Pricing Forwards, Futures, and Commodities

A trader sees gold spot at $2,400 and the six-month gold future at $2,460. Is that a 5% annualized arbitrage, a normal cost-of-carry market, or a signal about lease rates? The answer depends on whether the carry equation balances. Forward pricing on the FRM tests whether you can build that equation from spot, financing, income, and storage, and recognize when convenience yield breaks it.

Forward pricing rests on one idea. You have two ways to own an asset at time T. Buy it today and hold it. Or enter a forward contract today and pay the forward price at T. Both routes end at the same place. So the cost of each route must match. If they don't, you build a cash-and-carry trade and lock in riskless profit.

Borrow $S_0$ at rate r. Buy the asset. Sell a forward at $F_0$. At time T, deliver the asset and collect $F_0$. You owe the lender $S_0{e}^{rT}$. Profit equals $F_0-S_0{e}^{rT}$. If that gap is positive, the trade is a free lunch. Markets close it instantly.

Common mistakes

• Using the non-income formula on a dividend-paying asset. A stock index at 4,500 with a dividend yield of 2% and risk-free rate of 4% has a six-month forward of $4500\times e^{0.01}=4,545$, not $4500\times e^{0.02}=4,591$. Trap: 4,591 appears as the distractor for candidates who forget to net the yield.
• Confusing contango with a price forecast. A question states "the oil futures curve is in steep contango, what does the market expect?" The answer is "nothing about future prices; the curve reflects high financing and storage relative to convenience yield." Trap: "the market expects oil prices to rise" is wrong.
• Adding storage and lease rate. The lease rate already reflects net carry economics for a leasable commodity. Stacking u on top of -δ double-counts. Use $F_0=S_0{e}^{(r-\delta )T}$ when given the lease rate; use $F_0=S_0{e}^{(r+u-y)T}$ when given storage and convenience yield separately. Don't mix.

Bottom line

• No-arbitrage anchor: holding the asset must cost the same as locking in delivery via forward, so arbitrage sets the price.
• Non-income asset: $F_0=S_0{e}^{rT}$. The forward exceeds spot by exactly the financing cost of carrying the asset.
• Known cash income: $F_0=(S_0-I)e^{rT}$ where I is the present value of income to T. With continuous yield q: $F_0=S_0{e}^{(r-q)T}$.
• Cost of carry: $F_0=S_0{e}^{(r+u-y)T}$ (storage u, convenience yield y). Lease-rate form $F_0=S_0{e}^{(r-\delta )T}$ bundles both. Never stack one on the other.

Exam shortcut

When a forward problem lands, ask which carry inputs the stem gives. Risk-free rate alone → non-income form. Rate plus dividend or yield → $e^{(r-q)T}$. Rate plus storage plus convenience → cost-of-carry. Rate plus lease rate → lease-rate form. The trap is using the non-income formula by reflex when the asset pays income.

The full lesson (about 2,611 words, 17 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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