Explain how a synthetic commodity position is built, then use it to account for how the forward price relates to the future spot price that is expected.
The idea
A synthetic commodity position replicates owning the commodity without buying or storing it. Buy the commodity forward, and buy a zero-coupon bond that pays the forward price F at expiry, T years from now. At expiry the bond pays for the forward purchase and you hold the commodity, so the payoff is the spot price on that date. Its cost today is the price of the bond, the present value of F at the continuously compounded risk-free rate r: Cost = F × e^(−rT). If holding the commodity costs nothing and brings no lease income or convenience yield, that cost must equal the spot price S, giving F = S × e^(rT). The synthetic position earns the commodity's expected return α, so with continuous compounding, Expected spot = S × e^(αT). The forward price grows from spot at r instead, so Forward = Expected spot × e^(−(α − r)T). The forward price is therefore below the expected spot price when α exceeds r. The error is equating the forward price with the expected spot price.
Formulas
Cost of a synthetic commodity position
cost = F0 × e^(−r × T)
- F0
- the forward price, which the zero-coupon bond pays at T
- r
- the continuously compounded risk-free rate, per year
- T
- the time to delivery, in years
The forward costs nothing, so the synthetic costs the bond. With no carry costs or benefits this must equal spot, which gives F0 = S0 × e^(r × T).
Forward price from the expected future spot price
F0 = E(ST) × e^(−(α − r) × T)
- E(ST)
- the expected spot price at T
- α
- the commodity's expected continuously compounded return
- r
- the continuously compounded risk-free rate, per year
- T
- the time to delivery, in years
F0 is below expected spot when α exceeds r; equating the forward with expected spot is the error.
A practice question
Which combination creates a synthetic long position in a commodity that pays the spot price at time T?
- A short forward contract plus a zero-coupon bond paying the forward price at T.Why not: The short forward gives the negative of spot price exposure, so it cannot replicate a long position.
- A long forward contract plus borrowing the present value of the forward price.Why not: Borrowing instead of lending gives a debt to pay at T and leaves a net payoff of S - 2F.
- A long forward contract on the commodity plus a zero-coupon bond paying the forward price at T. - correct
- A long call and a short put with the same strike, and no bond.Why not: Call minus put replicates a forward, not the commodity itself; a bond covering the strike must be added.
At T the bond pays F, which pays for the commodity bought under the forward; you end up holding the commodity worth S_T. The strategy costs the present value of F.
A harder one
A commodity is expected to be worth USD 46 in one year (expected spot). Its expected return is 10% and the risk-free rate is 4%, both continuously compounded. Assuming no storage costs or convenience yield, the one-year forward price is closest to:
- USD 43.32 - correct
- USD 48.84Why not: Reverses the sign: the forward is below expected spot when the expected return exceeds the risk-free rate.
- USD 46.00Why not: Assumes the forward equals the expected spot price, which holds only if the risk premium is zero.
- USD 41.62Why not: Discounts at the full expected return, which gives today's spot price, not the forward.
F = E(S) x exp(-(alpha - r) x T) = 46 x exp(-0.06) = 43.32.
The hardest tier
A commodity with no carry costs is USD 100 spot. The two-year forward is USD 118 and the continuously compounded rate is 5%. Comparing the cost of the synthetic commodity (long forward plus a bond paying the forward price) with spot reveals an arbitrage. The present value of the arbitrage profit per unit, from selling the forward and buying the commodity, is closest to:
- USD 18.00Why not: Ignores the discounting of the forward price.
- USD 6.77 - correct
- USD 7.48Why not: Gives the profit at expiry rather than the present value.
- USD -6.77Why not: Gets the sign wrong; the forward is above the carry value so the trade of selling forward earns a gain.
Synthetic cost = 118 x exp(-0.10) = 106.77, which exceeds spot 100 by 6.77. Sell the synthetic (short forward, borrow) and buy the commodity for a present value gain of 6.77.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Commodity Forwards and Futures
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