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32
Equilibrium Forward Curves for Commodities
 Journal of Finance
, 2000
"... We develop an equilibrium model of the term structure of forward prices for storable commodities. As a consequence of a nonnegativity constraint on inventory, the spot commodity has an embedded timing option that is absent in forward contracts. This option’s value changes over time due to both endog ..."
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Cited by 40 (2 self)
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We develop an equilibrium model of the term structure of forward prices for storable commodities. As a consequence of a nonnegativity constraint on inventory, the spot commodity has an embedded timing option that is absent in forward contracts. This option’s value changes over time due to both endogenous inventory and exogenous transitory shocks to supply and demand. Our model makes predictions about volatilities of forward prices at different horizons and shows how conditional violations of the “Samuelson effect ” occur. We extend the model to incorporate a permanent second factor and calibrate the model to crude oil futures data. COMMODITY MARKETS IN RECENT YEARS have experienced dramatic growth in trading volume, the variety of contracts, and the range of underlying commodities. Market participants are also increasingly sophisticated about recognizing and exercising operational contingencies embedded in delivery contracts. 1 For all of these reasons, there is a widespread interest in models for pricing and hedging commoditylinked contingent claims. In this paper we present an equilibrium model of commodity spot and forward prices. By explicitly incorporating the microeconomics of supply, demand, and storage, our model captures some fundamental differences between commodities and financial assets. Empirically, commodities are strikingly different from stocks, bonds and other conventional financial assets. Among these differences are:
Equilibrium Commodity Prices with Irreversible Investment and NonLinear Technologies
, 2005
"... We model equilibrium spot and futures oil prices in a general equilibrium production economy. In our model production of the consumption good requires two inputs: the consumption good and a commodity, e.g., Oil. Oil is produced by wells whose flow rate is costly to adjust. Investment in new Oil well ..."
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Cited by 11 (2 self)
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We model equilibrium spot and futures oil prices in a general equilibrium production economy. In our model production of the consumption good requires two inputs: the consumption good and a commodity, e.g., Oil. Oil is produced by wells whose flow rate is costly to adjust. Investment in new Oil wells is costly and irreversible. As a result in equilibrium, investment in Oil wells is infrequent and lumpy. Even though the state of the economy is fully described by a onefactor Markov process, the spot oil price is not Markov (in itself). Rather it is best described as a regimeswitching process, the regime being an investment ‘proximity’ indicator. The resulting equilibrium oil price exhibits meanreversion and heteroscedasticity. Further, the risk premium for exposure to commodity risk is timevarying, positive in the farfrominvestment regime but negative in the nearinvestment regime. Further, our model captures many of the stylized facts of oil futures prices, such as backwardation and the ‘Samuelson effect.’ The futures curve exhibits backwardation as a result of a convenience yield, which arises endogenously. We estimate our model using the Simulated Method of
Unspanned stochastic volatility and the pricing of commodity derivatives
"... Commodity derivatives are becoming an increasingly important part of the global derivatives market. Here we develop a tractable stochastic volatility model for pricing commodity derivatives. The model features unspanned stochastic volatility, quasianalytical prices of options on futures contracts, ..."
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Cited by 6 (0 self)
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Commodity derivatives are becoming an increasingly important part of the global derivatives market. Here we develop a tractable stochastic volatility model for pricing commodity derivatives. The model features unspanned stochastic volatility, quasianalytical prices of options on futures contracts, and dynamics of the futures curve in terms of a lowdimensional affine state vector. We estimate the model on NYMEX crude oil derivatives using an extensive panel data set of 45,517 futures prices and 233,104 option prices, spanning 4082 business days. We find strong evidence for two, predominantly unspanned, volatility factors. JEL Classification: G13
An equilibrium analysis of exhaustable resources investments
, 2002
"... Abstract. We develop a general equilibrium model of an extractable resource market where both the prices and extraction choices are determined endogenously. The model generates price dynamics that are roughly consistent with observed oil and gas forward and option prices as well as with the twofact ..."
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Cited by 5 (0 self)
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Abstract. We develop a general equilibrium model of an extractable resource market where both the prices and extraction choices are determined endogenously. The model generates price dynamics that are roughly consistent with observed oil and gas forward and option prices as well as with the twofactor price processes that were calibrated in Schwartz (1997). However, the subtle di erences between the endogenous price process determined within our general equilibrium model and the exogenous processes considered in earlier papers can generate signi cant di erences in both nancial and real option values. 1.
Equilibrium forward contracts on nonstorable commodities in the presence of market power
 Operations Research
"... We consider an equilibrium forward contract on a nonstorable commodity when forward market participants have market powers. The forward contract is negotiated through a Nash bargaining process due to market powers. We derive the unique equilibrium forward contract in closed form and provide an exten ..."
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We consider an equilibrium forward contract on a nonstorable commodity when forward market participants have market powers. The forward contract is negotiated through a Nash bargaining process due to market powers. We derive the unique equilibrium forward contract in closed form and provide an extensive comparative statics analysis. We show in particular that the introduction of a forward market may increase both the production of the commodity and the trading volume in the spot market. We then calibrate our model to an electricity data set and conduct a numerical analysis. We find that in contrast to the forward price on a storable commodity, the forward price on a nonstorable can be nonmonotonic in the spot price. We show that the forward price can be a downward or an upward biased predictor of the spot price, depending on the convenience yield level and the market power. In addition, both the forward price volatility and the open interest volatility decrease with the time to maturity. Furthermore, for commodities with low (high) convenience yield, the open interest is greater for a shorter (longer) maturity forward.
Spot convenience yield models for the energy markets
 In Mathematics of finance, volume 351 of Contemp. Math
, 2004
"... Abstract. We review that part of the literature on energy spot price models which involves convenience yield as a factor, our goal being to document the shortcomings of the most commonly used models. From a mathematical point of view, the introduction of the convenience yield is usually justified by ..."
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Cited by 4 (2 self)
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Abstract. We review that part of the literature on energy spot price models which involves convenience yield as a factor, our goal being to document the shortcomings of the most commonly used models. From a mathematical point of view, the introduction of the convenience yield is usually justified by the desire to reconciliate dynamical models for the time evolution of commodity prices with standard arbitrage theory. Since the convenience yield appears as a factor which cannot be observed directly, stochastic filtering has been proposed as a strategy of choice for its estimation from observed market prices. We implement these ideas on the models we review, and on some natural extensions. We illustrate the inconsistencies of the spot models on readily available data, paving the way for the empirical analysis of models of the term structure of convenience yield recently proposed as a viable alternative. 1.
Market making and mean reversion
 In Proc. ACM Conf. on Elec. Commerce
, 2011
"... Market making refers broadly to trading strategies that seek to profit by providing liquidity to other traders, while avoiding accumulating a large net position in a stock. In this paper, we study the profitability of market making strategies in a variety of time series models for the evolution of a ..."
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Cited by 4 (2 self)
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Market making refers broadly to trading strategies that seek to profit by providing liquidity to other traders, while avoiding accumulating a large net position in a stock. In this paper, we study the profitability of market making strategies in a variety of time series models for the evolution of a stock’s price. We first provide a precise theoretical characterization of the profitability of a simple and natural market making algorithm in the absence of any stochastic assumptions on price evolution. This characterization exhibits a tradeoff between the positive effect of local price fluctuations and the negative effect of net price change. We then use this general characterization to prove that market making is generally profitable on mean reverting time series — time series with a tendency to revert to a longterm average. Mean reversion has been empirically observed in many markets, especially foreign exchange and commodities. We show that the slightest mean reversion yields positive expected profit, and also obtain stronger profit guarantees for a canonical stochastic mean reverting process, known as the OrnsteinUhlenbeck (OU) process, as well as other stochastic mean reverting series studied in the finance literature. We also show that market making remains profitable in expectation for the OU process even if some realistic restrictions on trading frequency are placed on the market maker.
Bargaining and Pricing in Networked Economic Systems
, 2011
"... Economic systems can often be modeled as games involving several agents or players who act according to their own individual interests. Our goal is to understand how various features of an economic system affect its outcomes, and what may be the best strategy for an individual agent. In this work, w ..."
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Cited by 1 (1 self)
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Economic systems can often be modeled as games involving several agents or players who act according to their own individual interests. Our goal is to understand how various features of an economic system affect its outcomes, and what may be the best strategy for an individual agent. In this work, we model an economic system as a combination of many bilateral economic opportunities, such as that between a buyer and a seller. The transactions are complicated by the existence of many economic opportunities, and the influence they have on each other. For example, there may be several prospective sellers and buyers for the same item, with possibly differing costs and values. Such a system may be modeled by a network, where the nodes represent players and the edges represent opportunities. We study the effect of network structure on the outcome of bargaining among players, through theoretical
November 1996
"... This paper presents an equilibrium model of the term structure of forward prices for storable commodities. Our approach differs from Brennan (1991) and Schwartz (1997) in that we do not explicitly assume an exogenous "convenience yield." Rather, our spot commodity has an embedded timing option th ..."
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This paper presents an equilibrium model of the term structure of forward prices for storable commodities. Our approach differs from Brennan (1991) and Schwartz (1997) in that we do not explicitly assume an exogenous "convenience yield." Rather, our spot commodity has an embedded timing option that is absent in forward contracts, which arises from a nonnegativity constraint on inventory. The value of this option changes over time as a function of both the endogenous inventory and exogenous transitory shocks to supply and demand. Our model makes predictions about the volatilities of forward prices at different horizons and shows how conditional violations of the "Samuelson effect" can occur. We address the related issue of dynamically trading neardated forward contracts to hedge a longdated position. We also present a tractable extension of the model with a permanent second factor and a calibration of the model to crude oil futures price data.
Real Options Meet Hotelling Problem: Optimal Investment in Extraction Capacity of Exhaustible Resources
, 2009
"... Preliminary Version Abstract: This paper extends the literature of real options and exhaustible resource economics by examining the investment decisions of an active exhaustible resource monopolist. With demand uncertainty and endogenous price dynamics, the monopolist optimally chooses both the prod ..."
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Preliminary Version Abstract: This paper extends the literature of real options and exhaustible resource economics by examining the investment decisions of an active exhaustible resource monopolist. With demand uncertainty and endogenous price dynamics, the monopolist optimally chooses both the production rates and times to build extra capacity. The capacity expansion option for such a firm is modeled as a twodimensional option on demand shocks and remaining reserves. Using a discretetime simulation, first the dynamics of option prices and its sensitivity to different parameters is calculated. Unlike previous literature which implies that all investments should happen in the beginning, I show that there is an optimal time to invest different from time zero. Furthermore, it is shown that the consideration of option value in investment decisions will lead the producer to choose a more conservative expansion policy and therefore causes higher prices in strong demand shock periods. Finally, the optimal production rate of the producer will change by the introduction of option feature to the problem. The findings of this paper may explain why we do not observe in practice the predictions of Hotelling rule regarding increasing prices and decreasing production rates.