Polynomial Diffusion Models for Life Insurance Liabilities

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.

[1]  Freddy Delbaen,et al.  An Interest Rate Model with Upper and Lower Bounds , 2002 .

[2]  Martin Larsson,et al.  Polynomial preserving diffusions on compact quadric sets , 2015 .

[3]  Eckhard Platen,et al.  A BENCHMARK APPROACH TO FINANCE , 2006 .

[4]  E. Platen.  A Benchmark Framework for Risk Management , 2003 .

[5]  F. Biagini.  Evaluating Hybrid Products: The Interplay Between Financial and Insurance Markets , 2013 .

[6]  Local risk minimization and numéraire , 1999 .

[7]  Thorsten Rheinländer,et al.  Risk-minimization for life insurance liabilities with basis risk , 2016 .

[8]  Eckhard Platen,et al.  Local risk-minimization under the benchmark approach , 2012, 1210.2337.

[9]  M. Dahl,et al.  Valuation and hedging of life insurance liabilities with systematic mortality risk , 2006 .

[10]  S. F. Jarner,et al.  PI-0902 Modelling Adult Mortality in Small Populations : The Saint Model , 2008 .

[11]  Mortality plateaus and directionality theory , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[12]  Christophe Stricker,et al.  Décomposition de Kunita-Watanabe , 1993 .

[13]  Alexander Szimayer,et al.  The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts , 2010 .

[14]  M. Schweizer,et al.  M-6-On Minimal Market Models and Minimal Martingale Measures , 2010 .

[15]  D. Heath,et al.  A Benchmark Approach to Quantitative Finance , 2006 .

[16]  P. Protter.  Stochastic integration and differential equations , 1990 .

[17]  C. Dellacherie,et al.  Probabilities and Potential B: Theory of Martingales , 2012 .

[18]  On Short Rate Processes and Their Implications for Term Structure Movements , 1994 .

[19]  D. Sondermann.  Hedging of non-redundant contingent claims , 1985 .

[20]  Jesper Lund,et al.  Non-Linear Kalman Filtering Techniques for Term-Structure Models , 1997 .

[21]  T. Bielecki,et al.  Credit Risk: Modeling, Valuation And Hedging , 2004 .

[22]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[23]  E. Platen.  Diversified Portfolios with Jumps in a Benchmark Framework , 2004 .

[24]  Francesca Biagini,et al.  Risk-Minimization for Life Insurance Liabilities , 2013, SIAM J. Financial Math..

[25]  Martin Larsson,et al.  Linear-Rational Term Structure Models , 2016 .

[26]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[27]  Huan Gao,et al.  A comonotonicity-based valuation method for guaranteed annuity options , 2013, J. Comput. Appl. Math..

[28]  M. Schweizer.  A guided tour through quadratic hedging approaches , 1999 .

[29]  Dirk Becherer.  The numeraire portfolio for unbounded semimartingales , 2001, Finance Stochastics.

[30]  Leonidas S. Rompolis,et al.  Forecasting the mean and volatility of stock returns from option prices , 2006 .

[31]  J. Barbarin.  Risk Minimizing Strategies for Life Insurance Contracts with Surrender Option , 2007 .

[32]  J. Duan,et al.  Série Scientifique Scientific Series Estimating and Testing Exponential-affine Term Structure Models by Kalman Filter Estimating and Testing Exponential-affine Term Structure Models by Kalman Filter , 2022 .

[33]  Analytic bond pricing for short rate dynamics evolving on matrix Lie groups , 2016 .

[34]  R. Durrett.  Stochastic Calculus: A Practical Introduction , 1996 .

[35]  G. Deelstra,et al.  The Role of the Dependence between Mortality and Interest Rates When Pricing Guaranteed Annuity Options , 2016 .

[36]  J. Jacod.  Calcul stochastique et problèmes de martingales , 1979 .

[37]  Walter Schachermayer,et al.  Stochastic Methods in Finance , 2004 .

[38]  Thomas Møller.  Risk-minimizing hedging strategies for insurance payment processes , 2001, Finance Stochastics.

[39]  M. Dacorogna,et al.  Exploring the Dependence between Mortality and Market Risks , 2015 .

[40]  Farshid Jamshidian,et al.  Valuation of credit default swaps and swaptions , 2004, Finance Stochastics.

[41]  E. Platen,et al.  Three-Benchmarked Risk Minimization for Jump Diffusion Markets , 2011 .

[42]  Thomas Møller,et al.  Risk-Minimizing Hedging Strategies for Unit-Linked Life Insurance Contracts , 1998, ASTIN Bulletin.

[43]  David Blake,et al.  Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk* , 2006, ASTIN Bulletin.

[44]  Martin Larsson,et al.  Polynomial diffusions and applications in finance , 2014, Finance Stochastics.

[45]  Martin Larsson,et al.  Polynomial diffusions on compact quadric sets , 2015, 1511.03554.

[46]  David Blake,et al.  Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities , 2006 .

[47]  Analytical Approximations for Short Rate Models , 2010 .