Efficient willow tree method for variable annuities valuation and risk management☆

Abstract Variable annuities (VAs) with various guarantees are popular retirement products in the past decades. However, due to the sophistication of the embedded guarantees, most existing methods only focus on the one of embedded guarantees underlying one specified stochastic model. The method to evaluate VAs with all guarantees and manage its risk is very limited, except for the Monte Carlo method. In this paper, we propose an efficient willow tree method to evaluate VAs embedded with all popular guarantees on the market underlying various stochastic models. Moreover, our tree structure is also applicable to compute dollar delta, value at risk (VaR) and conditional tail expectation (CTE) in hedging and risk-based capital calculation. Numerical experiments demonstrate the accuracy and efficiency of our method in pricing and managing the risk of VAs.

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

[2]  Runhuan Feng,et al.  Risk based capital for guaranteed minimum withdrawal benefit , 2017 .

[3]  Peter A. Forsyth,et al.  An optimal stochastic control framework for determining the cost of hedging of variable annuities , 2014 .

[4]  Pavel V. Shevchenko,et al.  Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization , 2014, 1411.5453.

[5]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[6]  Thomas S. Salisbury,et al.  Financial valuation of guaranteed minimum withdrawal benefits , 2006 .

[7]  Steven Haberman,et al.  Valuation of Guaranteed Annuity Conversion Options , 2003 .

[8]  A. Pelsser.  Pricing and Hedging Guaranteed Annuity Options via Static Option Replication , 2003 .

[9]  Mike Curran.  Willow Power: Optimizing Derivative Pricing Trees , 2001 .

[10]  Zhenyu Cui,et al.  Equity-Linked Annuity Pricing with Cliquet-Style Guarantees in Regime-Switching and Stochastic Volatility Models with Jumps , 2017 .

[11]  S. Vanduffel,et al.  Impact of Flexible Periodic Premiums on Variable Annuity Guarantees , 2016 .

[12]  Guojun Gan,et al.  Valuation of large variable annuity portfolios: Monte Carlo simulation and synthetic datasets , 2017 .

[13]  Guojun Gan,et al.  Valuation of Large Variable Annuity Portfolios Using Linear Models with Interactions , 2018, Risks.

[14]  Alexey Kuznetsov,et al.  Exponential Functionals of Levy Processes and Variable Annuity Guaranteed Benefits , 2016, Stochastic Processes and their Applications.

[15]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[16]  Thomas F. Coleman,et al.  Moment matching machine learning methods for risk management of large variable annuity portfolios , 2018 .

[17]  Daniel Bauer,et al.  A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities , 2008 .

[18]  Wei Xu,et al.  A Unified Willow Tree Framework for One-Factor Short-Rate Models , 2018 .

[19]  Eric R. Ulm,et al.  Optimal consumption and allocation in variable annuities with Guaranteed Minimum Death Benefits , 2012 .

[20]  Guojun Gan,et al.  Regression Modeling for the Valuation of Large Variable Annuity Portfolios , 2016 .

[21]  S. Ross,et al.  The valuation of options for alternative stochastic processes , 1976 .

[22]  P. Boyle,et al.  Guaranteed Annuity Options , 2003, ASTIN Bulletin.

[23]  Tian-Shyr Dai,et al.  A flexible tree for evaluating guaranteed minimum withdrawal benefits under deferred life annuity contracts with various provisions , 2013 .

[24]  Min Dai,et al.  GUARANTEED MINIMUM WITHDRAWAL BENEFIT IN VARIABLE ANNUITIES , 2007 .

[25]  Xiaochen Jing,et al.  Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits with Dynamic Policyholder Behavior , 2015, J. Comput. Appl. Math..

[26]  Peter Tufano,et al.  Playing with your Future: Who Gambles in Defined-Contribution Pension Plans? , 2018, International Review of Financial Analysis.

[27]  Pietro Millossovich,et al.  Variable Annuities: A Unifying Valuation Approach , 2011 .

[28]  Duy Nguyen,et al.  Continuous-Time Markov Chain and Regime Switching Approximations with Applications to Options Pricing , 2019 .

[29]  W. Xu,et al.  Efficient willow tree method for European-style and American-style moving average barrier options pricing , 2017 .

[30]  Xiaolin Luo,et al.  A Unified Pricing of Variable Annuity Guarantees Under the Optimal Stochastic Control Framework , 2016, 1605.00339.

[31]  O. Barndorff-Nielsen.  Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling , 1997 .

[32]  I. D. Hill,et al.  Fitting Johnson Curves by Moments , 1976 .

[33]  Charles Sutcliffe,et al.  Trading Death: The Implications of Annuity Replication for the Annuity Puzzle, Arbitrage, Speculation and Portfolios , 2013 .

[34]  Zhuliang Chen,et al.  A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) , 2008, Numerische Mathematik.

[35]  Wei Xu,et al.  A Simple and Efficient Two-Factor Willow Tree Method for Convertible Bond Pricing with Stochastic Interest Rate and Default Risk , 2017 .

[36]  Francois Quittard-Pinon,et al.  Pricing and Hedging Variable Annuities in a Lévy Market: A Risk Management Perspective , 2015 .

[37]  Wei Xu,et al.  A new sampling strategy willow tree method with application to path-dependent option pricing , 2013 .

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

[39]  Jonathan Ziveyi,et al.  Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method , 2018 .

[40]  Jonathan Ziveyi,et al.  FOURIER SPACE TIME-STEPPING ALGORITHM FOR VALUING GUARANTEED MINIMUM WITHDRAWAL BENEFITS IN VARIABLE ANNUITIES UNDER REGIME-SWITCHING AND STOCHASTIC MORTALITY , 2017, ASTIN Bulletin.

[41]  Wei Xu,et al.  Pricing American Options by Willow Tree Method Under Jump-Diffusion Process , 2014, The Journal of Derivatives.

[42]  Ioannis Kyriakou,et al.  Convertible bond valuation in a jump diffusion setting with stochastic interest rates , 2015 .

[43]  Mary R. Hardy,et al.  Valuation of a Guaranteed Minimum Income Benefit , 2010 .

[44]  Wai-Sum Chan,et al.  Modeling investment guarantees in Japan: A risk-neutral GARCH approach , 2011 .

[45]  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.

[46]  Yue Kuen Kwok,et al.  Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models , 2018, Quantitative Finance.