7.1 Forward contracts and the cost-of-carry price

7.1.1Definitions#

A forward contract is a bilateral agreement made at time 0 to exchange one unit of an asset at a fixed future date $T$ for a fixed price $K$, the delivery price. The buyer is long and the seller is short. Nothing changes hands at time 0. At $T$ the long's payoff is

The forward price $G_0$ at time 0 is the delivery price that makes a newly written contract worth zero to both sides. When the contract is struck, $K = G_0$; afterwards $K$ stays fixed while the forward price for the same delivery date moves.

A forward can be physically settled (the asset is delivered against $K$) or cash settled (the loser pays the winner $|S_T-K|$). For valuation the two are identical as long as the cash settlement uses the true $S_T$.

A futures contract is a standardised forward traded on an exchange, guaranteed by a clearing corporation and settled daily. 7.2 shows why daily settlement can make its price differ from the forward price, and why for equity index futures the difference is negligible. Until then we write $F_0$ for the forward price as well.

7.1.2Assumptions#

The derivations in 7.1 assume:

• A1. No frictions. No brokerage, taxes or bid–ask spreads. (Relaxed in Prop. 7.4.)
• A2. One interest rate. Anyone can borrow and lend any amount at the same continuously compounded rate $r$, constant over $[0,T]$. (Relaxed in Prop. 7.4 and 7.2.)
• A3. Short selling. Anyone can sell an asset they do not own and use the full proceeds, paying the lender any dividends that fall due.
• A4. No default. Counterparties honour the contract. (For exchange-traded futures, the clearing corporation enforces this; 7.2.)
• A5. Dividends. The asset pays a known continuous yield $q$, reinvested in the asset. (Discrete dividends: Prop. 7.2.)

The only economic principle used is no arbitrage: there is no self-financing strategy that costs nothing (or less than nothing) today and has a non-negative payoff that is strictly positive with positive probability. Equivalently: two portfolios with the same payoff at $T$ and no intermediate cash flows must cost the same today (the law of one price). If one were cheaper, buying it and selling the other would be an arbitrage.

7.1.3The no-arbitrage forward price#

Proof

We show that any other price allows an arbitrage, by constructing it explicitly.

First note the effect of reinvesting dividends. If you hold $n_t$ units and reinvest the dividends $qS_tn_t\,dt$ in the asset at price $S_t$, the holding grows by $dn_t = q n_t\,dt$, so $n_T = n_0e^{qT}$. Starting with $n_0=e^{-qT}$ units gives exactly one unit at $T$. The same argument in reverse holds for a short position: a short seller who must pay the lender $qS_tn_t\,dt$ can fund it by shorting $qn_t\,dt$ more units, so a short of $e^{-qT}$ units grows to a short of one unit.

Case 1: $F_0 > S_0e^{(r-q)T}$. Cash-and-carry arbitrage. At $t=0$:

Action at $t = 0$	Cash flow at 0	Cash flow at $T$
Borrow $S_0e^{-qT}$ at $r$	$+S_0e^{-qT}$	$-S_0e^{-qT}e^{rT}$
Buy $e^{-qT}$ units, reinvest dividends	$-S_0e^{-qT}$	hold 1 unit
Sell one forward at $F_0$	0	deliver 1 unit, receive $F_0$
Total	0	$F_0 - S_0e^{(r-q)T} > 0$

The terminal cash flow, the cash-and-carry profit

$$\Pi_T = F_0 - S_0e^{(r-q)T}, \tag{7.2}$$

does not depend on $S_T$, so it is riskless. A strategy that costs nothing and pays a positive certain amount is an arbitrage.

Case 2: $F_0 < S_0e^{(r-q)T}$. Reverse cash-and-carry arbitrage. At $t=0$:

Action at $t = 0$	Cash flow at 0	Cash flow at $T$
Short $e^{-qT}$ units, pay dividends by shorting more	$+S_0e^{-qT}$	short 1 unit
Lend $S_0e^{-qT}$ at $r$	$-S_0e^{-qT}$	$+S_0e^{-qT}e^{rT}$
Buy one forward at $F_0$	0	pay $F_0$, receive 1 unit, return it to close the short
Total	0	$S_0e^{(r-q)T} - F_0 > 0$

Again the payoff is certain and positive at zero cost. Since both strict inequalities lead to arbitrage, $F_0 = S_0e^{(r-q)T}$.∎

For a cash-settled contract, "deliver 1 unit, receive $F_0$" becomes "sell the unit at $S_T$ and pay $S_T - F_0$ on the short forward". The net is still $F_0$, so the proof is unchanged. The code m07_carry.carry_trade_pnl performs this accounting leg by leg and checks that the terminal P&L is the same for every $S_T$.

Remarks

1. $r$ enters because carrying the asset ties up money. $q$ enters because the holder of the asset receives dividends and the holder of the forward does not. Their difference $r-q$ is the net cost of carry.
2. Expected returns do not appear. A forward price that "builds in" a view of where the Nifty is going would be arbitraged. This is the first and simplest instance of the replication principle that M08 uses to price options.
3. With $r>q$ the forward is above spot (contango); with $q>r$ it is below (backwardation). Nifty futures usually trade above spot because Indian short rates exceed the index dividend yield. That is typical, not guaranteed (7.8.2).

7.1.4Discrete dividends#

Single-stock forwards are affected by individual dividends with known amounts and ex-dates, not by a smooth yield.

Proof

If $F_0 > (S_0 - I)e^{rT}$: borrow $S_0$; buy one unit; sell the forward at $F_0$. Split the loan into pieces: borrow $D_ie^{-rt_i}$ for maturity $t_i$ (each repaid exactly by the dividend $D_i$) and $S_0 - I$ for maturity $T$. At $T$ deliver the unit for $F_0$ and repay $(S_0-I)e^{rT}$. Profit $F_0 - (S_0-I)e^{rT}>0$ at zero cost.

*If $F_0 < (S_0 - I)e^{rT}$:* short one unit for $S_0$; lend $D_ie^{-rt_i}$ to each $t_i$ (to pay the dividends owed to the stock lender) and $S_0 - I$ to $T$; buy the forward. At $T$ receive $(S_0-I)e^{rT}$, pay $F_0$, take delivery and close the short. Profit $(S_0-I)e^{rT}-F_0>0$.∎

7.1.5Value of an existing forward#

Proof

Hold the old long forward and sell a new forward for the same date at today's forward price $F_t$, which costs nothing. At $T$ the two pay $(S_T-K) + (F_t - S_T) = F_t - K$, a sure amount known at $t$. By the law of one price, the pair is worth $(F_t-K)e^{-r\tau}$ today. The new forward is worth zero, so the old one is worth $(F_t-K)e^{-r\tau}$. Substituting $F_t = S_te^{(r-q)\tau}$ gives the second form.∎

Two consequences matter in practice. First, the value is linear in $S_t$ with slope $e^{-q\tau}$: a forward's delta is slightly below 1 for a dividend payer. Second, the value is the discounted difference of forward prices. A forward that is 100 points in the money with a year to go is worth less than 100 today. A futures position with the same price gain has already been paid the 100 in cash through daily settlement. That is the whole difference between the two contracts, and 7.2 builds on it.

7.1.6Frictions and the no-arbitrage band#

Real arbitrageurs borrow above the rate at which they lend, pay securities transaction tax (STT) and brokerage, and cross bid–ask spreads. The cost of carry then pins the forward price only to a band.

Proof

Repeat Prop. 7.1 with the true costs. Cash-and-carry: buying $e^{-qT}$ units costs $S_0(1+c)e^{-qT}$, all borrowed at $r_b$; at $T$ the trader owes $S_0(1+c)e^{(r_b-q)T}$ and receives $F_0$. The trade is profitable if and only if $F_0$ exceeds the right side of (7.5). Reverse: shorting $e^{-qT}$ units raises $S_0(1-c)e^{-qT}$, invested at $r_\ell$; at $T$ the trader holds $S_0(1-c)e^{(r_\ell-q)T}$ and pays $F_0$. Profitable if and only if $F_0$ is below the left side. Inside the band neither trade makes money.∎

The band is asymmetric in practice. The upper bound is enforced by anyone with capital: banks, proprietary desks and arbitrage mutual funds (a SEBI scheme category that holds stock and sells futures against it). The lower bound needs short selling. In India, institutions can short sell and retail investors can short intraday, but an overnight short of cash equities requires borrowing the stock through the securities lending and borrowing (SLB) mechanism, which is thinner and costlier. The lower bound is therefore enforced mostly by holders of the stock, who can sell what they own and buy futures instead. When those holders are themselves under pressure, futures can sit at a discount for days (7.8.2).