Chapter 30
Pricing of Energy Derivatives

Unpublished memorandum, 2002.

It was shown in Geman and Vasicek (2001) (Chapter 29 of this volume) that the price c30-math-0001 of a forward contract maturing at T is subject to

1 equation

where c30-math-0003 are Wiener processes under a risk-neutral probability measure c30-math-0004 equivalent to P.

Integrating Eq. (1) from 0 to T and taking into account that c30-math-0005 yields

2 equation

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 c30-math-0007. 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 c30-math-0008. That is, if c30-math-0009 is the price of a derivative security, then the quantity c30-math-0010 is a martingale.

Specifically, the forward contract is priced as

3 equation

A European option with a value c30-math-0012 at the expiration date T is priced as

4 equation

A compound option paying the amount c30-math-0014 at time T, which is dependent on the spot prices at times c30-math-0015, is valued as

5 equation

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 c30-math-0017 under the risk-neutral probability measure as

6 equation

Then

equation

where

equation

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, c30-math-0021. This imposes the condition

equation

for all c30-math-0023 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 c30-math-0024 is deterministic (so that interest rates are Gaussian under the risk-neutral measure) and assume that c30-math-0025 is also a deterministic function of t. Then the relationship of forward and future prices is given by

7 equation

Example B. If the commodity is storable, then

8 equation

and the futures contract price is given by

equation

For a nonstorable commodity, we have

equation

whenever Eq. (8) holds.

Example C. Assume that

equation

Then

equation

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

Example D. Suppose c30-math-0032 are deterministic, and the forward price volatilities are independent of the contract maturity date T,

equation

Then

equation

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

Reference

  1. Geman, H., and O. Vasicek. (2001). “Plugging into Electricity.” Risk, 14 (8), 93–106.
  2. Heath, D., R. Jarrow, and A. Morton. (1992). “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.” Econometrica, 60, pp. 77–105.
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