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.

http://www.cmap.polytechnique.fr/~valdo/papers/siam.pdf

Pricing and Hedging Spread Options

With the formation of a formal market for the trading of financial futures in October 1975, a renewed interest in the futures contract as an investment vehicle has emerged. The traditional approach was to view investing in futures as a way of off setting potential price risk associated with a given spot position. While these descriptive scenarios (see [3], [6], [10], [12], [13], [14], and [19]) adequately illustrate the traditional hedging strategy, their simplifying assumptions introduce a lack of realism into the investment process. The implication drawn from many of these articles is that, if one is interested in risk reduction, one should simply take the opposite position in the appropriate number of futures contracts to totally offset one's existing spot position.

A Risk-Return Measure of Hedging Effectiveness

This article investigates the long-term pricing relationship among crude oil, unleaded gasoline, and heating oil futures prices, and finds that these commodities futures prices are cointegrated. The study finds that the spreads between crude oil and its end products are stationary. Furthermore, this article investigates the risk arbitrage opportunities in three types of popularly traded petroleum futures spreads and finds that historically profitable risk arbitrage opportunities existed and were statistically significant. However, one cannot be certain that these opportunities still exist. The research also finds that moving averages are valid test variables for measuring spreads. Statistical and tabular constructions are used to illustrate findings. © 1999 John Wiley & Sons, Inc., Jrl Fut Mark 19: 931–955, 1999

Risk arbitrage opportunities in petroleum futures spreads

Beginning with Black and Scholes, it has been mathematically very convenient to model asset returns as logarithmic diffusions when developing pricing models for derivatives. But doing so causes a lot of trouble for instruments such as spread options, whose payoffs are a function of the arithmetic difference between two such assets, because the distribution of the price difference is not lognormal or any other familiar distribution. One could simply assume the difference is lognormal and obtain a simple model, but that would lead to an internal inconsistency with the models used for the component securities. Alternatively, one must develop numerical or approximation techniques to deal with this problem. In this article, the authors advance the technology by devising analytic formulas for spread option values, as well as their greeks, based on an approximation to the exercise boundary for one asset9s price, conditional on the other price. The result is much more efficient and accurate valuation than with previous approaches

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Closed-Form Approximations for Spread Option Prices and Greeks

We develop a new closed-form approximation method for pricing spread options. Numerical analysis shows that our method is more accurate than existing analytical approximations. Our method is also extremely fast, with computing time more than two orders of magnitude shorter than one-dimensional numerical integration. We also develop closed-form proximations for the greeks of spread options. In addition, we analyze the price sensitivities of spread options and provide lower and upper bounds for digital spread options. Our method enables the accurate pricing of a bulk volume of spread options with different specifications in real time, which offers traders a potential edge in financial markets. The closed-form approximations of greeks serve as valuable tools in financial applications such as dynamic hedging and Value-at-Risk calculations. The availability of a closed-form formula for spread options also helps us understand and design real and financial contracts with embedded spread-option-like features.

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Frank J. Jones

Papers

Spreads: Tails, turtles, and all that

The economics of futures and options contracts based on cash settlement

The integration of the cash and futures markets for treasury securities