Article2018

Pricing of guaranteed minimum withdrawal benefits in variable annuities under stochastic volatility, stochastic interest rates and stochastic mortality via the componentwise splitting method

Nikolay Gudkov · Katja Ignatieva · Jonathan Ziveyi

22 citations0/0 files storedQuantitative Finance

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Stochastic processes and financial applications Insurance and Financial Risk Management Insurance, Mortality, Demography, Risk Management

Abstract

This paper values guaranteed minimum withdrawal benefit (GMWB) riders embedded in variable annuities assuming that the underlying fund dynamics evolve under the influence of stochastic interest rates, stochastic volatility, stochastic mortality and equity risk. The valuation problem is formulated as a partial differential equation (PDE) which is solved numerically by employing the operator splitting method. Sensitivity analysis of the fair guarantee fee is performed with respect to various model parameters. We find that (i) the fair insurance fee charged by the product provider is an increasing function of the withdrawal rate; (ii) the GMWB price is higher when stochastic interest rates and volatility are incorporated in the model, compared to the case of static interest rates and volatility; (iii) the GMWB price behaves non-monotonically with changing volatility of variance parameter; (iv) the fair fee increases with increasing volatility of interest rates parameter, and increasing correlation between the underlying fund and the interest rates; (v) the fair fee increases when the speed of mean-reversion of stochastic volatility or the average long-term volatility increases; (vi) the GMWB fee decreases when the speed of mean-reversion of stochastic interest rates or the average long-term interest rates increase. We investigate both static and dynamic (optimal) policyholder's withdrawal behaviours; we present the optimal withdrawal schedule as a function of the withdrawal account and the investment account for varying volatility and interest rates. When incorporating stochastic mortality, we find that its impact on the fair guarantee fee is rather small. Our results demonstrate the importance of correct quantification of risks embedded in GMWBs and provide guidance to product providers on optimal hedging of various risks associated with the contract.

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References

1993A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options 1985A Theory of the Term Structure of Interest Rates 1825XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &c 1996Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options 2003The Impact of Jumps in Volatility and Returns 1987Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application 2006A Two‐Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration 2004Stochastic Calculus for Finance II

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