Forward Pricing

The fair forward price is the cost of carrying the underlying asset to delivery, derived purely from no-arbitrage replication.

1. The idea

A forward contract obligates the holder to buy the underlying at a fixed price $F_{0}$ at future time $T$ . To price it, replicate the forward's payoff by buying the asset today and financing the purchase to time $T$ .

With continuous compounding at risk-free rate $r$ , the fair forward price is $F_{0} = S_{0} e^{r T}$ . If the asset pays a continuous dividend yield $q$ , holding the asset also earns $q$ , so the forward price is reduced to $F_{0} = S_{0} e^{(r - q) T}$ .

If the actual traded forward price differs from this fair value, a cash-and-carry (or reverse cash-and-carry) arbitrage exists: borrow to buy the asset and sell the forward, or short the asset and buy the forward, locking in a riskless profit.

2. Why it matters

Forward pricing is the simplest replication argument and the direct building block for futures, FX forwards and the cost-of-carry model used across all asset classes.

3. Core mathematics

Forward price, no dividend

F_{0} = S_{0} e^{r T}

Forward price, dividend yield q

F_{0} = S_{0} e^{(r - q) T}

Value of an existing forward

V_{t} = (F_{0} - K) e^{- r (T - t)}

K is the contract's fixed delivery price

4. How to recognize it

Think of this when…

Any question giving spot price, rate, time to maturity and asking for a forward or futures price
Cash-and-carry arbitrage setups
FX forward pricing using domestic/foreign rates (special case of dividend yield)

5. Worked example

Worked example

A non-dividend stock trades at $50. The risk-free rate is 5% continuously compounded. Find the fair 6-month forward price.

1
Apply the no-dividend forward formula with $T = 0.5$ .
$F_{0} = 50 e^{0.05 (0.5)} = 50 e^{0.025}$
2
Evaluate the exponential.
$e^{0.025} \approx 1.0253$
3
Multiply through.
$F_{0} \approx 51.26$
Answer: Fair forward price ≈ $51.27

6. Common mistakes

Using simple interest formulas when the problem specifies continuous compounding, or vice versa
Forgetting to subtract the dividend yield $q$ when the asset pays dividends
Confusing the forward price $F_{0}$ (set at inception, zero value) with the value of an existing forward position later

7. Quick check

For a dividend-paying stock with yield q, the fair forward price is:

If the observed forward price is higher than fair value, the arbitrage is:

8. Practice

3 QuantSolve questions use this concept.

Practice this concept

9. Related methods

No-Arbitrage Replication

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