×

Polynomial diffusion models for life insurance liabilities. (English) Zbl 1371.91081

Summary: In this paper we study the pricing and hedging problem of a portfolio of life insurance products under the benchmark approach, where the reference market is modelled as driven by a state variable following a polynomial diffusion on a compact state space. Such a model can be used to guarantee not only the positivity of the OIS short rate and the mortality intensity, but also the possibility of approximating both pricing formula and hedging strategy of a large class of life insurance products by explicit formulas.

MSC:

91B30 Risk theory, insurance (MSC2010)
91G20 Derivative securities (option pricing, hedging, etc.)
60G44 Martingales with continuous parameter
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ansel, J. P.; Stricker, C., Décomposition de Kunita-Watanabe, Sém. Probab. Strasbg., 27, 30-32 (1993) · Zbl 0788.60057
[3] Back, K.; Bielecki, T. R.; Hipp, C.; Peng, S.; Schachermayer, W., (Stochastic Methods in Finance. Lectures Given at the C.I.M.E.-E.M.S. Summer School Held in Bressanone/Brixen, Italy, July 6-12, 2003 (2004), Springer)
[5] Biagini, F., Evaluating hybrid products: the interplay between financial and insurance markets, (Dalang, R.; Dozzi, M.; Russo, F., Stochastic Analysis, Random Fields and Applications VII. Stochastic Analysis, Random Fields and Applications VII, Progress in Probability, vol. 67 (2013), Birkhäuser Verlag) · Zbl 1281.91098
[6] Biagini, F.; Cretarola, A.; Platen, E., Local risk-minimization under the benchmark approach, Math. Financ. Econ., 8, 2, 109-134 (2014) · Zbl 1308.91157
[7] Biagini, F.; Pratelli, M., Local risk minimization and numéraire, J. Appl. Probab., 36, 4, 1126-1139 (1999) · Zbl 0993.91019
[8] Biagini, F.; Rheinländer, T.; Schreiber, I., Risk-minimization for life insurance liabilities with basis risk, Math. Financ. Econ., 10, 2, 151-178 (2016) · Zbl 1404.91136
[9] Biagini, F.; Schreiber, I., Risk-minimization for life insurance liabilities, SIAM J. Financial Math., 4, 243-264 (2013) · Zbl 1278.62165
[10] Bielecki, T. R.; Rutkowski, M., (Credit Risk: Modelling, Valuation and Hedging. Credit Risk: Modelling, Valuation and Hedging, Springer-Finance (2004), Springer) · Zbl 0979.91050
[11] Blake, D.; Cairns, A. J.G.; Dowd, K., Living with mortality: Longevity bonds and other mortality-linked securities, British Actuar. J,, 12, 153-228 (2008)
[12] Cairns, A. J.G.; Blake, D.; Dowd, K., Pricing death: frameworks for valuation and securitization of mortality risk, ASTIN Bull., 36, 1, 79-120 (2006) · Zbl 1162.91403
[14] Dahl, M.; Møller, T., Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance Math. Econom., 39, 2, 193-217 (2006) · Zbl 1201.91089
[16] Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. Ann., 300, 1, 463-520 (1994) · Zbl 0865.90014
[17] Delbaen, F.; Shirakawa, H., An interest rate model with upper and lower bounds, Asia-Pac. Financ. Markets, 9, 3, 191-209 (2002) · Zbl 1071.91020
[18] Dellacherie, C.; Meyer, P. A., Probabilities and Potential B: Theory of Martingales (1982), North Holland: North Holland Amsterdam
[19] Demetrius, L., Mortality plateaus and directionality theory, Proc. Roy. Soc. B: Biol. Sci., 268, 1480, 2029-2037 (2001)
[20] Duan, J. C.; Simonato, J. G., Estimating and testing exponential-affine term structure models by Kalman filter, Rev. Quant. Finance Account., 13, 2, 111-135 (1999)
[22] Durrett, R., (Stochastic Calculus: A Practical Introduction. Stochastic Calculus: A Practical Introduction, Probability and Stochastics Series (1996), CRC Press: CRC Press Boca Raton, FL) · Zbl 0856.60002
[23] Filipović, D.; Larsson, M., Polynomial diffusions and applications in finance, Finance Stoch. (2016), (forthcoming) · Zbl 1386.60237
[24] Filipović, D.; Larsson, M.; Trolle, A. B., Linear-rational term structure models, J. Finance, 14-15 (2016), (forthcoming): Swiss Finance Institute Research Paper
[25] Flesaker, B.; Hughston, L. P., Positive interest, Risk, 9, 1, 46-49 (1996)
[26] Föllmer, H.; Sondermann, D., Hedging of non-redundant contingent claims, (Hildenbrand, W.; Mas-Colell, A., Contributions to Mathematical Economics (1986), North Holland), 205-223 · Zbl 0663.90006
[27] Harvey, A. C., Forecasting, Structural Models and the Kalman Filter (1989), Cambridge University Press: Cambridge University Press New York
[28] Hulley, H.; Schweizer, M., \(M^6\)-on minimal market models and minimal martingale measures, (Chiarella, C.; Novikov, A., Contemporary Quantitative Finance. Essays in Honour of Eckhard Platen (2010), Springer), 35-51 · Zbl 1229.91376
[29] Jacod, J., Calcul Stochastique et Problèmes de Martingales (1979), Springer-Verlag · Zbl 0414.60053
[30] Jamshidian, F., Valuation of credit default swaps and swaptions, Finance Stoch., 8, 3, 343-371 (2004) · Zbl 1063.91034
[31] Jarner, S. F.; Kryger, E. M., Modelling adult mortality in small populations: the Saint model, ASTIN Bull., 41, 02, 377-418 (2011) · Zbl 1239.91128
[33] Lima, N.; Privault, N., Analytic bond pricing for short rate dynamics evolving on matrix Lie groups, Quant. Finance, 16, 1 (2016) · Zbl 1468.91182
[34] Li, J.; Szimayer, A., The uncertain mortality intensity framework: pricing and hedging unit-linked life insurance contracts, Insurance Math. Econom., 49, 471-486 (2011) · Zbl 1228.91041
[35] Liu, X.; Mamon, R.; Gao, H., A comonotonicity-based valuation method for guaranteed annuity options, J. Comput. Appl. Math., 250, 58-69 (2013) · Zbl 1285.91130
[37] Møller, T., Risk-minimizing hedging strategies for insurance payment processes, Finance Stoch., 5, 419-446 (1998) · Zbl 0983.62076
[38] Møller, T., Risk-minimizing hedging strategies for unit-linked life insurance contracts, ASTIN Bull., 28, 1, 17-47 (1998) · Zbl 1168.91417
[39] Platen, E., A benchmark framework for risk management, (Stochastic Processes and Applications to Mathematical Finance (2004), World Scientific), 305-335 · Zbl 1191.91048
[40] Platen, E., Diversified portfolios with jumps in a benchmark framework, Asia-Pac. Financ. Markets, 11, 1, 1-22 (2004) · Zbl 1075.91022
[41] Platen, E., A benchmark approach to finance, Math. Finance, 16, 1, 131-151 (2006) · Zbl 1128.91029
[42] Platen, E.; Heath, D., A Benchmark Approach to Quantitative Finance (2006), Springer: Springer Berlin · Zbl 1104.91041
[43] Rogers, L. C.G.; Williams, D., Diffusions, Markov Processes and Martingales (1994), Cambridge University Press · Zbl 0826.60002
[45] Schweizer, M., A guided tour through quadratic hedging approaches, (Option Pricing, Interest Rates and Risk Management (2001), Cambridge University Press: Cambridge University Press Cambridge, UK), 538-574 · Zbl 0992.91036
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.