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Explain why futures prices and spot prices converge.

FRM® Part I · Financial Markets and Products · Futures Markets · FRM1.FM.7.b

The idea

Futures and spot prices converge because at expiry the futures contract is an obligation to exchange the asset for the futures price, so it is economically the same as buying the asset immediately. If the futures price were above spot in the delivery month, an arbitrageur would buy spot, sell futures and deliver, and the selling pressure would push the futures price down. If it were below spot, an arbitrageur would buy futures, take delivery and sell the asset, pushing the futures price up. Basis is spot minus futures and tends to zero at expiry. A hedger who closes out before expiry still bears basis risk. So the effective price is the initial futures price plus the basis at close-out, not simply the initial futures price.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why do futures prices converge to the spot price as the delivery month arrives?

  1. The exchange sets the futures price equal to the spot price on each day of the delivery month.Why not: Wrong mechanism: exchanges do not set prices; market arbitrage drives convergence.
  2. Futures prices are unbiased forecasts of future spot prices, so they equal spot at all times and at every maturity.Why not: Confuses convergence with forecasting; an unbiased forecast would not equal spot before expiry.
  3. Daily marking to market forces the futures price to equal the spot price every day.Why not: Marking to market moves cash between accounts; it does not tie the futures price to spot before expiry.
  4. At expiry the contract becomes an obligation to buy or sell the asset, so any gap between the two prices creates an arbitrage that removes it. - correct

Near expiry, a futures price above spot invites buying spot, selling futures and delivering; a price below spot invites buying futures and taking delivery. These trades drive the prices together.

A harder one

A producer will sell an asset at expiry and shorts a futures contract at 103.5 when spot is 100. The position is held to expiry, when spot is 101 and the futures price has converged to spot. What is the producer's effective selling price per unit?

  1. 101.0Why not: Ignores the futures gain of 2.5 and counts only the spot sale.
  2. 100.0Why not: Uses the starting spot price, which has no role in the outcome of a hedge held to expiry.
  3. 103.5 - correct
  4. 104.5Why not: Adds the initial basis of 3.5 to the final spot price instead of the actual futures gain of 2.5.

Futures gain on the short = 103.5 - 101 = 2.5. Effective price = 101 + 2.5 = 103.5, the initial futures price, because the basis is zero at expiry.

The hardest tier

In the delivery month a futures contract for 5,000 units trades at 49.60 while spot is 50.10. Taking delivery and reselling in the spot market costs 0.30 per unit in handling. What is the arbitrage profit per contract from buying the futures?

  1. USD 1,000 - correct
  2. USD 2,500Why not: Ignores the 0.30 handling cost, giving the gross spread of 0.50 x 5,000.
  3. USD 4,000Why not: Adds the handling cost to the spread (0.80) instead of deducting it.
  4. USD 1,500Why not: Reports the handling cost (0.30 x 5,000) as if it were the profit.

Buy the futures at 49.60, take delivery, sell at 50.10: gross 0.50, less 0.30 costs = 0.20 per unit. 0.20 x 5,000 = USD 1,000.

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