Forwards and cost of carry

Definition A forward contract on one unit of an asset is an agreement made at time $0$ in which the long party will receive the unit at the maturity date $T$ and pay the delivery price $K$, fixed when the contract is made, while the short party delivers the unit and receives $K$. Nothing is paid when the contract is made or at any time before $T$. Valued in cash at $T$, the long position is worth

and the short position $K - S_T$, where $S_T$ is the ex-dividend spot price at $T$ in currency units per unit of the asset. A physically settled contract delivers the asset against $K$, and a cash-settled contract pays the difference $S_T - K$, which has the same value. In this lesson neither party defaults and collateral is ignored. The payoff is linear in $S_T$ and can be negative for either side, so a forward is an obligation for both parties, not an option.

Definition A prepaid forward on one unit of an asset is a contract in which the buyer pays the prepaid forward price $F^{P}_{0,T}$ at time $0$ and receives the unit at $T$. Assume a frictionless market in which the asset can be bought, held and sold short and zero-coupon bonds can be bought and sold. If the asset pays no income and has no holding costs before $T$, buying one unit at time $0$ and holding it delivers exactly the promised unit, so the absence of arbitrage forces

If the prepaid forward traded above $S_0$, selling it, buying the asset and investing the difference in a zero-coupon bond maturing at $T$ would cost nothing today and leave a positive amount at $T$ in every state. The opposite inequality is ruled out by the reverse trade, which needs a short sale of the asset. When the asset pays income before $T$, the holder of the asset receives it and the buyer of the prepaid forward does not, so the prepaid forward price is the spot price minus the present value of that income.

Definition Assume a frictionless, default-free market without transaction costs or taxes, in which zero-coupon bonds with prices $P(0,t) > 0$ can be bought and sold for every date $t \le T$ that is needed, so that borrowing and lending happen at the same rates, and in which the asset can be bought, held and sold short, with any income it pays passed on to the lender of a borrowed unit. The forward price $F_{0,T}$ is the delivery price that gives a newly made forward contract the value zero at time $0$. Under these assumptions

so for an asset without income or holding costs the forward price is $S_0/P(0,T)$. The argument is a cash-and-carry trade. If $K > S_0/P(0,T)$, buy one unit of the asset, pay for it by selling $S_0/P(0,T)$ zero-coupon bonds maturing at $T$, and enter a short forward at $K$. The cost at time $0$ is zero, and at $T$ the delivery brings in $K$ while the bonds require $S_0/P(0,T)$, so the surplus $K - S_0/P(0,T)$ is positive in every state, which is an arbitrage. If $K < S_0/P(0,T)$, the reverse trade, a short sale of the asset, a purchase of the bonds and a long forward, earns $S_0/P(0,T) - K$. The replication is static, so it needs no model for future interest rates or for the distribution of $S_T$. With a constant continuously compounded rate $r$, the bond price is $P(0,T) = e^{-rT}$ and the forward price is $S_0e^{rT}$. The forward price is an amount due at $T$, not a price paid today, and it is not a forecast of $S_T$.

Definition The cost of carry of an asset over $[0, T]$ is the net cost of holding it until delivery, the interest on the money tied up in its price plus holding costs such as storage, minus the income it pays. Suppose the asset pays known cash amounts $D_k$ at dates $t_k$ with $0 < t_k \le T$, fixed today and paid to whoever holds the asset under the dividend entitlement convention, and the unit delivered at $T$ is ex-dividend. The present value of this income is

and under the assumptions of the forward price card, with a short seller paying the dividends to the lender of the asset,

The cash-and-carry trade buys the asset, sells $D_k$ zero-coupon bonds maturing at each $t_k$ so that each dividend repays its bond, and finances the remaining $S_0 - I_0$ until $T$. Delivering the asset for $K$ at $T$ then leaves $K - F_{0,T}$, so a higher delivery price is an arbitrage, and the reverse trade with a borrowed asset rules out a lower one. Without the reverse trade, absence of arbitrage only bounds a quoted delivery price from above by $F_{0,T}$. Holding costs due at known dates enter like negative dividends and raise the forward price. Carrying every amount to $T$ gives the equivalent form

in which each dividend grows from its own date to $T$, by the factor $e^{r(T - t_k)}$ when the rate is a constant $r$.

Definition An asset has a continuous dividend yield $q$, a known constant, if each unit held pays income continuously at the rate $qS_t$ per year. Reinvesting all of this income in the asset makes a holding of $n_t$ units grow according to $dn_t = q\,n_t\,dt$, so $n_t = n_0e^{qt}$. A holding of $e^{-qT}$ units at time $0$ therefore becomes exactly one unit at $T$, and the prepaid forward price is $S_0e^{-qT}$. Under the assumptions of the forward price card, with the income reinvested at no cost and a short seller paying the yield to the lender,

which for a constant continuously compounded rate $r$ is

At a later time $t$, with $\tau = T - t$, the forward price is $F_{t,T} = S_te^{(r - q)\tau}$. The difference $r - q$ is the net carrying rate, the financing rate minus the income yield, and when $q > r$ the forward price is below the spot price for every $T > 0$. A proportional yield is a model for income spread over many small payments, such as the dividends of a broad index. A fixed cash dividend is a different case, because it does not grow with the size of the holding.

Definition At a time $t$ with $0 \le t \le T$, consider a long forward whose delivery price $K$ was fixed earlier. Under the assumptions of the forward price card its value is

and the short side is worth $-V_t$. To see this, add a new short forward at the current forward price $F_{t,T}$, which costs nothing. At $T$ the two contracts together pay $(S_T - K) - (S_T - F_{t,T}) = F_{t,T} - K$, an amount known at $t$, and that amount is worth $P(t,T)(F_{t,T} - K)$ at $t$. With a continuous dividend yield $q$ and a constant rate $r$, writing $\tau = T - t$,

A new contract with $K = F_{0,T}$ has $V_0 = 0$, and at maturity $P(T,T) = 1$ and $F_{T,T} = S_T$ give $V_T = S_T - K$. In between, the value moves with the spot price and the curve while $K$ stays fixed, so it can be positive or negative, and it is a different quantity from the forward price.