Charta is on Google Play: practise on the go. Get the app
Charta

Determine what a stock index futures contract is worth, and describe how index arbitrage works.

FRM® Part I · Financial Markets and Products · Pricing Financial Forwards and Futures · FRM1.FM.10.h

The idea

A stock index is treated as an asset paying a continuous dividend yield q. The futures price is F = S × e^((r − q)T). Here S is the index level, r the risk-free rate and T the years to expiry. The notional value of one contract is the multiplier times the futures price. If the futures price is above its theoretical value, index arbitrage sells futures and buys the stocks in the index, financed by borrowing. If it is below, the arbitrageur buys futures and sells the stocks short, investing the proceeds. In practice this is done by program trading, which buys or sells the whole basket at once. Transaction costs and imperfect tracking create a band within which arbitrage does not pay. The usual mistake is ignoring q, so that a futures price below spot, which is normal when q exceeds r, is wrongly read as an arbitrage.

Formulas

Theoretical stock index futures price

F0 = S0 × e^((r − q) × T)

S0
the spot price today
r
the continuously compounded risk-free rate, per year
q
the continuous income yield, such as an index's dividend yield or a currency's foreign rate
T
the time to delivery, in years

Above this, sell futures and buy the basket; below it, buy futures and short the basket. A futures price below spot is normal when q exceeds r, and transaction costs create a no-arbitrage band.

A practice question

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

For a stock index with spot level S, continuous dividend yield q, risk-free rate r and expiry T, the theoretical futures price is:

  1. S x e^((r - q)T) - correct
  2. S x e^(rT)Why not: Ignores the dividend yield of the index.
  3. S x e^((r + q)T)Why not: Adds the dividend yield; the dividends reduce the cost of carry.
  4. S x e^((q - r)T)Why not: Reverses the sign of the net cost of carry.

An index is treated as an asset paying a continuous yield q, so the cost of carry is r - q.

A harder one

An equity index stands at 4,200. The risk-free rate is 4.5% and the dividend yield is 1.8%, both continuously compounded. The three-month index futures price is closest to:

  1. 4,171.7Why not: Uses (q - r) instead of (r - q), reversing the net cost of carry.
  2. 4,228.4 - correct
  3. 4,247.5Why not: Ignores the dividend yield.
  4. 4,266.7Why not: Adds the dividend yield to the risk-free rate.

F = 4,200 x e^((0.045 - 0.018) x 0.25) = 4,200 x e^0.00675 = 4,228.4.

The hardest tier

An index is at 1,500. The risk-free rate is 2.8% and the dividend yield is 3.5%, both continuously compounded. The nine-month futures price is 1,490. Which statement is correct?

  1. There is no arbitrage, because a futures price below spot is expected when the dividend yield exceeds the risk-free rateWhy not: Treats futures below spot as automatically fair; the relevant test is the comparison with S x e^((r - q)T).
  2. Futures are cheap by 10 points relative to spot: buy futures, sell the basket short and lock in that amountWhy not: Compares the futures price with spot instead of with the cost-of-carry price.
  3. Futures are cheap relative to 1492.15: sell futures, buy the basket with borrowing, locking in about 2.15 pointsWhy not: Gets the pricing right but the direction of the trade wrong; cheap futures are bought.
  4. Futures are cheap relative to 1492.15: buy futures, sell the basket short and lend the proceeds to lock in about 2.15 points - correct

Theoretical F = 1,500 x e^(-0.007 x 0.75) = 1492.15, above the 1,490 quoted. Buy futures, short the stocks and invest the proceeds: profit about 2.15 points at expiry.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Pricing Financial Forwards and Futures