How cash-and-carry determines a forward price

What price should you agree on today to buy an asset a year from now?

Under the assumptions below, the cost of buying the asset now and carrying it to delivery pins the forward price. At any other price, a trader can lock in a profit with no risk and no money down. Stage 1 lets you find that profit yourself.

Before you start · 3 quick checks on continuous compounding and forward payoffs, about 2 min

Definition: forward contract

A forward contract is an agreement made at time $0$ to trade one unit of an asset at a later date $T$ for a price $K$ fixed today, the delivery price. The buyer holds the long position and the seller the short position. No money changes hands at time $0$.

At $T$ the long side pays $K$ and receives a unit worth $S_T$, so its payoff is $S_T - K$. The short side's payoff is $K - S_T$.

The forward price $F_0$ is the delivery price at which a new contract is worth zero to both sides today. $S_0$ is the asset's price today, $r$ the continuously compounded risk-free rate, and $T$ the time to delivery in years.

Market assumptions

Trading assumptions, used everywhere on this page (the predictions, the game, the explorer and the problems):

the asset price today is $S_0 \gt 0$, and the time to delivery is $T \gt 0$;
no transaction costs, taxes or bid-ask spreads;
anyone can borrow or lend any amount at the same constant, continuously compounded risk-free rate $r$;
short selling is allowed, and the short seller can use the proceeds;
a forward can be bought or sold at the quoted price, and both sides perform at delivery (no default).

Two kinds of income appear. A cash dividend has a known amount and a known payment date. A continuous yield is described by a known constant rate $q$: the cash paid per unit, $qS_t$ a year, changes with the price, but the rate $q$ does not. Yield income can be reinvested continuously, and fractions of a unit can be traded.

No arbitrage, used to derive the fair forward price: no position costs nothing today, never pays a negative amount later, and pays a positive amount in some state. The forward prices in the predictions, the explorer and P1, P3, P4 and P5 are the fair prices this gives. The dealer quotes in the stage 1 game and in P2 and P6 intentionally violate the no-arbitrage condition, so that you can find the arbitrage.

Fine print: units, rates and what is left out

Rates are continuously compounded: $1 lent at $r$ for $T$ years grows to $e^{rT}$, and $1 due at $T$ is worth $e^{-rT}$ today.
For a stock index, one "unit" is a basket of the index stocks worth the index level in dollars.
Both sides are assumed to perform at $T$: there is no default risk. Futures, which are settled daily through a margin account, are a different contract and are not covered here.
Storage costs (for commodities) work like negative income. They are not covered here.

1 See

You are the arbitrageur

A dealer quotes a forward on an asset that pays no income. You can trade three things: the asset, cash at the risk-free rate, and the forward at the dealer's quote. Choose your position, one setting for each leg, then check its cash flows. The page tells you whether your position locks in a profit with no risk.

How the cash flows are counted

Cash flows are from your point of view: money received is positive, money paid is negative.
Asset. Buying a unit costs $S_0$ today and leaves you with a unit worth $S_T$ at $T$. Shorting a unit means selling a borrowed unit for $S_0$ today; at $T$ you buy one for $S_T$ and return it.
Cash. Borrowing $S_0$ at $r$ brings in $S_0$ today and costs $S_0e^{rT}$ at $T$. Lending is the reverse.
Forward. Entering costs nothing. At $T$ a long forward pays $S_T - F$ and a short forward pays $F - S_T$, where $F$ is the quote. Delivering the unit you hold for $F$ gives the same total as selling it at $S_T$ and settling the forward in cash.
Rounds 1 and 2 use fixed markets. Later rounds draw a new market from a fixed seed, so the same sequence appears on every visit.

2 Try

What moves a forward price, and what does not

Explorer: reach the goals

The explorer prices a forward on an asset with a known constant income yield $q$, reinvested continuously, such as a stock index with dividend yield $q$. With $q = 0$ it is the no-income case of stage 1. The formula $F_0 = S_0e^{(r-q)T}$ is proved in stage 3.

The explorer opens after you answer the three questions above.

Key point The forward price uses only $S_0$, $r$, $T$ and the asset's income. Income lowers it, because the forward buyer does not receive the income that a holder of the asset collects.

3 Prove

Both mispricings give an arbitrage

Theorem (forward price, no income)

Assumptions.

$S_0 \gt 0$ and $T \gt 0$. The asset pays no income and costs nothing to hold between $0$ and $T$.
Anyone can borrow or lend any amount at the same constant, continuously compounded rate $r$ (of any sign).
There are no transaction costs, taxes or bid-ask spreads.
Short selling is allowed, and the short seller can use the proceeds.
A forward for delivery at $T$ can be bought or sold at $F_0$, and both sides perform at $T$.
The price $S_T$ at delivery is not known today and can be any positive number.
No arbitrage: no position costs nothing today, never pays a negative amount later, and pays a positive amount in some state.

Conclusion. The forward price for delivery at $T$ is

\[ F_0 = S_0e^{rT}. \]

Build the proof line by line. It has two halves: one for a forward that is too expensive and one for a forward that is too cheap.

Scratch work, not part of the proof. The cash flows of lines 1 and 2, the trade from round 1 of stage 1:

Leg	Today	At $T$
Borrow $S_0$	$+S_0$	$-S_0e^{rT}$
Buy one unit	$-S_0$	$+S_T$
Short forward at $F_0$	$0$	$F_0 - S_T$
Total	$0$	$F_0 - S_0e^{rT}$

Extension: known income and a continuous yield (8 lines)

Theorem (forward price with income)

Keep the assumptions above, except that the asset now pays income to whoever holds it, and a short seller must pay that income to the lender of the asset.

Known cash income. If the asset pays known amounts $D_1, \dots, D_m$ at times $0 \lt t_1 \lt \dots \lt t_m \lt T$, let $I = \sum_i D_i e^{-rt_i}$ be their present value. Then $F_0 = (S_0 - I)e^{rT}$.
Continuous yield. If the asset pays income continuously at a known constant rate $q$ times its price, the income can be reinvested continuously and fractions of a unit can be traded, then $F_0 = S_0e^{(r-q)T}$.

Known cash income

Continuous yield

Value of an existing forward (4 lines)

Theorem (value of a forward)

A long forward written earlier with delivery price $K$, with $T$ years left to delivery, is worth $f = (F_0 - K)e^{-rT}$ today, where $F_0$ is today's forward price for the same delivery date. The short side is worth $-f$. The proof uses the trading assumptions above, and also that the existing contract can be bought or sold today, so that either side of the pair in line 3 can be traded.

Key point A forward above $S_0e^{rT}$ allows a cash-and-carry arbitrage, and one below it allows a reverse cash-and-carry arbitrage. Only $F_0 = S_0e^{rT}$ is left. Income the holder collects lowers the forward price to $(S_0 - I)e^{rT}$ or $S_0e^{(r-q)T}$.

4 Apply

Problems

All prices are hypothetical. All rates are continuously compounded.

Leg	Today	At \(T\)
Borrow \(S_0\)	\(+S_0\)	\(-S_0e^{rT}\)
Buy one unit	\(-S_0\)	\(+S_T\)
Short forward at \(F_0\)	\(0\)	\(F_0 - S_T\)
Total	\(0\)	\(F_0 - S_0e^{rT}\)