Derivatives: Valuation and Risk Management

By David A. Dubofsky; Thomas W. Miller Jr. | Go to book overview

CHAPTER 5

Determining Forward Prices and
Futures Prices

This chapter explains the theory of forward pricing. The forward price of a commodity is the price that is quoted today for delivery of the commodity in the future; the price is contracted today but is paid when the good is delivered in the future. Forward exchange rates exist for the forward purchase or sale of foreign exchange. Forward interest rates are rates that exist for future borrowing and lending opportunities.

All the ideas introduced in this chapter also apply to the pricing of futures contracts. In Chapter 6, we explain why futures prices and forward prices might differ. Since, however, the forces that might cause futures prices to differ from forward prices do not seem to be important, we can fairly safely conclude that they ought to be identical.

In theory, forward prices are determined by the force of arbitrage. In other words, if forward prices are “wrong,” then arbitrage opportunities will exist. A fundamental assertion in the theory of finance is that well-functioning markets will not permit arbitrage. When arbitrage opportunities exist, markets are not in equilibrium (i.e., supply does not equal demand), and traders will quickly act to exploit arbitrage profits. (An arbitrage profit is riskless, involving a positive cash inflow at one or more dates, and zero cash flows at all dates. In other words, arbitrage requires no investment and no cash outlay. The arbitrageur generates only cash inflows at one or more dates.)

In the first section of this chapter, we discuss the cost-of-carry model for forward commodity prices. Under perfect markets assumptions, this model should determine one forward price. However, markets are imperfect; in particular, the convenience yield must be considered when one is determining the theoretical forward prices of commodities. The convenience yield is a unobservable theoretical construct that arises because there are problems in short selling commodities, and those who own the commodities may be reluctant to sell them. Because of this, it turns out that the cost-of-carry model determines only a maximum forward price that precludes arbitrage. Actual forward prices can be less than the theoretically correct forward price.

In contrast, however, this short selling problem does not exist for financial assets such as foreign currencies and interest rates. Thus, ignoring transaction costs, we will be able to compute one single forward price for these items. Theoretical forward exchange rates are computed in Section 5.2 and theoretical forward interest rates in Section 5.3.


5.1 FORWARD COMMODITY PRICES

5.1.1 The Cost-of-Carry Model

Assume that markets are perfect. This means that there are no transactions costs: no commissions or bid-ask spreads. There are no taxes. Market participants can buy or sell goods without affecting

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