Combined Fixed Point and Policy Iteration for Hamilton--Jacobi--Bellman Equations in Finance

Huang, Yu; Labahn, George

doi:10.1137/100812641

Cited by 27 publications

(79 citation statements)

References 34 publications

(82 reference statements)

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“…In an effort to minimize the work required to solve the linear system at each iteration, a fixed point policy iteration was suggested in [23]. The approach makes use of the splitting (4.2) and is given in Algorithm 5.2.…”

Section: Fixed Point Policy Iterationmentioning

confidence: 99%

“…We carry out a convergence analysis of the iterative method used to solve the nonlinear discretized algebraic equations. It is convenient to consider these equations as a special case of the general form of discretized Hamilton-Jacobi-Bellman (HJB) equations, as discussed in [19,23]. The previously mentioned numeric approaches (for regime switching) are all simply special cases of this general form.…”

Section: Introductionmentioning

confidence: 99%

“…This allows us to use a single framework to analyze the convergence of various iterative methods. These include full policy iteration [30], fixed point policy iteration [23], and a method whereby the regime coupling terms are lagged at each iteration, but the American option problem is solved to high accuracy within each regime [37]. In addition, using the same framework, we also analyze a global-in-time iteration procedure suggested in [31] (see also [5,6]) whereby a sequence of optimal stopping problems is solved.…”

Section: Introductionmentioning

confidence: 99%

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Methods for Pricing American Options under Regime Switching

Huang¹,

Forsyth²,

Labahn³

2011

SIAM J. Sci. Comput.

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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. Key words. regime switching, American options, iterative methods AMS subject classifications. 65N06, 93C20DOI. 10.1137/1108209201. Introduction. The standard approach to valuation of contingent claims (also known as derivatives) is to specify a stochastic process for the underlying asset and then construct a dynamic, self-financing hedging portfolio to minimize risk. The initial cost of constructing the portfolio is then considered to be the fair value of the contingent claim. This has been used with great success in the case of stochastic processes having constant volatility in the case of both European and (the more difficult) American options.However, it is well known that a financial model which follows a stochastic process having constant volatility is not consistent with market prices. Recent research has shown that models based on stochastic volatility, jump diffusion, and regime switching processes produce better fits to market data. A nonexhaustive list of regime switching applications includes insurance [22] [3], and interest rate dynamics [27]. Regime switching models are intuitively appealing, and computationally inexpensive compared to a stochastic volatility jump diffusion model.In this paper we study numerical techniques for the solution of American option contracts under regime switching. While our examples focus on problems with constant properties in each regime, the numerical methods developed can easily be applied to cases where the properties in each regime are more complex. An example would be the use of price-dependent regime switching (i.e., default hazard rates) in convertible bond pricing [4]. A number of different methods has been proposed for handling American options under regime switching models. Semianalytic approaches have been

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Section: Fixed Point Policy Iterationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Methods for Pricing American Options under Regime Switching

Huang¹,

Forsyth²,

Labahn³

2011

SIAM J. Sci. Comput.

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“…The direct control technique was previously suggested for solving American option type problems [16,5]. In addition, we consider the case where the underlying risky asset follows a jump diffusion process [9].…”

Section: Introductionmentioning

confidence: 99%

“…The method makes use of a fully implicit discretization which is shown to be monotone, consistent and l ∞ stable. We introduce a scaling factor in our direct control formulation, which allows us to solve the associated nonlinear algebraic equations using the fixed-point policy iteration method of [16]. We make use of some of the general results derived in [16].…”

Section: Introductionmentioning

confidence: 99%

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

Lu,

Dang

2023

Numerical Methods Partial

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We 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.

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Combined Fixed Point and Policy Iteration for Hamilton--Jacobi--Bellman Equations in Finance

Cited by 27 publications

References 34 publications

Methods for Pricing American Options under Regime Switching

Methods for Pricing American Options under Regime Switching

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

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

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