Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

A forward locks in a delivery price today for a transaction at expiration. Its price is the locked-in delivery rate. Its value is the mark-to-market gain or loss to one side. Confusing the two is the most common exam trap in derivatives.

KEY: Price is the contractual delivery rate negotiated at initiation. Value is the present worth of the contract to a counterparty at any moment. Price is fixed. Value moves.

At initiation, the forward price is set so neither party pays the other. The contract starts at zero value.

For a non-income, non-cost asset, no-arbitrage gives:

$$F_0(T)=S_0(1+r{)}^T$$

If the asset throws off income (dividends, coupons) with present value $I$, or imposes carry costs with present value $C$, adjust:

$$F_0(T)=(S_0-I+C)(1+r{)}^T$$

TRAP: Income reduces the forward price (you don't get the dividends if you wait). Storage costs raise it (the seller must be compensated for carrying the asset).

Common mistakes

• Confusing price with value at initiation. The price equals $S_0(1+r{)}^T$. The value is zero. Trap: computing $\$104$ and labeling it the contract's value.
• Forgetting to discount the locked-in price during the life. The mid-life value is $S_t-F_0(T)/(1+r{)}^{T-t}$, not $S_t-F_0(T)$. Trap: subtracting $104 from $108 directly and reporting $4.00 instead of $6.02.
• Sign errors on income and carry. Income (dividends, coupons) lowers the forward price. Storage costs raise it. Trap: adding dividends instead of subtracting.

Bottom line

• Forward price at initiation: $F_0(T)=S_0(1+r{)}^T$, adjusted for carry costs and benefits. Value at initiation = 0.
• Value during life (long): $V_t(T)=S_t-F_0(T)/(1+r{)}^{T-t}$; value at expiration (long): $S_T-F_0(T)$. The locked-in price must be discounted first.
• Income (dividends, coupons) lowers the forward price; storage costs raise it.
• Forward rates come from no-arbitrage between spot rates: $(1+S_B{)}^B=(1+S_A{)}^A(1+f_{A,B-A}{)}^{B-A}$, where the exponent is the gap $B-A$.

Exam shortcut

For mid-life value, always discount the locked-in price before subtracting from spot. For implied forwards, set the two financing strategies equal and solve algebraically (no memorization needed if you can write the no-arbitrage equation). For sign of income and carry, ask "would I rather hold the asset or hold the forward?" Income makes holding the asset more attractive, so the forward price falls.

The full lesson (about 1,696 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• pricing and valuation of forwards

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