Spot-Futures Parity
Kerry Back
Arbitrage and spot-futures parity
- In last session, looked at implications of the expectations hypothesis for the forward curve.
- In this session, we look at the implications of arbitrage.
- Question is: how do futures prices relate to spot (contango or backwardation) and why?
- Link between spot and futures is called spot-futures parity.
Overview of arbitrage
- Suppose the forward curve is extremely steep. At some point, it pays to buy spot and sell futures.
- You can buy spot and hold and then deliver it on the futures contract. So, you pay the spot price and receive the higher futures price.
- There are interest and other considerations we will discusss
- You can buy spot and hold and then deliver it on the futures contract. So, you pay the spot price and receive the higher futures price.
- Suppose the forward curve is extremely downwards sloping.
- If you can short sell the spot, you can sell spot and buy futures. Accept delivery on the futures to cover the short spot position.
- Or people who are long spot could sell it and buy futures to restore their long position.
- Again, there are interest and other considerations.
Synthetic long futures
- On a long futures (ignoring daily settlement), you pay at the delivery date and accept delivery then.
- To duplicate this, you can borrow money, buy spot and hold until the delivery date.
- At the delivery date, you will have the asset and owe money (so you pay at the delivery date).
- You can create a synthetic long futures and sell the actual futures.
Synthetic short futures
- On a short futures (ignoring daily settlement), you deliver at the delivery date and get paid then.
- To duplicate this, you can short the spot and invest the proceeds in T-bills.
- At the delivery date, you cover the short and have cash (T-bills).
- You can create a synthetic short futures and buy the actual futures.
Cost of carry and convenience yield
- Now, more details.
- On the synthetic long futures, you pay at delivery the spot price plus interest.
- Also, there may be storage costs.
- Cost of carry = interest + storage costs.
- On the other hand, you may earn dividends, etc. when holding the asset.
- Convenience yield = dividends or other income
- Arbitrage implies futures price should not be higher than cost of synthetic long futures, so
\[\text{futures price} \le \text{spot price} +\] \[\text{cost of carry} - \text{convenience yield}\]
- Also (when you can short) arbitrage implies futures price should not be less than cash generated by the synthetic short futures, so
\[\text{futures price} \ge \text{spot price} +\] \[\text{cost of carry} - \text{convenience yield}\]
Gold
- Convenience yield = 0
- Storage costs \(\approx\) 0
- Can probably short, but also lots of gold in storage (longs can sell and then restore, like shorting and covering)
- So,
\[\text{futures price} = \text{spot price} + \text{interest}\] \[= \text{spot price} \times (1+r)^n\]
Gold Forward Curve on 1-24-2022
Stock index futures
- Cost of carry = interest rate
- Convenience yield = dividends,
- Can short, so
\[\text{futures price} = \text{spot price} + \text{interest} - \text{dividends}\]
S&P 500 Index Futures on 1-22-2016
S&P 500 Index Futures on 1-24-2022
Commodities
- Storage costs can be very high \(\Rightarrow\) contango
- Example: in April 2020, the front-month crude futures contract went negative
- Forward curve was very steep
- There was no available storage in Cushing, OK
- But convenience yield can also be very high \(\Rightarrow\) backwardation
- In shortages, spot spikes and markets move into backwardation