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31
2008), “Guaranteed minimum withdrawal benefit in variable annuities
 Mathematical Finance
"... We develop a singular stochastic control model for pricing variable annuities with the guaranteed minimum withdrawal benefit. This benefit promises to return the entire initial investment, with withdrawals spread over the term of the contract, irrespective of the market performance of the underlying ..."
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Cited by 35 (7 self)
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We develop a singular stochastic control model for pricing variable annuities with the guaranteed minimum withdrawal benefit. This benefit promises to return the entire initial investment, with withdrawals spread over the term of the contract, irrespective of the market performance of the underlying asset portfolio. A contractual withdrawal rate is set and no penalty is imposed when the policyholder chooses to withdraw at or below this rate. Subject to a penalty fee, the policyholder is allowed to withdraw at a rate higher than the contractual withdrawal rate or surrender the policy instantaneously. We explore the optimal withdrawal strategy adopted by the rational policyholder that maximizes the expected discounted value of the cash flows generated from holding this variable annuity policy. An efficient finite difference algorithm using the penalty approximation approach is proposed for solving the singular stochastic control model. Optimal withdrawal policies of the holders of the variable annuities with the guaranteed minimum withdrawal benefit are explored. We also construct discrete pricing formulation that models withdrawals on discrete dates. Our numerical tests show that the solution values from the discrete model converge to those of the continuous model.
A semiLagrangian approach for natural gas storage valuation and optimal operation
, 2006
"... The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a HamiltonJacobiBellman (HJB) equation. In this paper, we present a semiLagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a ..."
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Cited by 30 (5 self)
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The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a HamiltonJacobiBellman (HJB) equation. In this paper, we present a semiLagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a wide class of spot price models that exhibit meanreverting, seasonality dynamics and price jumps. We develop fully implicit and CrankNicolson timestepping schemes based on a semiLagrangian approach and prove the convergence of fully implicit timestepping to the viscosity solution of the HJB equation. We show that fully implicit timestepping is equivalent to a discrete control strategy, which allows for a convenient interpretation of the optimal controls. The semiLagrangian approach avoids the nonlinear iterations required by an implicit finite difference method without requiring additional cost. Numerical experiments are presented for several variants of the basic scheme.
A numerical scheme for the impulse control formulation for pricing variable annuities with a Guaranteed Minimum Withdrawal Benefit (GMWB). Submitted to Numerische Mathematik
, 2007
"... In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a Guaranteed Minimum Withdrawal Benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a single numerical scheme for solving the HamiltonJacobiBellman (HJ ..."
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Cited by 24 (9 self)
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In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a Guaranteed Minimum Withdrawal Benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a single numerical scheme for solving the HamiltonJacobiBellman (HJB) variational inequality corresponding to the impulse control problem, and for pricing realistic discrete withdrawal contracts. We prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem, provided a strong comparison result holds. The convergence to the viscosity solution is also proved for the discrete withdrawal case. Numerical experiments are conducted, which show a region where the optimal control appears to be nonunique.
A HamiltonJacobiBellman approach to optimal trade execution
, 2009
"... The optimal trade execution problem is formulated in terms of a meanvariance tradeoff, as seen at the initial time. The meanvariance problem can be embedded in a LinearQuadratic (LQ) optimal stochastic control problem, A semiLagrangian scheme is used to solve the resulting nonlinear Hamilton Ja ..."
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Cited by 16 (3 self)
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The optimal trade execution problem is formulated in terms of a meanvariance tradeoff, as seen at the initial time. The meanvariance problem can be embedded in a LinearQuadratic (LQ) optimal stochastic control problem, A semiLagrangian scheme is used to solve the resulting nonlinear Hamilton Jacobi Bellman (HJB) PDE. This method is essentially independent of the form for the price impact functions. Provided a strong comparision property holds, we prove that the numerical scheme converges to the viscosity solution of the HJB PDE. Numerical examples are presented in terms of the efficient trading frontier and the trading strategy. The numerical results indicate that in some cases there are many different trading strategies which generate almost identical efficient frontiers.
Numerical solution of the HamiltonJacobiBellman formulation for continuous time mean variance asset allocation
 IN THE JOURNAL OF ECONOMIC DYNAMICS AND CONTROL
, 2009
"... We solve the optimal asset allocation problem using a mean variance approach. The original mean variance optimization problem can be embedded into a class of auxiliary stochastic LinearQuadratic (LQ) problems using the method in (Zhou and Li, 2000; Li and Ng, 2000). We use a finite difference metho ..."
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Cited by 14 (4 self)
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We solve the optimal asset allocation problem using a mean variance approach. The original mean variance optimization problem can be embedded into a class of auxiliary stochastic LinearQuadratic (LQ) problems using the method in (Zhou and Li, 2000; Li and Ng, 2000). We use a finite difference method with fully implicit timestepping to solve the resulting nonlinear HamiltonJacobiBellman (HJB) PDE, and present the solutions in terms of an efficient frontier and an optimal asset allocation strategy. The numerical scheme satisfies sufficient conditions to ensure convergence to the viscosity solution of the HJB PDE. We handle various constraints on the optimal policy. Numerical tests indicate that realistic constraints can have a dramatic
Pricing Options in Incomplete Equity Markets via the Instantaneous Sharpe Ratio
, 2007
"... Abstract: We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. First, we ..."
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Cited by 10 (2 self)
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Abstract: We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. First, we apply our method to price options on nontraded assets for which there is a traded asset that is correlated to the nontraded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and SaáRequejo (2000) and of Björk and Slinko (2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque, Papanicolaou, and Sircar (2000).
Maximal use of central differencing for HamiltonJacobiBellman PDEs in finance
 SIAM JOURNAL ON NUMERICAL ANALYSIS
, 2008
"... In order to ensure convergence to the viscosity solution, the standard method for discretizing HJB PDEs uses forward/backward differencing for the drift term. In this paper, we devise a monotone method which uses central weighting as much as possible. In order to solve the discretized algebraic eq ..."
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Cited by 9 (5 self)
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In order to ensure convergence to the viscosity solution, the standard method for discretizing HJB PDEs uses forward/backward differencing for the drift term. In this paper, we devise a monotone method which uses central weighting as much as possible. In order to solve the discretized algebraic equations, we have to maximize a possibly discontinuous objective function at each node. Nevertheless, convergence of the overall iteration can be guaranteed. Numerical experiments on two examples from the finance literature show higher rates of convergence for this approach compared to the use of forward/backward differencing only.
METHODS FOR PRICING AMERICAN OPTIONS UNDER REGIME SWITCHING ∗
"... Abstract. We analyze a number of techniques for pricing American options under a regime switching stochastic process. The techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable. In the case of implicit methods we also c ..."
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Cited by 5 (1 self)
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Abstract. We analyze a number of techniques for pricing American options under a regime switching stochastic process. The techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable. In the case of implicit methods we also compare a number of iterative procedures for solving the associated nonlinear algebraic equations. Numerical tests indicate that a fixed point policy iteration, coupled with a direct control formulation, is a reliable general purpose method. Finally, we remark that we formulate the American problem as an abstract optimal control problem; hence our results are applicable to more general problems as well.
Valuing the Guaranteed Minimum Death Benefit Clause with Partial Withdrawals
, 2008
"... In this paper we give a method for computing the fair insurance fee associated with the guaranteed minimum death benefit (GMDB) clause included in many variable annuity contracts. We allow for partial withdrawals, a common feature in most GMDB contracts, and determine how this affects the GMDB fair ..."
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Cited by 5 (1 self)
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In this paper we give a method for computing the fair insurance fee associated with the guaranteed minimum death benefit (GMDB) clause included in many variable annuity contracts. We allow for partial withdrawals, a common feature in most GMDB contracts, and determine how this affects the GMDB fair insurance charge. Our method models the GMDB pricing problem as an impulse control problem. The resulting quasivariational inequality is solved numerically using a fully implicit penalty method. The numerical results are obtained under both constant volatility and regimeswitching models. A complete analysis of the numerical procedure is included. We show that the discrete equations are stable, monotone and consistent and hence obtain convergence to the unique, continuous viscosity solution, assuming this exists. Our results show that the addition of the partial withdrawal feature significantly increases the fair insurance charge for GMDB contracts.
A penalty method for the numerical solution of HamiltonJacobiBellman (HJB) equations in finance
 SIAM Journal on Numerical Analysis
"... Abstract. We present a simple and easy to implement method for the numerical solution of a rather general class of HamiltonJacobiBellman (HJB) equations. In many cases, classical finite difference discretisations can be shown to converge to the unique viscosity solutions of the considered proble ..."
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Cited by 3 (2 self)
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Abstract. We present a simple and easy to implement method for the numerical solution of a rather general class of HamiltonJacobiBellman (HJB) equations. In many cases, classical finite difference discretisations can be shown to converge to the unique viscosity solutions of the considered problems. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of O(1/ρ), where ρ> 0 is the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.