055.91046 [9] Hans, F.; Schied, A., Stochastic finance. Walter de Gruyter & Co. (2011), Berlin, extended edition: An introduction in discrete time, Berlin, extended edition [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) · Zbl 0479.62076 [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 (1996) · Zbl 0859.60045 [13] Picard, J., On the existence of smooth densities for jump processes, Probab. Theory Related Fields, 105, 4, 481-511 (1996) · Zbl 0853.60064 [14] Privault, N., Stochastic analysis in discrete and continuous settings with normal martingales, volume 1982 of Lecture Notes in Mathematics (2009), Berlin: Springer-Verlag, Berlin · Zbl 1185.60005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.