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22
A semi-Lagrangian 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 Hamilton-Jacobi-Bellman (HJB) equation. In this paper, we present a semi-Lagrangian 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 Hamilton-Jacobi-Bellman (HJB) equation. In this paper, we present a semi-Lagrangian 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 mean-reverting, seasonality dynamics and price jumps. We develop fully implicit and Crank-Nicolson timestepping schemes based on a semi-Lagrangian 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 semi-Lagrangian 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 Hamilton-Jacobi-Bellman (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 Hamilton-Jacobi-Bellman (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 non-unique.
A Hamilton-Jacobi-Bellman approach to optimal trade execution
, 2009
"... The optimal trade execution problem is formulated in terms of a mean-variance tradeoff, as seen at the initial time. The mean-variance problem can be embedded in a Linear-Quadratic (LQ) optimal stochastic control problem, A semi-Lagrangian scheme is used to solve the resulting non-linear 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 mean-variance tradeoff, as seen at the initial time. The mean-variance problem can be embedded in a Linear-Quadratic (LQ) optimal stochastic control problem, A semi-Lagrangian scheme is used to solve the resulting non-linear 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.
Robust Numerical Valuation of European and American Options under the CGMY Process
, 2007
"... We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processe ..."
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Cited by 16 (1 self)
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We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semi-Lagrangian scheme. The resulting Partial Integro-Differential Equation (PIDE) is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.
Characterization of optimal stopping regions of American Asian and lookback options
- Mathematical Finance
, 2006
"... A general framework is developed to analyze the optimal stopping (exercise) regions of American path dependent options with either Asian feature or lookback feature. We examine the monotonicity properties of the option values and stopping regions with respect to the interest rate, dividend yield and ..."
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Cited by 11 (3 self)
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A general framework is developed to analyze the optimal stopping (exercise) regions of American path dependent options with either Asian feature or lookback feature. We examine the monotonicity properties of the option values and stopping regions with respect to the interest rate, dividend yield and time. From the ordering properties of the values of American lookback options and American Asian options, we deduce the corresponding nesting relations between the exercise regions of these American options. We illustrate how some properties of the exercise regions of the American Asian options can be inferred from those of the American lookback options.
Optimal Trade Execution: A Mean–Quadratic-Variation Approach
, 2009
"... We propose the use of a mean–quadratic-variation criteria to determine an optimal trading strategy in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian Motio ..."
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Cited by 8 (0 self)
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We propose the use of a mean–quadratic-variation criteria to determine an optimal trading strategy in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian Motion (GBM). We also derive the HJB PDE assuming that the trading horizon is small and that the underlying process can be approximated by Arithmetic Brownian Motion (ABM). The exact solution of the ABM formulation is in fact identical to the price-independent approximate optimal control for the mean-variance objective function in [2]. The GBM mean–quadratic-variation optimal trading strategy is in general a function of the asset price. However, for short term trading horizons, the control determined under the ABM assumption is an excellent approximation.
Penalty Methods for Continuous-Time Portfolio Selection with Proportional Transaction Costs
"... This paper is concerned with numerical solutions to a singular stochastic control problem arising from the continuous-time portfolio selection with proportional transaction costs. The associated value function is governed by a variational inequality with gradient constraints. We propose a penalty me ..."
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Cited by 7 (4 self)
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This paper is concerned with numerical solutions to a singular stochastic control problem arising from the continuous-time portfolio selection with proportional transaction costs. The associated value function is governed by a variational inequality with gradient constraints. We propose a penalty method to deal with the gradient constraints and employ a finite difference discretization. Convergence analysis is presented. We also show that the standard penalty method can be applied in the case of a single risky asset where the problem can be reduced to a standard variational inequality. Numerical results are given to demonstrate the efficiency of the methods and to examine the behaviour of the optimal trading strategy. 1
Regime Switching in Stochastic Models of Commodity Prices: An Application to an Optimal Tree Harvesting Problem
- Journal of Economic Dynamics and Control
"... problem ..."
Hedging with a Correlated Asset: Solution of a Nonlinear Pricing PDE ∗
, 2005
"... Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk. Even if the residual risk is considered diversifiable, the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the und ..."
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Cited by 4 (1 self)
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Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk. Even if the residual risk is considered diversifiable, the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument. If the residual risk is not considered diversifible, then this risk can be priced using an actuarial standard deviation principle in infinitesmal time. In both cases, these models result in the same nonlinear partial differential equation (PDE). A fully implicit, monotone discretization method is developed for solution of this pricing PDE. This method is shown to converge to the viscosity solution. Certain grid conditions are required to guarantee monotonicity. An algorithm is derived which, given an initial grid, inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied. At each timestep, the nonlinear discretized algebraic equations are solved using an iterative algorithm, which is shown to be globally convergent. Monte Carlo hedging examples are given to illustrate the standard deviation of the profit and loss distribution at the expiry of the option.
Efficient solution of backward jump-diffusion PIDEs with splitting and matrix exponentials
- Journal of Computational
"... We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on financial pro-cesses (diffusion and jumps) is applied to thes ..."
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Cited by 3 (3 self)
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We propose a new, unified approach to solving jump-diffusion partial integro-differential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a second-order operator splitting on financial pro-cesses (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finite-difference methods, which for multi-dimensional problems could also include splitting on various dimensions. For the jump part, we transform the jump integral into a pseudo-differential operator. Then for various jump models we show how to construct an appropriate first and second order approximation on a grid which supersets the grid that we used for the diffusion part. These approximations make the scheme to be unconditionally stable in time and preserve positivity of the solution which is computed either via a matrix exponential, or via Páde approximation of the matrix exponent. Various numerical experiments are provided to justify these results. 1
