Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB)

Huang, Yu; Forsyth, Peter

doi:10.1093/imanum/drq044

Cited by 41 publications

(60 citation statements)

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“…These earlier pricing models of GMWB take the assumption of a geometric Brownian motion (GBM) for the underlying fund value process and continuous withdrawal process as the control in the derivation of the singular stochastic control models. More recently, Huang and Forsyth (2012) and Huang and Kwok (2014) manage to perform a complete characterization of the optimal withdrawal policies of the GMWB under the simplifying assumption of GBM and continuous withdrawal. The more recent research works on pricing GMWB show the departure from the simple GBM framework for the fund value processes to that under stochastic volatilities and / or stochastic interest rates.…”

Section: Introductionmentioning

confidence: 99%

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

Huang

Kwok

2015

Quantitative Finance

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We present the regression-based Monte Carlo simulation algorithms for solving the stochastic control models associated with pricing and hedging of the Guaranteed Lifelong Withdrawal Benefit (GLWB) in variable annuities, where the dynamics of the underlying fund value is assumed to evolve according to the stochastic volatility model. The GLWB offers a lifelong withdrawal benefit even when the policy account value becomes zero while the policyholder remains alive. Upon death, the remaining account value will be paid to the beneficiary as a death benefit. The bang-bang control strategy analyzed under the assumption of maximization of the policyholder's expected cash flow reduces the strategy space of optimal withdrawal policies to three choices: zero withdrawal, withdrawal at the contractual amount or complete surrender. The impact on the GLWB value under various withdrawal behaviors of the policyholder is examined. We also analyze the pricing properties of GLWB subject to different model parameter values and structural features.JEL classification: G22, C50

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Section: Introductionmentioning

confidence: 99%

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

Huang

Kwok

2015

Quantitative Finance

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“…In contrast, discretization of singular control formulations requires the solution of a system of nonlinear algebraic equations at each timestep, which is a significant computational hurdle. In this article, we consider the singular control formulation of a particular financial contingent claim, the Guaranteed Minimum Withdrawal Benefit (GMWB) contract [21,10,15] and provide an efficient method for numerically pricing such a contract. Although we focus specifically on GMWB problems here, our results and methods of analysis are applicable to many other singular control problems in finance.…”

Section: Introductionmentioning

confidence: 99%

“…Generally, the viscosity solution of the HJB VI can be shown to be the solution of the contract valuation problem posed as a dynamic program, which is the financially relevant solution. In [10,15], a penalty method is used to discretize the HJB VI arising from a GMWB contract, assuming Geometric Brownian Motion. The main focus of paper [15] is to show that the discretization scheme is monotone, consistent and l ∞ stable, and hence converges to the viscosity solution as the mesh parameter becomes small.…”

Section: Introductionmentioning

confidence: 99%

“…In [10,15], a penalty method is used to discretize the HJB VI arising from a GMWB contract, assuming Geometric Brownian Motion. The main focus of paper [15] is to show that the discretization scheme is monotone, consistent and l ∞ stable, and hence converges to the viscosity solution as the mesh parameter becomes small. The authors in [15] use a fully implicit timestepping method which guarantees an unconditionally monotone discretization.…”

Section: Introductionmentioning

confidence: 99%

“…Example computations verify that these estimates are of the correct order. Finally, we compare the scaled direct control formulation to a formulation based on a block version of the penalty method [15]. We show that the scaled direct control method has some advantages over the penalty method.…”

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Iterative methods for the solution of a singular control formulation of a GMWB pricing problem

Huang

Labahn

2012

Numer. Math.

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Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a Guaranteed Minimum Withdrawal Benefit (GMWB), where the underlying asset is assumed to follow a jump diffusion process. We use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges. Our methods take advantage of the special structure of the GMWB problem in order to obtain a rapidly convergent iteration. The direct control method has a scaling parameter which must be set by the user. We give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round-off error. Example computations verify that these estimates are of the correct order. Finally, we compare the scaled direct control formulation to a formulation based on a block version of the penalty method [15]. We show that the scaled direct control method has some advantages over the penalty method.

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Continuous time mean‐variance optimal portfolio allocation under jump diffusion: An numerical impulse control approach

Dang

2013

Numerical Methods Partial

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We present efficient partial differential equation methods for continuous time mean‐variance portfolio allocation problems when the underlying risky asset follows a jump‐diffusion. The standard formulation of mean‐variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one‐dimensional (1D) nonlinear Hamilton–Jacobi–Bellman (HJB) partial integrodifferential equation (PIDE) with the control present in the integrand of the jump term, and thus is difficult to solve efficiently. To preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 2D impulse control problem, 1D for each asset in the portfolio, namely the bond and the stock. We then develop a numerical scheme based on a semi‐Lagrangian timestepping method, which we show to be monotone, consistent, and stable. Hence, assuming a strong comparison property holds, the numerical solution is guaranteed to converge to the unique viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of realistic financial modeling, such as different borrowing and lending interest rates, transaction costs, and constraints on the portfolio, such as maximum limits on borrowing and solvency. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 664–698, 2014

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Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB)

Abstract: 5The no arbitrage pricing of Guaranteed Minimum Withdrawal Benefits (GMWB) contracts 6 results in a singular stochastic control problem which can be formulated as a Hamilton Jacobi

Cited by 41 publications

References 14 publications

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

Regression-based Monte Carlo methods for stochastic control models: variable annuities with lifelong guarantees

Iterative methods for the solution of a singular control formulation of a GMWB pricing problem

Continuous time mean‐variance optimal portfolio allocation under jump diffusion: An numerical impulse control approach

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