Some numerical methods for solving stochastic impulse control in natural gas storage facilities

The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP

Regime switching in stochastic models of commodity prices: An application to an optimal tree harvesting problem

This paper investigates a regime switching model of stochastic lumber prices in the context of an optimal tree harvesting problem. Using lumber derivatives prices, two lumber price models are calibrated: a regime switching model and a single regime model. In the regime switching model, the lumber price can be in one of two regimes in which different mean reverting price processes prevail. An optimal tree harvesting problem is specified in terms of a linear complementarity problem which is solved using a fully implicit finite difference, fully-coupled, numerical approach. The land value and critical harvesting prices are found to be significantly different depending on which price model is used. The regime switching model shows promise as a parsimonious model of timber prices that can be incorporated into forestry investment problems.optimal tree harvesting, regime switching, calibration, lumber derivatives prices, fully implicit finite difference approach

Numerical Methods for Optimal Stochastic Control in Finance

In this thesis, we develop partial differential equation (PDE) based numerical methods to solve certain optimal stochastic control problems in finance. The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. The HJB equation corresponds to the case when the controls are bounded while the HJB variational inequality corresponds to the unbounded control case. As a result, the solution to the stochastic control problem can be computed by solving the corresponding HJB equation/variational inequality as long as the convergence to the viscosity solution is guaranteed. We develop a unified numerical scheme based on a semi-Lagrangian timestepping for solving both the bounded and unbounded stochastic control problems as well as the discrete cases where the controls are allowed only at discrete times. Our scheme has the following useful properties: it is unconditionally stable; it can be shown rigorously to converge to the viscosity solution; it can easily handle various stochastic models such as jump diffusion and regime-switching models; it avoids Policy type iterations at each mesh node at each timestep which is required by the standard implicit finite difference methods. In this thesis, we demonstrate the properties of our scheme by valuing natural gas storage facilities---a bounded stochastic control problem, and pricing variable annuities with guaranteed minimum withdrawal benefits (GMWBs)---an unbounded stochastic control problem. In particular, we use an impulse control formulation for the unbounded stochastic control problem and show that the impulse control formulation is more general than the singular control formulation previously used to price GMWB contracts

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer's utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.Comment: 32 pages, 5 figure

Characterizations of long-run producer optima and the short-runapproach to long-run market equilibrium: a general theory withapplications to peak-load pricing

This is a new formal framework for the theory of competitive equilibrium and its applications.Our "short-run approach" means the calculation of long-run producer optimaand general equilibria from the short-run solutions to the producer's profit maximizationprogramme and its dual. The marginal interpretation of the dual solution means that itcan be used to value the capital and other fixed inputs, whose levels are then adjustedaccordingly (where possible). But short-run profit can be a nondifferentiable function ofthe fixed quantities, and the short-run cost is nondifferentiable whenever there is a rigidcapacity constraint. Nondifferentiability of the optimal value requires the introductionof nonsmooth calculus into equilibrium analysis, and subdifferential generalizations ofsmooth-calculus results of microeconomics are given, including the key Wong-Viner EnvelopeTheorem. This resolves long-standing discrepancies between "textbook theory"and industrial experience. The other tool employed to characterise long-run produceroptima is a primal-dual pair of programmes. Both marginalist and programming characterizationsof producer optima are given in a taxonomy of seventeen equivalent systemsof conditions. When the technology is described by production sets, the most usefulsystem for the short-run approach is that using the short-run profit programme andits dual. This programme pair is employed to set up a formal framework for long-rungeneral-equilibrium pricing of a range of commodities with joint costs of production.This gives a practical method that finds the short-run general equilibrium en route tothe long-run equilibrium, exploiting the operating policies and plant valuations that mustbe determined anyway. These critical short-run solutions have relatively simple formsthat can greatly ease the fixed-point problem of solving for equilibrium, as is shownon an electricity pricing example. Applicable criteria are given for the existence of theshort-run solutions and for the absence of a duality gap. The general analysis is speltout for technologies with conditionally fixed coefficients, a concept extending that of thefixed-coefficients production function to the case of multiple outputs. The short-run approachis applied to the peak-load pricing of electricity generated by thermal, hydro andpumped-storage plants. This gives, for the first time, a sound method of valuing thefixed assets-in this case, river flows and the sites suitable for reservoirs.general equilibrium, fixed-input valuation, nondifferentiable joint costs,Wong-Viner Envelope Theorem, public utility pricing