Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit Under Optimal Withdrawal Strategy

Luo, Xiaolin; Shevchenko, Pavel V.

doi:10.2139/ssrn.2517094

Cited by 7 publications

(15 citation statements)

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“…Below we present numerical results for fair fee of GMWB with surrender option under optimal and suboptimal bang-bang withdrawal strategies. For convenience we denote results for optimal withdrawal strategy without surrender option as GMWB, and with surrender option as GMWB-S. As discussed in Luo and Shevchenko (2015a), only very few results for GMWB under dynamic policyholder behavior can be found in the literature, and these results are for GMWB without the surrender option. For validation purposes, perhaps the most accurate results are found in Chen and Forsyth (2008), which were obtained with a very fine mesh in a detailed convergence study.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Variable Annuity with GMWB: Surrender Or Not, That Is the Question

Luo

Shevchenko

2015

SSRN Journal

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Abstract:A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB) promises to return the entire initial investment through cash withdrawals during the policy life plus the remaining account balance at maturity, regardless of the portfolio performance. We assume that market is complete in financial risk and also there is no mortality risk (in the event of policyholder death, the contract is maintained by beneficiary), thus the annuity price can be expressed as an appropriate expectation. Under the optimal withdrawal strategy of a policyholder, the pricing of variable annuities with GMWB is an optimal stochastic control problem. The surrender feature available in marketed products allows termination of the contract before maturity, making it also an optimal stopping problem.Although the surrender feature is quite common in variable annuity contracts, there appears to be no published analysis and results for this feature in GMWB under optimal policyholder behavior -results found in the literature so far are consistent with the absence of such a feature. Recently, Azimzadeh and Forsyth (2014) prove the existence of an optimal bang-bang control for a Guaranteed Lifelong Withdrawal Benefits (GLWB) contract. In particular, they find that the holder of a GLWB can maximize a writers losses by only ever performing nonwithdrawal, withdrawal at exactly the contract rate, or full surrender. This dramatically reduces the optimal strategy space. However, they also demonstrate that the related GMWB contract is not convexity preserving, and hence does not satisfy the bang-bang principle other than in certain degenerate cases. For GMWB under optimal withdrawal assumption, the numerical algorithms developed by Dai et al. (2008), Chen and Forsyth (2008) and Luo and Shevchenko (2015a) appear to be the only ones found in the literature, but none of them actually performed calculations with surrender option on top of optimal withdrawal strategy. Also, it is of practical interest to see how the much simpler bang-bang strategy, although not optimal for GMWB, compares with optimal GMWB strategy with surrender option.Recently, in Luo and Shevchenko (2015a), we have developed a new efficient numerical algorithm for pricing GMWB contracts in the case when transition density of the underlying asset between withdrawal dates or its moments are known. This algorithm relies on computing the expected contract value through a high order Gauss-Hermite quadrature applied on a cubic spline interpolation and much faster than the standard partial differential equation methods. In this paper we extend our algorithm to include surrender option in GMWB and compare prices under different policyholder strategies: optimal, static and bang-bang. Results indicate that following a simple but sub-optimal bang-bang strategy does not lead to significant reduction in the price or equivalently in the fee, in comparison with the optimal strategy. We also observed that the extra value added by the surrender option strongly depends on volatility an...

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Section: Numerical Resultsmentioning

confidence: 99%

“…A very detailed description of the algorithm that we adapt for pricing GMWB with surrender can be found in Luo and Shevchenko (2015a). Below we outline the main steps.…”

Section: Numerical Algorithmmentioning

confidence: 99%

Variable Annuity with GMWB: Surrender Or Not, That Is the Question

Luo

Shevchenko

2015

SSRN Journal

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“…The variable annuities with GMWB feature have been considered in e.g. Milevsky & Salisbury (2006), Bauer et al (2008), Dai et al (2008), Chen & Forsyth (2008), Bacinello et al (2011) and Luo & Shevchenko (2014b).…”

Section: Introductionmentioning

confidence: 99%

“…Recently we have developed a very efficient new algorithm for pricing variable annuities with GMWB under both static and dynamic policyholder behaviors solving an equivalent stochastic control problem; see Luo & Shevchenko (2014b). Here the definition of "dynamic" is similar to the one used by Bauer et al (2008), Dai et al (2008) and Chen & Forsyth (2008), i.e.…”

Section: Introductionmentioning

confidence: 99%

“…It relies on computing the expected option values in a backward time-stepping between withdrawal dates through Gauss-Hermite integration quadrature applied on a cubic spline interpolation (referred to as GHQC). In Luo & Shevchenko (2014b) it is demonstrated that in pricing GMWB under the optimal policyholder behavior the GHQC algorithm can achieve similar accuracy as the finite difference method, but it is significantly faster because it requires less number of steps in time. This method can be applied when transition density of the underlying asset between withdrawal dates or it's moments are known in closed form and required expectations are 1d integrals.…”

Section: Introductionmentioning

confidence: 99%

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Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization

Luo

Shevchenko

2014

SSRN Journal

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In this paper we present a numerical valuation of variable annuities with combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum Death Benefit (GMDB) under optimal policyholder behavior solved as an optimal stochastic control problem. This product simultaneously deals with financial risk, mortality risk and human behavior. We assume that market is complete in financial risk and mortality risk is completely diversified by selling enough policies and thus the annuity price can be expressed as appropriate expectation. The computing engine employed to solve the optimal stochastic control problem is based on a robust and efficient Gauss-Hermite quadrature method with cubic spline. We present results for three different types of death benefit and show that, under the optimal policyholder behavior, adding the premium for the death benefit on top of the GMWB can be problematic for contracts with long maturities if the continuous fee structure is kept, which is ordinarily assumed for a GMWB contract. In fact for some long maturities it can be shown that the fee cannot be charged as any proportion of the account value -there is no solution to match the initial premium with the fair annuity price. On the other hand, the extra fee due to adding the death benefit can be charged upfront or in periodic instalment of fixed amount, and it is cheaper than buying a separate life insurance.

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A flexible lattice framework for valuing options on assets paying discrete dividends and variable annuities embedding GMWB riders

Angelis

Marchis

Martire

et al. 2022

Decisions Econ Finan

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In a market where a stochastic interest rate component characterizes asset dynamics, we propose a flexible lattice framework to evaluate and manage options on equities paying discrete dividends and variable annuities presenting some provisions, like a guaranteed minimum withdrawal benefit. The framework is flexible in that it allows to combine financial and demographic risk, to embed in the contract early exercise features, and to choose the dynamics for interest rates and traded assets. A computational problem arises when each dividend (when valuing an option) or withdrawal (when valuing a variable annuity) is paid, because the lattice lacks its recombining structure. The proposed model overcomes this problem associating with each node of the lattice a set of representative values of the underlying asset (when valuing an option) or of the personal subaccount (when valuing a variable annuity) chosen among all the possible ones realized at that node. Extensive numerical experiments confirm the model accuracy and efficiency.

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Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit Under Optimal Withdrawal Strategy

Cited by 7 publications

References 10 publications

Variable Annuity with GMWB: Surrender Or Not, That Is the Question

Variable Annuity with GMWB: Surrender Or Not, That Is the Question

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization

A flexible lattice framework for valuing options on assets paying discrete dividends and variable annuities embedding GMWB riders

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