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59
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 31 (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 Fourier transform method for spread option pricing. Working paper
, 2009
"... Spread options are a fundamental class of derivative contract written on multiple assets, and are widely used in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient and flexi ..."
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Cited by 24 (1 self)
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Spread options are a fundamental class of derivative contract written on multiple assets, and are widely used in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient and flexible enough to apply in general models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the spread option payoff function. Our detailed investigation proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of prices. It is found to be easy to implement, stable, efficient and applicable in a wide variety of asset pricing models.
Optimal switching with application to energy tolling agreements
, 2005
"... We consider the problem of optimal switching with finite horizon. This special case of stochastic impulse control naturally arises during analysis of operational flexibility of exotic energy derivatives. The current practice for such problems relies on Markov decision processes that have poor dimens ..."
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Cited by 24 (4 self)
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We consider the problem of optimal switching with finite horizon. This special case of stochastic impulse control naturally arises during analysis of operational flexibility of exotic energy derivatives. The current practice for such problems relies on Markov decision processes that have poor dimensionscaling properties, or on strips of spark spread options that ignore the operational constraints of the asset. To overcome both of these limitations, we propose a new framework based on recursive optimal stopping. Our model demonstrates that the optimal dispatch policies can be described with the aid of ‘switching boundaries’, similar to standard American options. In turn, this provides new insight regarding the qualitative properties of the value function. Our main contribution is a new method of numerical solution based on Monte Carlo regressions. The scheme uses dynamic programming to simultaneously approximate the optimal switching times along all the simulated paths. Convergence analysis is carried out and numerical results are illustrated with a variety of concrete
Pricing asset scheduling flexibility using optimal switching
 Applied Mathematical Finance
, 2008
"... www.umich.edu / mludkov We study the financial engineering aspects of operational flexibility of energy assets. The current practice relies on a representation that uses strips of European sparkspread options, ignoring the operational constraints. Instead, we propose a new approach based on a stoch ..."
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Cited by 21 (5 self)
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www.umich.edu / mludkov We study the financial engineering aspects of operational flexibility of energy assets. The current practice relies on a representation that uses strips of European sparkspread options, ignoring the operational constraints. Instead, we propose a new approach based on a stochastic impulse control framework. The model reduces to a cascade of optimal stopping problems and directly demonstrates that the optimal dispatch policies can be described with the aid of ‘switching boundaries’, similar to the free boundaries of standard American options. Our main contribution is a new method of numerical solution relying on Monte Carlo regressions. The scheme uses dynamic programming to efficiently approximate the optimal dispatch policy along the simulated paths. Convergence analysis is carried out and results are illustrated with a variety of concrete examples. We benchmark and compare our scheme to alternative numerical methods.
Schleicher (2006), Pricing multivariate currency options with copulas
 in Copulas, from Theory to Application in Finance
"... Multivariate options are widely used when there is a need to hedge against a number of risks simultaneously; such as when there is an exposure to several currencies or the need to provide cover against an index such as the FTSE100, or indeed any portfolio of assets. ..."
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Cited by 16 (1 self)
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Multivariate options are widely used when there is a need to hedge against a number of risks simultaneously; such as when there is an exposure to several currencies or the need to provide cover against an index such as the FTSE100, or indeed any portfolio of assets.
Electricity price modelling and asset valuation: A multifuel structural approach. Working paper
, 2012
"... ABSTRACT. We introduce a new and highly tractable structural model for spot and derivative prices in electricity markets. Using a stochastic model of the bid stack, we translate the demand for power and the prices of generating fuels into electricity spot prices. The stack structure allows for a ran ..."
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Cited by 11 (2 self)
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ABSTRACT. We introduce a new and highly tractable structural model for spot and derivative prices in electricity markets. Using a stochastic model of the bid stack, we translate the demand for power and the prices of generating fuels into electricity spot prices. The stack structure allows for a range of generator efficiencies per fuel type and for the possibility of future changes in the merit order of the fuels. The derived spot price process captures important stylized facts of historical electricity prices, including both spikes and the complex dependence upon its underlying supply and demand drivers. Furthermore, under mild and commonly used assumptions on the distributions of the input factors, we obtain closedform formulae for electricity forward contracts and for spark and dark spread options. As merit order dynamics and fuel forward prices are embedded into the model, we capture a much richer and more realistic dependence structure than can be achieved by classical reducedform models. We illustrate these advantages by comparing with Margrabe’s formula and a simple cointegration model, and highlight important implications for the valuation of power plants. Electricity markets and structural model and forward prices and spread options and power plant valuation
A survey of commodity markets and structural models for electricity prices
 Financial Engineering for Energy Asset Management and Hedging in Commodity Markets; Proceedings from the special thematic year at the Wolfgang Pauli Institute
, 2012
"... Abstract. The goal of this survey is to review the major idiosyncrasies of the commodity markets and the methods which have been proposed to handle them in spot and forward price models. We devote special attention to the most idiosyncratic of all: electricity markets. Following a discussion of trad ..."
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Cited by 9 (1 self)
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Abstract. The goal of this survey is to review the major idiosyncrasies of the commodity markets and the methods which have been proposed to handle them in spot and forward price models. We devote special attention to the most idiosyncratic of all: electricity markets. Following a discussion of traded instruments, market features, historical perspectives, recent developments and various modeling approaches, we focus on the important role of other energy prices and fundamental factors in setting the power price. In doing so, we present a detailed analysis of the structural approach for electricity, arguing for its merits over traditional reducedform models. Building on several recent articles, we advocate a broad and flexible structural framework for spot prices, incorporating demand, capacity and fuel prices in several ways, while calculating closedform forward prices throughout. 1.
1 Long Term Spread Option Valuation and Hedging
"... This paper investigates the valuation and hedging of spread options on two commodity prices which in the long run are cointegrated. For long term option pricing the spread between the two prices should therefore be modelled directly. This approach offers significant advantages relative to the tradit ..."
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Cited by 7 (0 self)
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This paper investigates the valuation and hedging of spread options on two commodity prices which in the long run are cointegrated. For long term option pricing the spread between the two prices should therefore be modelled directly. This approach offers significant advantages relative to the traditional multifactor spread option pricing model since the correlation between two asset returns is notoriously hard to model. In this paper, we propose one and two factor models for spot spread processes under both the riskneutral and market measures. We develop pricing and hedging formulae for options on spot and futures spreads. Two examples of spread options in energy markets—the crack spread between heating oil and WTI crude oil and the location spread between Brent blend and WTI crude oil – are analyzed to illustrate the results. JEL classification: G12
Tail behavior of sums and differences of lognormal random variables, available at arXiv:1309.3057
, 2013
"... This article is dedicated to the memory of Peter Laurence We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated lognormal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior tur ..."
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Cited by 7 (1 self)
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This article is dedicated to the memory of Peter Laurence We present sharp tail asymptotics for the density and the distribution function of linear combinations of correlated lognormal random variables, that is, exponentials of components of a correlated Gaussian vector. The asymptotic behavior turns out to be determined by a subset of components of the Gaussian vector, and we identify the relevant components by relating the asymptotics to a tractable quadratic optimization problem. As a corollary, we characterize the limiting behavior of the conditional law of the Gaussian vector, given a linear combination of the exponentials of its components. Our results can be used either to estimate the probability of tail events directly or to construct efficient variance reduction procedures for precise estimation of these probabilities by Monte Carlo methods. They lead to important qualitative and quantitative insights concerning the behavior of individual stocks and portfolios during market downturns in the multidimensional BlackScholes model.
Energy spot price models and spread options pricing
 International Journal of Theoretical and Applied Finance
, 2007
"... In this article, we construct forward price curves and value a class of two asset exchange options for energy commodities. We model the spot prices using an affine twofactor meanreverting process with and without jumps. Within this modeling framework, we obtain closed form results for the forward ..."
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Cited by 6 (2 self)
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In this article, we construct forward price curves and value a class of two asset exchange options for energy commodities. We model the spot prices using an affine twofactor meanreverting process with and without jumps. Within this modeling framework, we obtain closed form results for the forward prices in terms of elementary functions. Through measure changes induced by the forward price process, we further obtain closed form pricing equations for spread options on the forward prices. For completeness, we address both an Actuarial and a riskneutral approach to the valuation problem. Finally, we provide a calibration procedure and calibrate our model to the NYMEX Light Sweet Crude Oil spot and futures data, allowing us to extract the implied market prices of risk for this commodity. 1.