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30
An Approximate Dynamic Programming Approach to Benchmark Practicebased Heuristics for Natural Gas Storage Valuation
, 2008
"... The valuation of the real option to store natural gas is a practically important problem that entails dynamic optimization of inventory trading decisions with capacity constraints in the face of uncertain natural gas price dynamics. Stochastic dynamic programming is a natural approach to this valuat ..."
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Cited by 29 (10 self)
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The valuation of the real option to store natural gas is a practically important problem that entails dynamic optimization of inventory trading decisions with capacity constraints in the face of uncertain natural gas price dynamics. Stochastic dynamic programming is a natural approach to this valuation problem, but it does not seem to be widely used in practice because it is at odds with the highdimensional naturalgas price evolution models that are widespread among traders. According to the practicebased literature, practitioners typically value natural gas storage heuristically. The effectiveness of the heuristics discussed in this literature is currently unknown, because good upper bounds on the value of storage are not available. We develop a novel and tractable approximate dynamic programming method that coupled with Monte Carlo simulation computes lower and upper bounds on the value of storage, which we use to benchmark these heuristics on a set of realistic instances. We find that these heuristics are extremely fast but significantly suboptimal as compared to our upper bound, which appears to be fairly tight and much tighter than a simpler perfect information upper bound; our lower bound is slower to compute than these heuristics but substantially outperforms them in terms of valuation. Moreover, with periodic reoptimizations embedded in Monte Carlo simulation, the practicebased heuristics become nearly optimal, with one exception, at the expense of higher computational effort. Our lower bound with reoptimization is also nearly optimal, but exhibits a higher computational requirement than these heuristics. Besides natural gas storage, our results are potentially relevant for the valuation of the real option to store other commodities, such as metals, oil, and petroleum products.
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.
Valuation of energy storage: an optimal switching approach. Quantitative Finance
"... We consider the valuation of energy storage facilities within the framework of stochastic control. Our two main examples are natural gas dome storage and hydroelectric pumped storage. Focusing on the timing flexibility aspect of the problem we construct an optimal switching model with inventory. Thu ..."
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Cited by 24 (3 self)
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We consider the valuation of energy storage facilities within the framework of stochastic control. Our two main examples are natural gas dome storage and hydroelectric pumped storage. Focusing on the timing flexibility aspect of the problem we construct an optimal switching model with inventory. Thus, the manager has a constrained compound American option on the intertemporal spread of the commodity prices. Extending the methodology from Carmona and Ludkovski (2008), we then construct a robust numerical scheme based on Monte Carlo regressions. Our simulation method can handle a generic Markovian price model and easily incorporates many operational features and constraints. To overcome the main challenge of the pathdependent storage levels two numerical approaches are proposed. The resulting scheme is compared to the traditional quasivariational framework and illustrated with several concrete examples. We also consider related problems of interest, such as supply guarantees and mines management. Key words: gas storage; optimal switching; least squares Monte Carlo; hydro pumped storage; impulse control, commodity derivatives
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.
Optimal Trade Execution: A Mean–QuadraticVariation Approach
, 2009
"... We propose the use of a mean–quadraticvariation 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–quadraticvariation 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 priceindependent approximate optimal control for the meanvariance objective function in [2]. The GBM mean–quadraticvariation 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.
Comparison between the mean variance optimal and the mean quadratic variation optimal trading strategies.
, 2011
"... Abstract We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton Jacobi Bellman (HJB) partial differential equations (PDE). In particular, we compare the path dependent, timeconsistent meanquadraticvariation strategy with ..."
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Cited by 7 (0 self)
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Abstract We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton Jacobi Bellman (HJB) partial differential equations (PDE). In particular, we compare the path dependent, timeconsistent meanquadraticvariation strategy with the pathindependent, timeinconsistent (precommitment) meanvariance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the optimal trading velocities, the meanquadraticvariation strategy is much less sensitive to changes in asset price and varies more smoothly. In terms of the liquidation profiles, the meanvariance strategy strategy is much more variable, although the mean liquidation profiles for the two strategies are surprisingly similar. On a numerical note, we show that using an interpolation scheme along a parametric curve in conjunction with the semiLagrangian method results in significantly better accuracy than standard axisaligned linear interpolation. We also demonstrate how a scaled computational grid can improve solution accuracy.
Mine valuations in the presence of a Stochastic oregrade
 Int. Assoc. Eng. III
, 2010
"... AbstractMining companies worldwide are faced with the problem of how to accurately value and plan extraction projects subject to uncertainty in both future price and ore grade. Whilst the methodology of modelling price uncertainty is reasonably well understood, modelling oregrade uncertainty is ..."
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Cited by 5 (5 self)
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AbstractMining companies worldwide are faced with the problem of how to accurately value and plan extraction projects subject to uncertainty in both future price and ore grade. Whilst the methodology of modelling price uncertainty is reasonably well understood, modelling oregrade uncertainty is a much harder problem to formulate, and when attempts have been made the solutions take unfeasibly long times to compute. By treating the grade uncertainty as a stochastic variable in the amount extracted from the resource, this paper provides a new approach to the problem. We show that this method is wellposed, since it can realistically re
ect the geology of the situation, and in addition it enables solutions to be derived in the order of a few seconds. A comparison is made between a real mine valuation where the prior estimate of ore grade variation is taken as fact, and our approach, where we treat it as an uncertain estimate.
The expected lifetime of an extraction project
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467 244–263. doi:10.1098/rspa.2010.0247. URL http://rspa.royalsocietypublishing.org/content/ early/2010/07/22/rspa.2010.0247.abstract
, 2011
"... When a mining company begins extraction from a finite resource, it does so in the presence of numerous uncertainties. One key uncertainty is the future price of the commodity being extracted, since a large enough drop in price can make a resource no longer costeffective to extract, resulting in the ..."
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Cited by 5 (5 self)
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When a mining company begins extraction from a finite resource, it does so in the presence of numerous uncertainties. One key uncertainty is the future price of the commodity being extracted, since a large enough drop in price can make a resource no longer costeffective to extract, resulting in the mine being closed down. By specifying a stochastic price process, and implementing a financialtype model which leads to the use of partial differential equations, this paper creates the framework for efficiently capturing the probability of a mine remaining open throughout its planned extraction period, and derives the associated expected lifetime of extraction. An approximation to the abandonment price is described, which enables a closedform solution to be derived for the probability of operational success and expected lifetime. This approximation compares well with the full solution obtained using a semiLagrangian numerical technique.
Relaxations of Approximate Linear Programs for the Real Option Management of Commodity Storage
, 2012
"... The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs). This is due primarily to the high dimensionality of a commodity forward curve, which is part of the MDP state when using high dimensional models of the evolution of this curve, as co ..."
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Cited by 4 (3 self)
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The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs). This is due primarily to the high dimensionality of a commodity forward curve, which is part of the MDP state when using high dimensional models of the evolution of this curve, as commonly done in practice. Focusing on commodity storage, we develop a novel approximate dynamic programming methodology that hinges on the relaxation of approximate linear programs (ALPs) obtained using value function approximations based on reducing the number of futures prices that are part of the MDP state. We derive equivalent approximate dynamic programs (ADPs) for a class of these ALPs, also subsuming a known ADP. We obtain two new ADPs, the value functions of which induce feasible policies for the original MDP, and lower and upper bounds, estimated via Monte Carlo simulation, on the value of an optimal policy of this MDP. We investigate the performance of our ADPs on existing natural gas instances and new crude oil instances. Our approach has potential relevance for the approximate solution of MDPs that arise in the real option management of other commodity conversion assets, as well as the valuation and management of real and financial options that depend on forward curve dynamics.
The role of price spreads and reoptimization in the real option management of commodity storage assets. Working paper
"... The real option management of commodity storage assets is an important practical problem. Practitioners approach the resulting stochastic optimization model using heuristic policies that rely on sequential reoptimization of linear programs. Used in conjunction with Monte Carlo simulation, these poli ..."
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Cited by 3 (1 self)
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The real option management of commodity storage assets is an important practical problem. Practitioners approach the resulting stochastic optimization model using heuristic policies that rely on sequential reoptimization of linear programs. Used in conjunction with Monte Carlo simulation, these policies typically yield near optimal lower bound estimates on the value of storage. This paper reveals that a simple one stage lookahead policy is optimal for a fast storage asset without frictions. Thus, in this (not entirely realistic) case the problem is easy and the reoptimization policies are unnecessary, albeit optimal. In contrast, this paper provides numerical and structural justification for the use of these policies in the general case. Further, the use of price spreads simplifies the estimation of near tight dual upper bounds on the value of storage. This approach relies on using the fast and frictionless asset optimal value function to estimate dual upper bounds in the general case. Monte Carlo simulation and linear programming Storable commodity industries include storage assets embedded in physical markets for the commodity, and financial markets for commodity derivatives. These markets can be fairly competitive, as exemplified by the natural gas industry in North America and parts of Europe