Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T14:08:26.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Extended reduced-form framework for non-life insurance

Part of: Mathematical finance Stochastic analysis

Published online by Cambridge University Press:  14 June 2022

Francesca Biagini Open the ORCID record for Francesca Biagini [Opens in a new window]  and
Yinglin Zhang
Francesca Biagini*
Affiliation:
LMU Munich
Yinglin Zhang*
Affiliation:
LMU Munich
*
*Postal address: Department of Mathematics, LMU Munich, Theresienstraße, 39, 80333 Munich, Germany.
*Postal address: Department of Mathematics, LMU Munich, Theresienstraße, 39, 80333 Munich, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we propose a general framework for modeling an insurance liability cash flow in continuous time, by generalizing the reduced-form framework for credit risk and life insurance. In particular, we assume a nontrivial dependence structure between the reference filtration and the insurance internal filtration. We apply these results for pricing and hedging non-life insurance liabilities in hybrid financial and insurance markets, while taking into account the role of inflation under the benchmarked risk-minimization approach. This framework offers at the same time a general and flexible structure, and an explicit and treatable pricing-hedging formula.

Keywords

Non-life insurancereduced-form frameworkmarked point processbenchmark approachfiltration dependencemarket-consistent valuationrisk mitigation

MSC classification

Primary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 945 - 973
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arjas, E. (1989). The claims reserving problem in non-life insurance: some structural ideas. ASTIN Bull. 19, 139–152.10.2143/AST.19.2.2014905CrossRefGoogle Scholar
Barbarin, J. (2007). Risk-minimizing strategies for life insurance contracts with surrender option. Available at http://ssrn.com/abstract=1334580 or http://dx.doi.org/10.2139/ssrn.1334580.CrossRefGoogle Scholar
Barbarin, J., De Launois, T. and Devolder, P. (2009). Risk minimization with inflation and interest rate risk: applications to non-life insurance. Scand. Actuarial J. 2009, 119–151.CrossRefGoogle Scholar
Biagini, F. (2013). Evaluating hybrid products: the interplay between financial and insurance markets. In Stochastic Analysis, Random Fields and Applications VII, eds R. Dalang, M. Dozzi and F. Russo, Birkhäuser, Basel, pp. 285–304.10.1007/978-3-0348-0545-2_15CrossRefGoogle Scholar
Biagini, F., Cretarola, A. and Platen, E. (2014). Local risk-minimization under the benchmark approach. Math. Financial Econom. 8, 109–134.CrossRefGoogle Scholar
Biagini, F. and Pratelli, M. (1999). Local risk minimization and numéraire. J. Appl. Prob. 36, 1126–1139.CrossRefGoogle Scholar
Biagini, F., Rheinländer, T. and Schreiber, I. (2016). Risk-minimization for life insurance liabilities with basis risk. Math. Financial Econom. 10, 151–178.CrossRefGoogle Scholar
Biagini, F., Rheinländer, T. and Widenmann, J. (2013). Hedging mortality claims with longevity bonds. ASTIN Bull. 43, 123–157.10.1017/asb.2013.12CrossRefGoogle Scholar
Biagini, F. and Schreiber, I. (2013). Risk-minimization for life insurance liabilities. SIAM J. Financial Math. 4, 243–264.CrossRefGoogle Scholar
Biagini, F. and Zhang, Y. (2016). Polynomial diffusion models for life insurance liabilities. Insurance Math. Econom. 71, 114–129.CrossRefGoogle Scholar
Bielecki, T. R. and Rutkowski, M. (2004). Credit Risk: Modelling, Valuation and Hedging, 2nd edn. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Briys, É. and De Varenne, F. (2001). Insurance: from Underwriting to Derivatives: Asset Liability Management in Insurance Companies. John Wiley, New York.Google Scholar
Cairns, A. J. G., Blake, D. and Dowd, K. (2006). Pricing death: frameworks for valuation and securitization of mortality risk. ASTIN Bull. 36, 79–120.CrossRefGoogle Scholar
Dahl, M. and Møller, T. (2006). Valuation and hedging of life insurance liabilities with systematic mortality risk. Insurance Math. Econom. 39, 193–217.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Delong, L. (2005). Optimal investment strategy for a non-life insurance company: quadratic loss. Applicationes Math. 32, 263–277.CrossRefGoogle Scholar
Dettweiler, E. (2005). On the construction of point processes. Preprint, Technische Universität Dresden.Google Scholar
Happ, S., Merz, M. and Wüthrich, M. V. (2015). Best-estimate claims reserves in incomplete markets. Europ. Actuarial J. 5, 55–77.CrossRefGoogle Scholar
Hulley, H. and Schweizer, M. (2010). $M^6$ —On minimal market models and minimal martingale measures. In Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, eds C. Chiarella and A. Novikov, Springer, Berlin, Heidelberg, pp. 35–51.CrossRefGoogle Scholar
Jacobsen, M. (2006). Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston.Google Scholar
Jacod, J. (1975). Multivariate point processes: predictable projection, Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
Jewell, W. S. (1989). Predicting IBNYR events and delays: I. Continuous time. ASTIN Bull. 19, 25–55.CrossRefGoogle Scholar
Kalashnikov, V. and Norberg, R. (2003). On the sensitivity of premiums and reserves to changes in valuation elements. Scand. Actuarial J. 2003, 238–256.CrossRefGoogle Scholar
Larsen, C. R. (2007). An individual claims reserving model. ASTIN Bull. 37, 113–132.CrossRefGoogle Scholar
Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line: the Dynamic Approach. Springer, New York.Google Scholar
Mikosch, T. (2009). Non-Life Insurance Mathematics: an Introduction with the Poisson Process. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Møller, T. (1998). Risk-minimizing hedging strategies for insurance payment processes. Finance Stoch. 5, 419–446.Google Scholar
Møller, T. (1998). Risk-minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bull. 28, 17–47.CrossRefGoogle Scholar
Norberg, R. (1993). Prediction of outstanding liabilities in non-life insurance. ASTIN Bull. 23, 95–115.CrossRefGoogle Scholar
Norberg, R. (1999). Prediction of outstanding liabilities II: model variations and extensions. ASTIN Bull. 29, 5–25.CrossRefGoogle Scholar
Norberg, R. (2013). Quadratic hedging: an actuarial view extended to solvency control. Europ. Actuarial J. 3, 45–68.CrossRefGoogle Scholar
Norberg, R. and Savina, O. (2012). A quadratic hedging approach to comparison of catastrophe indices. Internat. J. Theoret. Appl. Finance 15, 1250030.CrossRefGoogle Scholar
Parodi, P. (2014). Pricing in General Insurance. CRC Press, Boca Raton.CrossRefGoogle Scholar
Platen, E. and Heath, D. (2006). A Benchmark Approach to Quantitative Finance. Springer, Berlin.CrossRefGoogle Scholar
Schmidt, T. (2014). Catastrophe insurance modelled with shot-noise processes. Risks 2, 3–24.CrossRefGoogle Scholar
Wüthrich, M. V. (2016). Market-Consistent Actuarial Valuation, 3rd edn. Springer, Cham.CrossRefGoogle Scholar
Wuthrich, M. V. and Merz, M. (2008). Stochastic Claims Reserving Methods in Insurance. John Wiley, Chichester.Google Scholar
Zhang, Y. (2018). Insurance modeling in continuous time. Doctoral Thesis, LMU Munich.Google Scholar