Unpublished memorandum, 2002.

It was shown in Geman and Vasicek (2001) (Chapter 29 of this volume) that the price of a forward contract maturing at T is subject to

1

where are Wiener processes under a risk-neutral probability measure equivalent to P.

Integrating Eq. (1) from 0 to T and taking into account that yields

2

Eq. (2) represents a complete specification of the forward/spot process. It is fully described by the forward contract volatilities, and it only includes processes whose stochastic properties under the measure P^* are known. Therefore, the prices of energy derivatives and contingent claims can be calculated without recourse to the market prices of risk, which are not directly observable. In this sense, it is akin to the Heath/Jarrow/Morton (1992) model of interest rates (their Eq. (26)).

The price of any derivative contract (e.g., a futures or a swap) is a martingale under the measure . The price of any derivative security (such as options, whether simple or compound, European, American, or Asian, etc.) expressed in units of the money market fund is also a martingale under . That is, if is the price of a derivative security, then the quantity is a martingale.

Specifically, the forward contract is priced as

3

A European option with a value at the expiration date T is priced as

4

A compound option paying the amount at time T, which is dependent on the spot prices at times , is valued as

5

These valuation relationships, applied to Eq. (2), give an exact meaning to the phrase that energy derivatives are priced off the forward price curve.

Write the dynamics of the spot price under the risk-neutral probability measure as

6

Then

where

is the slope of the forward price curve at the present date. If the commodity can be stored, the expected rate of return on the commodity under the risk-neutral measure is the risk-free rate, . This imposes the condition

for all that must be satisfied by the forward price curves. This is not so for nonstorable commodities, and the forward prices can be specified without restrictions.

Examples

Example A. Suppose is deterministic (so that interest rates are Gaussian under the risk-neutral measure) and assume that is also a deterministic function of t. Then the relationship of forward and future prices is given by

7

Example B. If the commodity is storable, then

8

and the futures contract price is given by

For a nonstorable commodity, we have

whenever Eq. (8) holds.

Example C. Assume that

Then

This is the Example 4 in Geman and Vasicek (2001).

Example D. Suppose are deterministic, and the forward price volatilities are independent of the contract maturity date T,

Then

This corresponds to the Example 3 in Geman and Vasicek (2001).