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.
Downloads
References
H. Albrecher, J. C. Araujo-Acuna and J. Beirlant: Fitting non-stationary Cox processes: an application to fire insurance data. North American Actuarial Journal, (2020). https://doi.org/10.1080/10920277.2019.1703752 DOI: https://doi.org/10.1080/10920277.2019.1703752
H. Albrecher, J. Beirlant and J. Teugels: Reinsurance: Actuarial and Statistical Aspects. Wiley (2017). https://doi.org/10.1002/9781119412540 DOI: https://doi.org/10.1002/9781119412540
S. Asmussen and H. Albrecher: Ruin probabilities. World Scientific (2010). https://doi.org/10.1142/7431 DOI: https://doi.org/10.1142/7431
R. Cont and P. Tankov: Financial Modelling with jump processes. Chapman-Hall (2004).
G. Di Nunno and J. Vives: A Malliavin-Skorohod calculus in L0 and L1 for additive and Volterra-type processes. Stochastics 89 (1): 142-170 (2017). https://doi.org/10.1080/17442508.2016.1140767 DOI: https://doi.org/10.1080/17442508.2016.1140767
G. Di Nunno, B. Oksendal and F. Proske: Malliavin calculus for L'evy processes with applications to Finance. Springer (2009). https://doi.org/10.1007/978-3-540-78572-9 DOI: https://doi.org/10.1007/978-3-540-78572-9
A. Dassios and J. W. Jang (2003): Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance and Stochastics 7: 73-95. https://doi.org/10.1007/s007800200079 DOI: https://doi.org/10.1007/s007800200079
P. Embrechts, C. Kl¨uppelberg and T. Mikosch: Modelling Extremal Events for Insurance and Finance. Springer (1997). https://doi.org/10.1007/978-3-642-33483-2 DOI: https://doi.org/10.1007/978-3-642-33483-2
H. Follmer and A. Schied: Stochastic Finance: An Introduction in Discrete Time. Walter De Gruyter (2011). https://doi.org/10.1515/9783110218053 DOI: https://doi.org/10.1515/9783110218053
C. Hillairet, Y. Jiao and A. R'eveillac: Pricing formulae for derivatives in insurance using the Malliavin calculus. Probability, uncertainty and Quantitative Risk 3:7 (2018). https://doi.org/10.1186/s41546-018-0028-9 DOI: https://doi.org/10.1186/s41546-018-0028-9
K. Ito: Spectral type of shift transformations of differential processes with stationary increments. Transactions of the American Mathematical Society 81: 252-263 (1956). https://doi.org/10.1090/S0002-9947-1956-0077017-0 DOI: https://doi.org/10.1090/S0002-9947-1956-0077017-0
T. Mikosch: Non-Life Insurance Mathematics, An Introduction with Stochastic Processes. Springer (2004).
H. Panjer and G. Willmot: Insurance Risk Models. Society of actuaries, USA (1992).
J. Picard: Formules de dualite sur l'espace de Poisson. Annales de l'IHP, section B, 32 (4) (1996).
K. I. Sato: Levy processes and infinitely divisible distributions. Cambridge (1999).
J. L. Sole, F. Utzet and J. Vives: Canonical L'evy processes and Malliavin calculus. Stochastic Processes and their Applications 117: 165-187 (2007). https://doi.org/10.1016/j.spa.2006.06.006 DOI: https://doi.org/10.1016/j.spa.2006.06.006
A. L. Yablonski: The Malliavin Calculus for Processes with Conditionally Independent Increments. In F. E. Benth, G. Di Nunno, T. Lindstrøm, B. Øksendal and T. Zhang (editors), Stochastic Analysis and Applications, Abel Symposyum 2005. Springer (2007). https://doi.org/10.1007/978-3-540-70847-6_30 DOI: https://doi.org/10.1007/978-3-540-70847-6_30
A. L. Yablonski: The Calculus of variation for Processes with Independent Increments. Rocky Mountain Journal of Mathematics 38 (2): 669-701 (2008). https://doi.org/10.1216/RMJ-2008-38-2-669 DOI: https://doi.org/10.1216/RMJ-2008-38-2-669
Copyright (c) 2022 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
