Guaranteed Minimum Withdrawal Benefit Research Articles

A semi‐Lagrangian ε$$ \varepsilon $$‐monotone Fourier method for continuous withdrawal GMWBs under jump‐diffusion with stochastic interest rate

AbstractWe develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump‐diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no‐arbitrage GMWB pricing problem as a time‐dependent Hamilton‐Jacobi‐Bellman (HJB) Quasi‐Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi‐Lagrangian method and the Green's function of an associated linear partial integro‐differential equation, we develop an ‐monotone Fourier pricing method, where is a monotonicity tolerance. Together with a provable strong comparison result for the HJB‐QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB‐QVI as . We present a comprehensive study of the impact of simultaneously considering jumps in the subaccount process and stochastic interest rate on the no‐arbitrage prices and fair insurance fees of GMWBs, as well as on the holder's optimal withdrawal behaviors.

Valuation of general GMWB annuities in a low interest rate environment

Variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees.

Two-stage nested simulation of tail risk measurement: A likelihood ratio approach

Estimating tail risk measures is an important task in many financial and actuarial applications and often requires nested simulations, with the outer simulations representing real world scenarios, and the inner simulations typically used for risk neutral pricing or conditional risk measurement. The standard nested simulation method is highly flexible, able to incorporate complex asset models and exotic payoff structures, but it is also computationally highly burdensome, particularly in a multi-period setting, when inner simulation paths are required at each time step of each outer level scenario. In this study, we propose and analyze a two-stage simulation procedure that efficiently estimates the conditional tail expectation of cost of a dynamic hedging program for a Variable Annuity Guaranteed Minimum Withdrawal Benefit (GMWB), under a multi-period nested simulation. In each of the two stages, the method re-uses the same set of inner level simulation paths for each outer scenario at each time point, using a likelihood ratio method to re-weight the probabilities of each individual path for the different outer scenarios. Our numerical study shows that our two-stage, likelihood ratio weighted procedure can offer a very significant improvement in efficiency, of the order of 95% as measured by the RMSE, compared with a standard nested simulation with the same computational cost.

The GMWB guarantee embedded in Life Insurance Contracts: Fair Value Pricing Problem

Due to the complexity of the variable annuities products, we rely on numerical resolution to fairly price the guaranteed minimum withdrawal benefit (GMWB) rider. This policy, known as an embedded put option in the life insurance contracts, gives the contract owner the possibility to withdraw a fixed amount for a fixed period. We assume a single premium payment upfront. We use a Monte Carlo simulation to approximate the fair fees under specific assumptions related to the contract design and conception, the policyholder characteristics and lapse behavior, and different market parameters. More notably, our algorithm allows to perform a sensitivity analysis to seek the impact of different parameters on the value of the policy as recommended by the Solvency II regularities. We find that the GMWB fair fees are intensively sensitive and responsive to any sudden change in the contract inputs.

TAXATION OF A GMWB VARIABLE ANNUITY IN A STOCHASTIC INTEREST RATE MODEL

AbstractModeling taxation of Variable Annuities has been frequently neglected, but accounting for it can significantly improve the explanation of the withdrawal dynamics and lead to a better modeling of the financial cost of these insurance products. The importance of including a model for taxation has first been observed by Moenig and Bauer (2016) while considering a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity. In particular, they consider the simple Black–Scholes dynamics to describe the underlying security. Nevertheless, GMWB are long-term products, and thus accounting for stochastic interest rate has relevant effects on both the financial evaluation and the policyholder behavior, as observed by Goudenège et al. (2018). In this paper, we investigate the outcomes of these two elements together on GMWB evaluation. To this aim, we develop a numerical framework which allows one to efficiently compute the fair value of a policy. Numerical results show that accounting for both taxation and stochastic interest rate has a determinant impact on the withdrawal strategy and on the cost of GMWB contracts. In addition, it can explain why these products are so popular with people looking for a protected form of investment for retirement.

Gaussian process regression for pricing variable annuities with stochastic volatility and interest rate

In this paper, we investigate value and Greeks computation of a guaranteed minimum withdrawal benefit (GMWB) variable annuity, when both stochastic volatility and stochastic interest rate are considered together in the Heston–Hull–White model. In addition, as an insurance product, a guaranteed minimum death benefit is embedded in the contract. We consider a numerical method that solves the dynamic control problem due to the computing of the optimal withdrawal. Moreover, in order to speed up the computation, we employ Gaussian process regression (GPR), a machine learning technique that allows one to compute very fast approximations of a function from training data. In particular, starting from observed prices previously computed for some known combinations of model parameters, it is possible to approximate the whole value function on a defined domain. The regression algorithm consists of algorithm training and evaluation. The first step is the most time demanding, but it needs to be performed only once, while the latter is very fast and it requires to be performed only when predicting the target function. The developed method, as well as for the calculation of prices and Greeks, can also be employed to compute the no-arbitrage fee, which is a common practice in the variable annuities sector. Numerical experiments show that the accuracy of the values estimated by GPR is high with very low computational cost. Finally, we stress out that the analysis is carried out for a GMWB annuity, but it could be generalized to other insurance products.

Willow tree algorithms for pricing Guaranteed Minimum Withdrawal Benefits under jump-diffusion and CEV models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even to surrender the contract. For numerical valuation of the GMWB rider, we use willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve a similar level of numerical accuracy. The design of our pricing algorithm also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine the effectiveness of delta hedging when the fund dynamics exhibit various jump levels.

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

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.

Willow Tree Algorithms for Pricing Guaranteed Minimum Withdrawal Benefits Under Jump-Diffusion and CEV Models

This paper presents the willow tree algorithms for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB), where the underlying fund dynamics evolve under the Merton jump-diffusion process or constant-elasticity-of-variance (CEV) process. The GMWB rider gives the policyholder the right to make periodic withdrawals from his policy account throughout the life of the contract. The dynamic nature of the withdrawal policy allows the policyholder to decide how much to withdraw on each withdrawal date, or even surrender the contract. For numerical valuation of the GMWB rider, we use the willow tree algorithms that adopt more effective placement of the lattice nodes based on better fitting of the underlying fund price distribution. When compared with other numerical algorithms, like the finite difference method and fast Fourier transform method, the willow tree algorithms compute GMWB prices with significantly less computational time to achieve similar level of numerical accuracy. The design of our pricing algorithms also includes an efficient search method for the optimal dynamic withdrawal policies. We perform sensitivity analysis of various model parameters on the prices and fair participating fees of the GMWB riders. We also examine effectiveness of hedging when the fund dynamics exhibit various levels of jump.

VIX-linked fees for GMWBs via explicit solution simulation methods

In a market with stochastic volatility and jumps, we consider a VIX-linked fee structure (see Cui et al. 2017) for variable annuity contracts with guaranteed minimum withdrawal benefits (GMWB). Our goal is to assess the effectiveness of the VIX-linked fee structure in decreasing the sensitivity of the insurer’s liability to volatility risk. Since the GMWB payoff is highly path-dependent, it is particularly sensitive to volatility risk, and can also be challenging to price, especially in the presence of the VIX-linked fee. In this paper, following Kouritzin, 2018, we present an explicit weak solution for the value of the VA account and use it in Monte Carlo simulations to value the GMWB guarantee. Numerical examples are provided to analyze the impact of the VIX-linked fee on the sensitivity of the liability to changes in market volatility.

Valuation of GMWB under stochastic volatility

In this paper, we consider the pricing of the variable annuities with Guaranteed Minimum Withdrawal Benefit (GMWB) options, which are deferred, fund-linked annuity contracts, usually with a single premium payment up front. We use the Heston model for the financial market, which assumes the risky asset dynamic follows a stochastic volatility model. The structure of the GMWB policy enables us to adopt the pricing problem of a barrier option. We derive a GMWB pricing partial differential equation (PDE) and the insurance fee is computed by solving an optimization problem. The computed insurance fee is found to be underpriced in the market with a stochastic volatility model. A sensitivity analysis is performed to see the impacts of various parameters on the value of the policy, and the sensitivity of the pricing function with respect to the market risk.

Pricing and hedging GMWB in the Heston and in the Black–Scholes with stochastic interest rate models

Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Yang and Dai, the Black and Scholes framework seems to be inappropriate for such a long maturity products. Also Chen Vetzal and Forsyth in showed that the price of these products is very sensitive to interest rate and volatility parameters. We propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GMWB variables annuities: a hybrid tree-finite difference method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a Standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GMWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal, optimal surrender and optimal withdrawal strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.

Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method

This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Lévy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. [Scand. Actuar. J., 2014, 1–20], and Luo and Shevchenko [Int. J. Financ. Eng., 2014, 2, 1–24], to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. [op. cit.]. We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging intra-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Lévy framework. While the second moment and variance have been considered in existing literature, we show that the Value-at-Risk (VaR) may also be of interest as a risk measure to minimise risk in variable annuities portfolios.

FOURIER SPACE TIME-STEPPING ALGORITHM FOR VALUING GUARANTEED MINIMUM WITHDRAWAL BENEFITS IN VARIABLE ANNUITIES UNDER REGIME-SWITCHING AND STOCHASTIC MORTALITY

AbstractThis paper introduces the Fourier Space Time-Stepping algorithm to the valuation of variable annuity (VA) contracts embedded with guaranteed minimum withdrawal benefit (GMWB) riders when the underlying fund dynamics evolve under the influence of a regime-switching model. Mortality risk is introduced to the valuation framework by incorporating a two-factor affine stochastic mortality model proposed in Blackburn and Sherris (2013). The paper considers both, static and dynamic policyholder withdrawal behaviour associated with GMWB riders and assesses how model parameters influence the fees levied on providing such guarantees. Our numerical experiments reveal that the GMWB fees are very sensitive to regime-switching parameters; a percentage increase in the force of interest results in significant decrease in guarantee fees. The guarantee fees increase substantially with increasing volatility levels. Numerical experiments also highlight an increasing importance of mortality as maturity of the VA contract increases. Mortality has less impact on shorter maturity contracts regardless of the policyholder's withdrawal behaviour. As much as mortality influences pricing results for long maturities, the associated guarantee fees are decreasing functions of maturities for the VA contracts. Robustness checks of the Fourier Space Time-Stepping algorithm are performed by making numerical comparisons with several existing valuation approaches.

Valuation of variable annuities with Guaranteed Minimum Withdrawal Benefit under stochastic interest rate

This paper develops an efficient direct integration method for pricing of the variable annuity (VA) with guarantees in the case of stochastic interest rate. In particular, we focus on pricing VA with Guaranteed Minimum Withdrawal Benefit (GMWB) that promises to return the entire initial investment through withdrawals and the remaining account balance at maturity. Under the optimal (dynamic) withdrawal strategy of a policyholder, GMWB pricing becomes an optimal stochastic control problem that can be solved using backward recursion Bellman equation. Optimal decision becomes a function of not only the underlying asset but also interest rate. Presently our method is applied to the Vasicek interest rate model, but it is applicable to any model when transition density of the underlying asset and interest rate is known in closed-form or can be evaluated efficiently. Using bond price as a numéraire the required expectations in the backward recursion are reduced to two-dimensional integrals calculated through a high order Gauss–Hermite quadrature applied on a two-dimensional cubic spline interpolation. The quadrature is applied after a rotational transformation to the variables corresponding to the principal axes of the bivariate transition density, which empirically was observed to be more accurate than the use of Cholesky transformation. Numerical comparison demonstrates that the new algorithm is significantly faster than the partial differential equation or Monte Carlo methods. For pricing of GMWB with dynamic withdrawal strategy, we found that for positive correlation between the underlying asset and interest rate, the GMWB price under the stochastic interest rate is significantly higher compared to the case of deterministic interest rate, while for negative correlation the difference is less but still significant. In the case of GMWB with predefined (static) withdrawal strategy, for negative correlation, the difference in prices between stochastic and deterministic interest rate cases is not material while for positive correlation the difference is still significant. The algorithm can be easily adapted to solve similar stochastic control problems with two stochastic variables possibly affected by control. Application to numerical pricing of Asian, barrier and other financial derivatives with a single risky asset under stochastic interest rate is also straightforward.

Pricing and Hedging Guaranteed Minimum Withdrawal Benefits under a General LLvy Framework Using the COS Method

This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Levy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. (2014) and Luo and Shevchenko (2014) to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. (2014). We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging inter-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Levy framework. While the second moment and variance have been considered in existing literature, we show that the value-at-risk may also be of interest as a risk measure to minimise risk in variable annuities portfolios.

Analytical Approximation of Variable Annuities for Small Volatility and Small Withdrawal

This work is concerned with asymptotic approximation applied to insurance valuation issues. Namely we present a method to derive explicit approximations for the valuation of variable annuities of guaranteed minimum withdrawal benefit (GMWB) type in a hybrid model (Black--Scholes plus Hull--White). It leads to explicit and tractable approximation formulas, whose evaluation is almost instantaneous on a computer and that achieve about 0.01% accuracy in our experiments. Error estimates support the convergence of our approximations, as the withdrawal and volatilities are small.

Risk based capital for guaranteed minimum withdrawal benefit

The guaranteed minimum withdrawal benefit (GMWB), which is sold as a rider to variable annuity contracts, guarantees the return of total purchase payment regardless of the performance of the underlying investment funds. The valuation of GMWB has been extensively covered in the previous literature, but a more challenging problem is the computation of the risk based capital for risk management and regulatory reasons. One needs to find the tail distribution of the profit–loss function, which differs from its expected payoff required for pricing the GMWB contract. GMWB has embedded two option-like features: Management fees are proportional to the current value of the policyholder’s account which results in an average price of the account. Thus the contract resembles an Asian option. However, the fees are charged only up to the time of the account hitting zero which resembles a barrier option payoff. Thus the GMWB is mathematically more complicated than Asian or barrier options traded on the financial markets. To the authors’ best knowledge, this is the first paper in the literature to formulate and analyse profit–loss distribution using PDE methods of such a product with intricate option-like features. Our approach is much more efficient than the current market practice of rather intensive and expensive Monte Carlo simulations due to the lack of samples for extreme cases.

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit Under Stochastic Interest Rate

A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB) promises to return the entire initial investment through cash withdrawals during the contract plus the remaining account balance at maturity, regardless of the portfolio performance. Under the optimal (dynamic) withdrawal strategy of a policyholder, GMWB pricing becomes an optimal stochastic control problem that can be solved by backward recursion of Bellman equation. In this paper we develop a very efficient new algorithm for pricing these contracts in the case of stochastic interest rate not considered previously in the literature. Presently our method is applied to the Vasicek interest rate model, but it is generally applicable to any model when transition density or moments of the underlying asset and interest rate are known in closed form or can be evaluated efficiently. Using bond price as a numeraire the required expectations in the backward recursion are reduced to two-dimensional integrals calculated through a high order Gauss-Hermite quadrature applied on a two-dimensional cubic spline interpolation. The quadrature is applied after a rotational transformation to the variables corresponding to the principal axes of the bivariate transition density, which empirically was observed to be more accurate than the the use of Cholesky transformation.Numerical results from the new algorithm for a series of GMWB contracts for both static and optimal cases are presented. As a validation, results of the algorithm are compared with the closed form solutions for simple vanilla options, and with Monte Carlo and finite difference results for the static GMWB. The comparison demonstrates that the new algorithm is significantly faster than finite difference or Monte Carlo for all the two-dimensional problems tested so far. For dynamic GMWB pricing, we found that for positive correlation between the underlying asset and interest rate, the GMWB price under the stochastic interest rate is significantly higher compared to the case of deterministic interest rate, while for negative correlation the difference is less but still significant. In the case of static GMWB, for negative correlation, the difference in prices between stochastic and deterministic interest rate cases is not material while for positive correlation the difference is still significant. The algorithm can be easily adapted to solve similar stochastic control problems with two state variables possibly affected by control. Also applications to numerical pricing of Asian, barrier and other financial derivatives under stochastic interest rate are straightforward.

Valuation of Guaranteed Minimum Withdrawal Benefits in Variable Annuities Under Stochastic Mortality, Stochastic Volatility and Stochastic Interest Rates

Extend the existing literature on GMWB pricing and hedging by incorporating simultaneously: I equity risk, I mortality risk, I interest rate risk, I volatility risk.