Question

Suppose today S&P 500 index is 1800 and the continuously compounded annual dividend yield on the index is 2%.

Assume that it is possible to lend at 4 % and borrow at 7 %, annually continuously compounded.

a) Above what futures price is there arbitrage?

b) Below what futures price is there arbitrage?

Please show the cash flows, which make each type of arbitrage and explain what you would

do at each relevant date. Consider time to maturity of 6 months

.

Answer 1

Theoretical futures price, F = S * EXP ((r - q) *T)

Where S is current index price, r is risk free rate, q is dividend yield and T is year fraction of maturity

Taking risk free borrowing rate as 7%,

F = 1800 * EXP( (7% - 2%) * 0.5) = $1845.57

Assuming risk free borrowing rate as 7%, there will be arbitrage whether futures price is above or below the above price.

a) If the futures price is above 1845.57, there is a possibility of arbitrage. Let us say F=1900

Arbitrages steps:

1. At time = 0, borrow $1800*EXP(2%*0.5)/EXP(7%*0.5), purchase spot index and sell futures on index

The above step leads to obligation to pay loan plus interest after six months = 1800*EXP(2%*0.5) = 1818.09

2.At time =0.5 year, receive dividends, repay the loan with interest, closure futures position by delivering index

Net cash flow =receive dividends + sell index + repay loan + close future position

Net cash flow = 36.36 + 1900 – 1818.09 - (1900-1845.57)

= $63.84

Thus, there is net arbitrage profit to be made in the above portfolio

B) If the futures price is 1700.

Net cash flow =receive dividends + sell index + repay loan + close future position

Net cash flow = 36.36 + 1700 – 1818.09 - (1700-1845.57) = $63.84

Hence, arbitrage profit is made irrespective of whether futures price is above or below the theortetical price.