Question

Consider a three month futures contract on the S&P 500 index. The value of the index is 1000; the dividend yield is 1% and the three month interest rate is 4% continuously compounded. (a) Explain how to compute the futures price making sure to define all terms and assumptions. In particular carefully explain why the formula holds. Then compute the fair futures price. (b) Suppose the actual futures price is 1010.0. In great detail describe a strategy that creates guaranteed arbitrage free profits. What is the profit and when is it realized.

Answer 1

The value of the Index is 1000

Dividend yield = 1%

The rate is 4% compounded continuously

a) The future index value would be computed by

Future price = Spot price * e^ (rate – dividend yield) * time period

Future price is the price as to what should be the expected price. The spot price is the current cost and interest rate is the borrowing cost and dividend yield is an income. The reason why this formula holds is because we are calculating the future cost or price of an asset so we have to add the cost and subtract the expected income from the asset.

Here we are calculating the future value on continuous basis so the (e^) formula and the dividend yield is the income so it would be subtracted and the time period given in the question is 3 months so we will calculate the future price by using the above formula

Future price = 1000 *e^ (0.04 – 0.01) *3/12

= 1000 * 1.007528 (Use financial calculator to solve the value of e)

= 1007.53

b) Now the arbitrage opportunities opportunity comes up when there is a difference between the price. The future price is 1010

So, the future price is higher than the price we calculated in the part a)

Here you can short the future at 1010 and buy the underlying asset. Now here you can not buy Index so you can an ETF that replicates the performance of the index and it would be less costly to apply rather than buying stocks in the S&P 500.

The profit should be the difference between the price of (1010 – 1007.53) = 2.47, in arbitrage there is no capital investment so you realize the profit when you make the transaction but, in many countries, you can realize the net profit from the transaction at the end of contract.