Forwards, futures and put-call parity explained
Derivatives look abstract until you see what they are made of. A forward, a future and a swap are all agreements to do something at a later date at a price fixed today. An option is the right, but not the obligation, to do it. Almost everything in the derivatives part of the Level I exam comes from three questions: what is the contract, what price is fair, and what is it worth along the way?
Forwards, futures and swaps
A forward contract is an agreement between two parties to buy or sell an asset at a set price on a set date. Nothing is paid up front. The buyer is said to be long the forward and the seller short. At the end, the long gains if the market price is above the agreed price, and the short gains if it is below. One side's gain is exactly the other side's loss.
A futures contract is a forward that has been standardised and moved onto an exchange. The contract size, the delivery date and the asset are fixed by the exchange, and a clearing house stands between buyer and seller.
A swap is an agreement to exchange a series of payments. In a plain interest rate swap, one party pays a fixed rate and receives a floating rate on the same notional amount. It helps to see a swap as a string of forward commitments, one for each payment date.
How a forward price is set
The forward price is not a forecast of where the asset will trade. It is the price that leaves neither side with a free profit, which is the no-arbitrage idea. If you can buy the asset today, borrow to pay for it and deliver it under the forward, the forward price has to be what that costs.
For an asset that pays nothing while you hold it, the forward price is the spot price grown at the risk-free rate:
F = S × (1 + r)T
With a spot price of 100, a risk-free rate of 5% and one year to delivery, the forward price is 100 × 1.05 = 105. If the forward were 108, you could buy the asset, borrow 100, agree to sell at 108 and lock in 3 with no risk. If it were 102, you would do the reverse. Either way trading would push the price back to 105.
Carrying costs and benefits
Holding the asset can cost money, such as storage, or pay money, such as a dividend. Costs push the forward price up, because the seller must be paid for carrying them. Benefits push it down, because the buyer would otherwise collect them by holding the asset. If the stock above pays a dividend of 3 on the delivery date, the forward price falls to 105 − 3 = 102.
What a forward is worth before it ends
At the start, a fairly priced forward is worth nothing to either side. Afterwards its value moves with the spot price. The long's value at time t is the spot price minus the forward price discounted back from the delivery date:
Value = S(t) − F ÷ (1 + r)T − t
Suppose you went long at 105 and six months later the spot price is 110, with the same 5% rate. The discounted forward price is 105 ÷ 1.050.5 = 102.47, so the forward is worth 110 − 102.47 = 7.53 to the long, and −7.53 to the short. At delivery the discounting disappears and the value is simply the spot price minus 105.
Futures against forwards
The contract is the same idea; the plumbing differs.
- Settlement. A forward settles once, at the end. A futures contract is marked to market every day: gains are paid in and losses paid out, so the contract is reset to zero value each day.
- Margin. Futures traders post margin and top it up if losses build. Forwards usually need none.
- Credit risk. A forward carries the risk that the other side fails to pay at the end. A clearing house guarantees futures, which removes most of that risk.
- Flexibility. A forward can be tailored to the exact amount and date. A future is standard and trades on an exchange.
Because futures pay out daily, their price can differ a little from a forward's. When interest rates are constant and unrelated to the asset's price, the two are equal.
Put-call parity
A call gives the right to buy at the strike price, and a put the right to sell. Put-call parity links them. Take a European call and a risk-free bond that pays the strike price at expiry. Together they give the same outcome at expiry as a European put and the underlying asset. Two portfolios with the same payoff must cost the same today:
C + K ÷ (1 + r)T = P + S
Here C is the call price, P the put price, S the spot price and K the strike. Say the spot price is 50, the strike is 50, the rate is 4% and expiry is a year away. The present value of the strike is 50 ÷ 1.04 = 48.08. If the call costs 5.00, then the put must cost 5.00 + 48.08 − 50 = 3.08. A put quoted at 4.00 could be bought against a sold call to lock in a riskless gain, and trading would close the gap.
Rearranged, the same equation shows that a long call plus a short put is the same as a long forward, which is why parity is sometimes called put-call-forward parity.
Mistakes to avoid
- Reading the forward price as a prediction. It is a cost-of-carry number, not a view on the market.
- Forgetting carrying benefits. A dividend or a coupon lowers the forward price.
- Mixing up the long and the short. The long gains when the spot price ends above the forward price.
- Treating a forward's value and its price as the same thing. The price is fixed at the start. The value moves.
- Using put-call parity for American options. The exact relationship holds for European options.
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Where this is tested
Forward pricing, forward value, futures against forwards and put-call parity are all in the derivatives topic area, and the swap idea returns in fixed income. The learning outcomes below set out the formulas and the cases you are expected to handle.