Commodity Swap:

Swap in which at least one set of payments is based on the price of a commodity. The other set of payments can be either fixed or determined by some other floating price or rate. While payments could be made by delivering actual units of underlying commodity, in practice cash is exchanged instead.

http://www.russell.com/US/glossary/derivatives/commodity_swap.htm

Valuation Equation:

Let Si be the spot price of a commodity at the beginning of period i. Party A receives the spot price for N units of the commodity and pays a fixed amount, X, per period. We will assume that payments take place at the beginning of the period and there will be a total of M payments, beginning one period from now. The cash-flow as seen by the party that is long the swap is

C=N(0,S_{1}-X,S_{2}-X,…,S_{M}-X)

Note that this cash-flow is stochastic and so we cannot compute its present value directly by discounting. However, we can decompose C into a stream of fixed payments (of ¡NX) that we can easily price, and a stochastic stream, N(0; S1; S2; : : : ; SM). The stochastic stream is easily seen to be equivalent to a stream of forward contacts on N units of the commodity. We then see that receiving NSi at period i has the same value of receiving NFi at period i where Fi is the date 0 forward price for delivery of one unit of the commodity at date i. As the forward prices, Fi, are deterministic and known at date 0, we can see that the value of the commodity swap is given by

V=∑^{M}_{i=1}d(0,i)(F_{i}-X)

V = Value of the commodity swap

N = Units of commodity

M = # of payments (beginning one period from now)

i = period where forward price equals 0 for delivery of one unit of the commodity.

Fi = forward price for delivery of one unit of the commodity

X = Fixed amount paid per period