We develop a model to value options on commodity futures in the presence of stochastic interest rates as well as stochastic convenience yields. In the development of the model, we distinguish between forward and future convenience yields, a distinction that has not been recognized in the literature. Assuming normality of continuously compounded forward interest rates and convenience yields and log-normality of the spot price of the underlying commodity, we obtain closed-form solutions generalizing the Black-Scholes/Merton's formulas. We provide numerical examples with realistic parameter values showing that both the effect of introducing stochastic convenience yields into the model and the effect of having a short time lag between the maturity of a European call option and the underlying futures contract have significant impact on the option prices.

I propose several mean-reversion jump-di#usion models to describe spot prices of energy commodities that maybevery costly to store. I incorporate multiple jumps, regime-switching and stochastic volatilityinto these models in order to capture the salient features of energy commodity prices due to physical characteristics of energy commodities. Prices of various energy commodity derivatives are derived under each model using the Fourier transform methods. In the context of deregulated electric power industry, I construct a real options approachtovalue physical assets such as generation and transmission facilities. The implications of modeling assumptions to the valuation of real assets are also examined.

This paper presents and applies a methodology for valuing electricity derivatives by constructing replicating portfolios from electricity futures and the risk free asset. Futures based replication is argued to be made necessary by the non-storable nature of electricity, which rules out the traditional spot mar-ket, storage-based method of valuing commodity derivatives. Using the futures based approach, valuation formulae are derived for both spark and locational spread options for both geometric Brownian motion and mean reverting price processes. These valuation results are in turn used to construct real options based valuation formulae for generation and transmission assets. Finally, the valuation formula derived for generation assets is used to value a sample of

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 commodity-linked 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:

Most firms do not make explicit use of real option techniques in evaluating investments. Nevertheless, real option considerations can be a significant component of value, and firms which approximately take them into account should outperform firms which do not. This paper asks whether the use of seemingly arbitrary investment criteria, such as hurdle rates and profitability indexes, can proxy for the use of more sophisticated real options valuation. We find that for a variety of parameters, particular hurdle-rate and profitability index rules can provide close-to-optimal investment decisions. Thus, it may be that firms using seemingly arbitrary “rules of thumb ” are approximating optimal decisions.

Abstract. We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.

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The value of a gas fired power plant depends on the spark spread, defined as the difference between the unit price of electricity and the cost of gas. We model the spark spread using two-factor model, allowing mean-reversion in short-term variations and uncertainty in the equilibrium price to which prices revert. We analyze two types of gas plants. The first type is a base load plant, generating electricity at all levels of spark spread. The second type is also a base load plant from the outset, but can be upgraded, at a cost, to a peak load plant generating electricity only when spark spread exceeds emission costs. We compute optimal building and upgrading thresholds for such plants when the plant types are mutually exclusive. Our results indicate that selecting a project which is first profitable leads to a nonoptimal investment policy, and that increase in short-term volatility preempts upgrading whereas increase in equilibrium volatility delays upgrading. Key words: Real options, spark spread, gas fired power plants, investment flexibility, mutually exclusive projects

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