10.2 The theory of storage

Suppose $F_0(T) > S_0 e^{(r+u)T}$. At $t=0$: borrow $S_0 e^{uT}$; buy $e^{uT}$ units of the commodity; sell one futures contract at $F_0(T)$. Net cost zero. Storage paid in kind leaves $e^{uT}\cdot e^{-uT} = 1$ unit at $T$. At $T$: deliver it against the futures, receive $F_0(T)$, repay $S_0 e^{uT} e^{rT}$. Profit $F_0(T) - S_0 e^{(r+u)T} > 0$ in every state, with zero initial outlay: an arbitrage. With rupee storage cost: borrow $S_0 + U$, buy one unit and prepay storage, deliver at $T$; profit $F_0(T) - (S_0 + U)e^{rT}$.∎

The agent's static portfolio is $x \ge 0$ units of the commodity bought at $0$ (stored in kind), $f$ futures (long if $f>0$), and $\beta$ rupees in the bond. Zero cost: $\beta = -x S_0$. Payoff at $T$:

$V_T$ is affine in $S_T$, so $V_T \ge 0$ for every $S_T \in [0, \infty)$ if and only if (i) the slope is non-negative, $x e^{-uT} + f \ge 0$, and (ii) the intercept is non-negative, $-fF_0 - xS_0 e^{rT} \ge 0$. Take $0 < F_0 < S_0 e^{(r+u)T}$. Condition (ii) gives $f \le -x S_0 e^{rT}/F_0$, and (i) gives $f \ge -x e^{-uT}$. Both hold only if $-x e^{-uT} \le -x S_0 e^{rT}/F_0$, i.e. $x e^{-uT} \ge x S_0 e^{rT}/F_0$, i.e. $x\,(F_0 - S_0e^{(r+u)T}) \ge 0$, which with $F_0 < S_0 e^{(r+u)T}$ forces $x = 0$. Then (i) and (ii) read $f \ge 0$ and $-fF_0 \ge 0$, so $f = 0$. The only zero-cost portfolio with a non-negative payoff is the empty one: no arbitrage, however low $F_0$ is.

What about $F_0 \le 0$? Then buying futures ($x = 0$, $f > 0$) has payoff $f(S_T - F_0) > 0$, which looks like an arbitrage. But that payoff assumes the long can take delivery and dispose of the commodity at $S_T$. A long without access to storage or transport at the delivery point cannot; at expiry it must pay someone to take the barrels. Its true payoff is $f(-k_T - F_0)$, where $k_T$ is the cost of disposal, which is negative when $k_T > |F_0|$. So even the bound $F \ge 0$ is not enforced by arbitrage when storage is unavailable.

For the inventory holder, $x$ may be negative down to $-I$. Selling one unit from inventory at $S_0$, investing at $r$, saving the storage, and buying one futures yields at $T$ the same unit back plus $S_0 e^{(r+u)T} - F_0$, minus the value of the services the unit would have provided between $0$ and $T$. Writing those services as a continuous yield $\ell$ on the unit's value gives (10.3) by the same algebra as Proposition 10.1 with signs reversed. For gold, $\ell$ is observable: it is the lease rate at which central banks and bullion banks lend metal.∎

A storer who buys one more unit at $t$ pays $S_t + k$ and expects $\mathbb{E}_t S_{t+1}/(1+r)$ in present value. Free entry means expected profit cannot be positive: if it were, storers would buy more, raising $S_t$ (consumption falls) and lowering $\mathbb{E}_t S_{t+1}$ (more carried into $t+1$) until the gap closes. If expected profit is negative and $I_t > 0$, storers release stocks: carry-out falls, this period's price falls and next period's expected price rises, until either the gap closes or $I_t = 0$. These are exactly the Kuhn–Tucker conditions (10.5).∎

$s'(I) = -y'(I) > 0$; $s''(I) = -y''(I) < 0$; $\lim_{I\to\infty} s(I) = r + u - 0$, and since $y > 0$, $s < r + u$ everywhere; $y$ is continuous and strictly decreasing from $\bar y > r + u$ toward $0 < r + u$, so by the intermediate value theorem there is a unique $I^*$ with $s(I^*) = 0$ and $s < 0$ on $[0, I^*)$.∎