VALUING EQUITY-LINKED DEATH BENEFITS IN A REGIME-SWITCHING FRAMEWORK

Abstract In this article, we consider the problem of computing the expected discounted value of a death benefit, e.g. in Gerber et al. (2012, 2013), in a regime-switching economy. Contrary to their proposed discounted density approach, we adopt the Laplace transform to value the contingent options. By this alternative approach, closed-form expressions for the Laplace transforms of the values of various contingent options, such as call/put options, lookback options, barrier options, dynamic fund protection and the dynamic withdrawal benefits, have been obtained. The value of each contingent option can then be recovered by the numerical Laplace inversion algorithm, and this efficient approach is documented by several numerical illustrations. The strength of our methodology becomes apparent when we tackle the valuations of exotic contingent options in the cases when (1) the contracts have a finite expiry date; (2) when the time-until-death variable is uniformly distributed in accordance with De Moivre's law.

[1]  Robert J. Elliott,et al.  On Markov‐modulated Exponential‐affine Bond Price Formulae , 2009 .

[2]  James D. Hamilton.  A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[3]  Rama Cont.  Encyclopedia of quantitative finance , 2010 .

[4]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[5]  H. Gerber,et al.  Valuing equity-linked death benefits and other contingent options: A discounted density approach , 2012 .

[6]  Anna Rita Bacinello.  Fair Pricing of Life Insurance Participating Policies with a Minimum Interest Rate Guaranteed , 2001, ASTIN Bulletin.

[7]  John B. Moore,et al.  Hidden Markov Models: Estimation and Control , 1994 .

[8]  Carl M. Harris,et al.  Approximation with generalized hyperexponential distributions: Weak convergence results , 1986, Queueing Syst. Theory Appl..

[9]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[10]  Eduardo S. Schwartz,et al.  Abstract: Alternative Investment Strategies for the Issuers of Equity-Linked Life Insurance Policies with an Asset Value Guarantee , 1977, Journal of Financial and Quantitative Analysis.

[11]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[12]  M. Schweizer.  Minimal Entropy Martingale Measure , 2010 .

[13]  蕭 志中.  Financial modeling with Markov additive processes , 2012 .

[14]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[15]  Eric R. Ulm.  ANALYTIC SOLUTION FOR RETURN OF PREMIUM AND ROLLUP GUARANTEED MINIMUM DEATH BENEFIT OPTIONS UNDER SOME SIMPLE MORTALITY LAWS , 2008 .

[16]  The Effect of the Real Option to Transfer on the Value of Guaranteed Minimum Death Benefits , 2006 .

[17]  Xiaoming Liu,et al.  Markov Aging Process and Phase-Type Law of Mortality , 2007 .

[18]  Mary R. Hardy,et al.  A Regime-Switching Model of Long-Term Stock Returns , 2001 .

[19]  Hans U. Gerber,et al.  Risk theory for the compound Poisson process that is perturbed by diffusion , 1991 .

[20]  Ward Whitt,et al.  The Fourier-series method for inverting transforms of probability distributions , 1992, Queueing Syst. Theory Appl..

[21]  S. Asmussen,et al.  Russian and American put options under exponential phase-type Lévy models , 2004 .

[22]  Hui Wang,et al.  First passage times of a jump diffusion process , 2003, Advances in Applied Probability.

[23]  H. Gerber,et al.  Valuing equity-linked death benefits in jump diffusion models , 2013 .

[24]  S. Posner,et al.  The Titanic Option: Valuation of the Guaranteed Minimum Death Benefit in Variable Annuities and Mutual Funds , 2001 .

[25]  Zhengjun Jiang,et al.  On perpetual American put valuation and first-passage in a regime-switching model with jumps , 2008, Finance Stochastics.

[26]  B. Chan.  Ruin Probability for Translated Combination of Exponential Claims , 1990, ASTIN Bulletin.

[27]  Phelim P. Boyle,et al.  Pricing exotic options under regime switching , 2007 .

[28]  Bernt Øksendal,et al.  Advanced mathematical methods for finance , 2011 .

[29]  Steven Kou,et al.  Option Pricing Under a Mixed-Exponential Jump Diffusion Model , 2011, Manag. Sci..

[30]  Hailiang Yang,et al.  Option Pricing in a Jump-Diffusion Model with Regime Switching , 2009 .

[31]  Hans U. Gerber,et al.  Maximizing Dividends without Bankruptcy , 2006, ASTIN Bulletin.

[32]  Eduardo S. Schwartz,et al.  The pricing of equity-linked life insurance policies with an asset value guarantee , 1976 .

[33]  Steven Kou,et al.  Option Pricing Under a Double Exponential Jump Diffusion Model , 2001, Manag. Sci..

[34]  T. Siu.  Fair valuation of participating policies with surrender options and regime switching , 2005 .

[35]  Phelim P. Boyle,et al.  Equilibrium Prices of Guarantees Under Equity-Linked Contracts , 1977 .

[36]  Bangwon Ko,et al.  Pricing maturity guarantee with dynamic withdrawal benefit , 2010 .

[37]  Hélyette Geman,et al.  Interest Rate Risk Management and Valuation of the Surrender Option in Life Insurance Policies , 1994 .

[38]  Ehud I. Ronn,et al.  Pricing Risk‐Adjusted Deposit Insurance: An Option‐Based Model , 1986 .

[39]  G. Petrella.  An extension of the Euler Laplace transform inversion algorithm with applications in option pricing , 2004, Oper. Res. Lett..

[40]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[41]  Offer Kella,et al.  A multi-dimensional martingale for Markov additive processes and its applications , 2000, Advances in Applied Probability.

[42]  Hans U. Gerber,et al.  The probability and severity of ruin for combinations of exponential claim amount distributions and their translations , 1988 .

[43]  Hangsuck Lee.  Pricing equity-indexed annuities with path-dependent options , 2003 .

[44]  Xin Guo.  An explicit solution to an optimal stopping problem with regime switching , 2001, Journal of Applied Probability.

[45]  Daniel Dufresne,et al.  Fitting combinations of exponentials to probability distributions , 2007 .