10.1 What is different about commodities

Motivation

An equity is a claim on cash flows. A barrel of crude or a kilogram of silver pays nothing; it costs money to store, insure and move, and its value depends on where and when it is. Every commodity price you see is therefore a price for a place and a date. Most of the trading is in futures, and the relation between today's physical price and the futures curve carries more information than either price alone.

Setup and notation

We keep M07's notation.

$S_t$ : spot price, the price for immediate delivery at a named location (for crude, "physical" or "dated" prices; for gold, London spot or an Indian bullion rate).
$F_t(T)$ : futures price at $t$ for delivery at $T$ ; $\tau = T - t$ .
$r$ : continuously compounded risk-free rate; $u$ : storage cost as a proportional rate per year (warehouse, insurance, losses); $y$ : convenience yield, defined in 10.2.
Cost of carry $c = r + u - y$ , so that M07's relation reads $F_t(T) = S_t\, e^{c\,(T-t)} = S_t\, e^{(r+u-y)(T-t)}. \tag{10.1}$
Contango: $F_t(T)$ increasing in $T$ (here $c > 0$ ). Backwardation: $F_t(T)$ decreasing in $T$ ( $c < 0$ ). A curve can be backwardated at the front and in contango further out.
Basis: we use $b_t = F_t(T) - S_t$ (positive in contango). Hull uses $S - F$ ; watch the sign when reading other sources. The log basis $\ln(F_t/S_t) = c\tau$ .

As in M07, we treat futures and forwards as equal, which is exact when interest rates are deterministic. Futures are marked to market daily; forwards are not. Where the difference matters (for example, during a margin squeeze) we say so.

Three facts that drive the rest of the module

You cannot short-sell most physical commodities, because there is no general market for borrowing them. Gold and silver are exceptions: there is a lease market. This asymmetry kills the lower half of the no-arbitrage argument (10.2).
Inventories cannot be negative. When they run out, the spot price is set by current supply and demand alone, and it can rise far above the futures price. This is why spikes happen in the spot and the front of the curve.
Futures expire. A long position held to the last day becomes an obligation to take delivery at a specific place. That is how WTI printed $-\$37.63$ in April 2020 (10.11.1).