Optimal Surrender Research Articles

Optimal surrender policy for reverse mortgage loans

Abstract This study conducts an optimal surrender analysis of reverse mortgage (RM) loans offered to elderly homeowners as a financing option. Recent market evidence on borrower early surrenders has raised concerns about the marketability of RM products and their impact on the program viability. In this article, we derive the borrower optimal surrender strategy as a function of the underlying value of the home used as collateral for RM contracts with tenure payment option. Using a probabilistic approach to American option pricing, we present a decomposition result for the value of the contract as the sum of its European counterpart without the surrendering provision and an early exercise premium. The methodology allows policymakers to assess the financial incentive of their policy design, from which we explain the existing market evidence about borrower rational lapse by means of the resulting surrender boundary and reference probabilities.

Enhancing valuation of variable annuities in Lévy models with stochastic interest rate

This paper extends the valuation and optimal surrender framework for variable annuities with guaranteed minimum benefits in a Lévy equity market environment by incorporating a stochastic interest rate described by the Hull-White model. This approach frames a more dynamic and realistic financial setting compared to previous literature. We exploit a robust valuation mechanism employing a hybrid numerical method that merges tree methods for interest rate modeling with finite difference techniques for the underlying asset price. This method is particularly effective for addressing the complexities of variable annuities, where periodic fees and mortality risks are significant factors. Our findings reveal the influence of stochastic interest rates on the strategic decision-making process concerning the surrender of these financial instruments. Through comprehensive numerical experiments, and by comparing our results with those obtained through the Longstaff-Schwartz Monte Carlo method, we illustrate how our refined model can guide insurers in designing contracts that equitably balance the interests of both parties. This is particularly relevant in discouraging premature surrenders while adapting to the realistic fluctuations of financial markets. Lastly, a comparative statics analysis with varying interest rate parameters underscores the impact of interest rates on the cost of the optimal surrender strategy, emphasizing the importance of accurately modeling stochastic interest rates.

Variational inequality arising from variable annuity with mean reversion environment

In this paper, we study a variational inequality arising from variable annuity (VA) to find the optimal surrender strategy for a VA investor when the underlying asset follows a mean reverting process. We formulate the problem as a free boundary partial differential equation (PDE) to obtain the optimal strategy. The PDE is solved analytically by the Mellin transform approach. Using the Mellin transform, we derive the integral equations for the value of the VA and the optimal surrender boundary. Since the solutions are obtained as the integral equations, we use the recursive integration method to determine the optimal surrender strategy. Finally, we provide the optimal surrender boundaries and values of VA with respect to some significant parameters to show the impacts of mean reversion.

Valuation of variable annuities with guaranteed minimum maturity benefits and periodic fees

This paper focuses on the valuation of variable annuities with a guaranteed minimum maturity benefit under a regime-switching Lévy model. The model allows policyholders to surrender their annuities and receive a surrender benefit at predetermined tenor times before maturity. Additionally, we consider a state-dependent periodic fee structure where fees are deducted from the policyholder's account if it exceeds a certain level at discrete time points. Incorporating this fee structure, the Fourier cosine series expansion method based on characteristic functions is employed to determine the values and optimal surrender strategies for variable annuity contracts. Finally, we provide a comprehensive set of numerical examples to demonstrate and assess the effectiveness of our approach thoroughly.

Valuation and optimal surrender of variable annuities with guaranteed minimum benefits and periodic fees

This work studies the valuation and optimal surrender of variable (equity-linked) annuities under a Lévy-driven equity market with mortality risk. We consider a practical periodic fee structure which can vary over time and is assessed as a proportion of the fund value. At maturity, the fund value is returned to the policyholder according to a guaranteed minimum accumulation benefit (GMAB). Mortality risk is also modeled discretely, and the contract offers a guaranteed minimum death benefit (GMBD) prior to maturity. The benefits accommodate caps on the growth of funds (in addition to the rising floor) to reduce the fee level and as a disincentive to early surrender. Interest rates are modeled via a deterministic discounting term structure, which can be calibrated (bootstrapped) to the rates market, according to market convention. An efficient and accurate valuation framework is developed, along with closed form pricing formulas in the case where policy surrender is not permitted. Numerous experiments are conducted to illustrate the interplay between contract parameters and the decision to surrender, and we provide an extensive analysis that investigates how to structure contracts to disincentivize early surrender.

Surrender contagion in life insurance

Modern life insurance policies embed a surrender option that entitles policyholders to terminate their policy early for its cash value. Mass surrenders–spiking surrenders of life insurance policies within a short period–are widely documented prior to many life insurer failures, but their impacts on insurers are rarely quantified. This paper builds a model of mass surrenders in a heterogeneous pool consisting of a minority of financially guided policyholders who make rational surrender decisions and a majority of individual policyholders who are subject to herd. By revising the self-excitation in the Hawkes process to a contingent event-accumulation-excitation, we introduce a history-dependent surrender process for the individuals whose contagious behavior can be triggered contingent on past surrenders becoming salient. We then incorporate it into a life insurance pricing framework and study mass surrenders’ impacts on contract valuation and default risk of the insurer, allowing for its structural default under regulatory solvency intervention. Our numerical results show that while contagion aligns individual policyholders’ surrender behavior with the optimal surrender of financially guided ones, it jeopardizes their financial positions in favor of equity holders. Surprisingly, surrender contagion only mildly enlarges an insurer’s bankruptcy rate, which is not driven by strict solvency regulation. Despite its public perception of posing a significant risk to insurers as similar to a bank run, surrender contagion does not critically threaten insurers’ solvency: To prevent insurers’ bankruptcies, their investment riskiness and leverage pose more of a concern to regulators.

An analytical study of participating policies with minimum rate guarantee and surrender option

We perform a detailed theoretical study of the value of a class of participating policies with four key features: (i) the policyholder is guaranteed a minimum interest rate on the policy reserve; (ii) the contract can be terminated by the holder at any time until maturity (surrender option); (iii) at the maturity (or upon surrender), a bonus is credited to the holder if the portfolio backing the policy outperforms the current policy reserve; (iv) due to solvency requirements, the contract ends if the value of the underlying portfolio of assets falls below the policy reserve.Our analysis is probabilistic and relies on optimal stopping and free boundary theory. We find a structure of the optimal surrender strategy which was undetected by previous (mostly numerical) studies on the topic. Optimal surrender of the contract is triggered by two ‘stop-loss’ boundaries and by a ‘too-good-to-persist’ boundary (in the language of Ekström and Vaicenavicius in Stoch. Process. Appl. 130: 806–823, 2020). Financial implications of this strategy are discussed in detail and supported by extensive numerical experiments.

A variational inequality arising from optimal surrender of variable annuity with lookback benefit

We introduce a variable annuity (VA) contract with a surrender option and lookback benefit, that is, the benefit of the VA contract is linked to the maximum process of the policyholder’s account value. In contrast to the constant guarantee model provided in Bernard et al. (Insur. Math. Econ. 55:116–128, 2014), it is optimal for the policyholder of the VA contract with lookback benefit to surrender the VA contract when the policyholder’s account value is below or equal to the optimal surrender boundary. Thus, from the perspective of the insurer to construct a portfolio of VA contracts, utilizing the VA contracts with lookback benefit along with VA contracts with constant guarantee provides the diversification of early surrenders. The valuation of this contract can be described as a two-dimensional parabolic variational inequality. By converting this into the one-dimensional problem, we obtain the integral equations for the value function and the free boundary. The recursive integration method is applied to obtain the numerical solutions. We also provide comparative statics of the optimal surrender boundaries with respect to various parameters.

Valuation and Optimal Surrender of Variable Annuities with Guaranteed Minimum Benefits and Periodic Fees

Valuation and Optimal Surrender of Variable Annuities with Guaranteed Minimum Benefits and Periodic Fees

Optimal Surrender Policy of Guaranteed Minimum Maturity Benefits in Variable Annuities with Regime-Switching Volatility

This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.

Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation

We consider the pricing of variable annuities (VAs) with general fee structures under a class of stochastic volatility models which includes the Heston, Hull-White, Scott, α-Hypergeometric, 3/2, and 4/2 models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

Pricing variable annuity with surrender guarantee

In this paper we present a variable annuity (VA) contract embedded with a guaranteed minimum accumulated benefit rider that can be chosen to surrender the contract anytime before the maturity. In contrast to the model considered by Bernard et al. (2014), the surrender benefit in our problem is linked to the maximum value between the policyholder’s account value and the guaranteed minimum accumulated benefit. Thus, the surrender benefit of our model provides a protection against the downside risk of financial market throughout the life of contract, and thus it can be referred to as surrender guarantee. Under this circumstance with surrender guarantee, the VA contract has double surrender regions, that is, there exist two optimal surrender boundaries such that if the policyholder’s account value hits one of these boundaries, then the policyholder immediately surrenders the VA contract. Based on the Mellin transform techniques, we derive the coupled integral equations for the two optimal surrender boundaries. We solve numerically these coupled integral equations by using the recursive integration method and provide comparative statics analysis with respect to various parameters.

배경위험 하에서의 최적보험계약과 해약환급금에 관한 연구

배경위험 하에서의 최적보험계약과 해약환급금에 관한 연구

Surrender Contagion in Life Insurance: Modeling and Valuation

Mass surrenders—spiking surrenders of life insurance policies within a short period— are widely documented prior to many life insurers failures, but their impacts on insurers are rarely quantified. This paper builds a model of mass surrenders in a heterogeneous pool consisting of a minority of financially guided policyholders and a majority of individual policyholders with contagious behavior that can be triggered contingent on past surrenders becoming salient. By incorporating it into a life insurance pricing framework, we study mass surrenders’ impacts on contract valuation and early bankruptcy of the insurer, allowing for its structural default under regulatory solvency intervention. Our numerical results show that while contagion aligns individual policyholders’ surrender behavior with the optimal surrender of financially guided ones, it jeopardizes their financial positions in favor of equity holders. Surprisingly, contagion at worst only marginally enlarges the insurer’s bankruptcy rate, which is not driven by strict solvency regulation. Insurers have incentives to encourage contagion. Despite its public perception of endangering market stability as similar to a bank run, surrender contagion does not threaten insurers’ solvency, but can improve it instead with the assets left behind by individual policyholders.

Efficient valuation of a variable annuity contract with a surrender option

We study the partial differential equation (PDE) approach for efficient and accurate valuation of a variable annuity (VA) contract with a surrender option. Specifically, using the Laplace–Carson Transform (LCT), we derive an analytic pricing formula for a VA contract with a surrender option which is formulated as a PDE with an optimal surrender boundary. To demonstrate the efficiency and accuracy of our approach, we show that our pricing formula efficiently provides the exact value of a VA contract. Moreover, we compare the performance of three numerical methods for Laplace inversion to find the most efficient method. Among these methods, we found that the Gaver–Stehfest method is the most efficient pricing for a VA contract with a surrender option.

Pricing European continuous-installment strangle options

This paper investigates the valuation of European continuous-installment strangle options written on dividend-paying underlying assets in the standard Black-Scholes framework. In this pricing problem, the premium of the strangle option is paid continuously instead of up-front. Since the holder of this option has the right to surrender installment payments at any time, the valuation of installment strangle option can be formulated as an optimal stopping problem with two surrender boundaries. Based on the Mellin transform approaches, we derive the integral equation representations for the value function and the two optimal surrender boundaries. By using the recursive integration method, we obtain efficiently the numerical solution for the integral equations and illustrate the optimal surrender boundaries with respect to the significant parameters.

Optimal surrender strategies and valuations of path-dependent guarantees in variable annuities

In this paper, we consider variable annuity contracts with guaranteed minimum maturity benefits in which the embedded guarantees can be surrendered any time before the maturity date has been reached. We derive the optimal surrender strategies and the values of three different types of guarantees embedded in VA contracts: (1) constant guarantees, (2) constant guarantees with an up-and-out barrier, and (3) path-dependent guarantees with a lookback feature. The valuation of the embedded guarantees with a surrender option can be formulated as a free boundary problem, and we can obtain a corresponding non-homogeneous partial differential equation (PDE). We apply the Mellin transform to the non-homogeneous PDE in order to obtain an integral equation that produces the value of the embedded guarantee as well as the optimal surrender boundary in a variable annuity contract. We present the properties of the fair fee rate and the optimal surrender strategy with numerical examples.

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.

Optimal surrender of guaranteed minimum maturity benefits under stochastic volatility and interest rates

In this paper we analyse how the policyholders’surrender behaviour is influenced by changes in various sources of risk impacting a variable annuity (VA) contract embedded with a guaranteed minimum maturity benefit rider that can be surrendered anytime prior to maturity. We model the underlying mutual fund dynamics by combining a Heston (1993) stochastic volatility model together with a Hull and White (1990) stochastic interest rate process. The model is able to capture the smile/skew often observed on equity option markets (Grzelak and Oosterlee 2011) as well as the influence of the interest rates on the early surrender decisions as noted from our analysis. The annuity provider charges management fees which are proportional to the level of the mutual fund as a way of funding the VA contract. To determine the optimal surrender decisions, we present the problem as a 4-dimensional free-boundary partial differential equation (PDE) which is then solved efficiently by the method of lines (MOL) approach. The MOL algorithm facilitates simultaneous computation of the prices, fair management fees, optimal surrender boundaries and hedge ratios of the variable annuity contract as part of the solution at no additional computational cost. A comprehensive analysis on the impact of various risk factors in influencing the policyholder’s surrender behaviour is carried out, highlighting the significance of both stochastic volatility and interest rate parameters in influencing the policyholder’s surrender behaviour. With the aid of the hedge ratios obtained from the MOL, we construct an effective dynamic hedging strategy to mitigate the provider’s risk and compare different hedging performances when the policyholders’ surrender behaviour is either optimal or sub-optimal.

Optimal Surrender of Guaranteed Minimum Maturity Benefits Under Stochastic Volatility and Interest Rates

Optimal Surrender of Guaranteed Minimum Maturity Benefits Under Stochastic Volatility and Interest Rates