A pricing model for quantity contracts

Journal of Risk and Insurance, Dec, 2004 by Knut K. Aase

ABSTRACT

An economic model is proposed for a combined price futures and yield futures market. The innovation of the article is a technique of transforming from quantity and price to a model of two genuine pricing processes. This is required in order to apply modern financial theory. It is demonstrated that the resulting model can be estimated solely from data for a yield futures market and a price futures market. We develop a set of pricing formulas, some of which are partially tested, using price data for area yield options from the Chicago Board of Trade. Compared to a simple application of the standard Black and Scholes model, our approach seems promising.

INTRODUCTION

In the farming industry as well as for many other primary commodity producers it is possible to effectively manage price risk by the use of futures price contracts and options on futures. However, in many of these industries there is still considerable uncertainty left when it comes to revenue, since quantities produced can be volatile, depending on many factors, such as e.g., weather conditions in the growing season. Until recently, similar market-based instruments for managing yield risk have not been available. Instead, federal agricultural support programs and subsidized Crop Yield Insurance (CYI) programs have served as alternatives. In an important development in 1995, the Chicago Board of Trade (CBOT) has launched its CYI Futures and Options contracts. The first CYI contract that began trading on June 2, 1995 was Iowa Corn Yield Insurance Futures and Options. On January 19, 1996, the CBOT added a U.S. contract plus four additional state corn yield contracts for Illinois, Indiana, Ohio, and Nebraska. So far the trading volumes have been fairly modest. Regardless of the status of this particular market for the moment, we want to discuss such contracts from a principle point of view, and develop a pricing theory for this kind of markets. (1)

The CYI contracts are designed to provide a hedge for crop yield risk. For example, CYI futures users can lock in a certain crop yield several months into the future as a temporary substitute for a later yield-based commitment, or they can alternatively lock in the revenue of a given acreage by combining yield contracts with futures price contracts. (20

The emergence of markets like these can be thought of as a result of dynamic efficiency; if agents think that such instruments will improve economic efficiency, they will somehow be created.

The focus of this article is to construct a pricing model for yield futures and futures option contracts. The innovation is in the modeling stage. In order to apply modern financial theory, one has to start with genuine pricing models. The starting point here is, on the other hand, a model for yield and a model for the spot price of corn. A transformation is proposed in order to overcome this difficulty. It is demonstrated that the resulting technique is consistent with financial pricing theory, and also possible to implement in practice.

There is a large literature on non-market-based risk management and insurance of crop yield, which we will not address here. Yield contracts have been dealt with from the perspective of hedging, using a mean variance approach by Vukina, Li, and Holthausen (1996), while minimizing the variance of revenue was the objective in Li and Vukina (1998). In both these papers the yield contracts traded at CBOT are explained, so we need not elaborate on the market structure here.

There was another securitized insurance market at the CBOT centered around certain catastrophe indexes, these indexes playing a similar role to the yield index of the present article (e. g., Aase, 1999, 2001). The analysis of such markets must typically differ from the model chosen in the present article, since catastrophes cannot be modeled well by a continuous stochastic process.

The article is organized as follows. In the first section we present the economic model, which we develop in the subsequent section to a pricing model for any combination of yield and price futures and futures option contracts, like a futures contract on revenue (if it were to exist). In the third section, we specialize to pure yield contracts, where in Proposition 2 we present pricing formulas for yield futures and yield option contracts. These we calibrate and estimate from price data at the CBOT. Two proofs are relegated to Appendix A. The mechanism of using yield futures can best be illustrated by an example, which can be found in Appendix B. Some contract specifications are given in Appendix C. We round off the article with some remarks on risk management, related in particular to the example of Appendix B. The last section concludes.

AREA YIELD FUTURES AND OPTIONS

Introduction

Imagine a country, or another area, sectioned into regions that are uniform in terms of growing conditions for a certain crop, say corn. In each area there is a quantity index [y.sub.t], for time t running from 0 to T, where T is the time of sale and 0 is the time of sowing. As an example, for agricultural yield contracts in the United States traded at the CBOT the values of y are provided by the United States Department of Agriculture (USDA). One may think of [y.sub.t] as a forecast at each time t of quantity, measured in bushels per acre, up for sale in this specific region at the final time T. On this index we assume it is possible to trade futures, and futures options contracts. In order to bring in the quantum uncertainty, we assume that this index can be modeled as a stochastic process. A farmer in this region may have production uncertainty that is well represented by this index, where the relevant number of contracts can be determined from each farmer's production area.