Question

A stock index is at 1213.52. A futures contract on the index expires in 121 days. The price of the futures contract is 1358.50.The risk-free interest rate is 4.50 percent. The value of the dividends reinvested over the life of the futures is 245.0.

(a) Show that the futures contract above is mispriced by computing what the price of this futures contract should be.

(b) Show how an arbitrageur could take advantage of the mispricing.

Answer 1

Sol:

Stock index (So) = 1213.52

Risk free rate (r) = 4.50%

Futures contract on the index expires in 121 days

Future contract selling price = 1358.50

Dividends (d) reinvested over the life of the futures = 245

a) Fair value (F) = So x (1 + r x 121/365) - d

Fair value (F) = 1213.52 x (1+0.045 x 121/365) - 245

Fair value (F) = 986.62

b) Since there is a difference between the fair value and market value, an arbitrage opportunity exists.

We will buy the underlying stock at 1213.52 and sell the future contract at 1358.5.

We will borrow at risk free rate and repay it in expiry of contract.

So 1213.52 x (1+0.045 x 121/365) = 1231.62

Dividend recieved (d) = 245

So net amount to be paid = 1231.62 - 245 = 986.62

Therefore a arbitrageur could take advantage of the mispricing and earn profit = Future contract selling price - Net amount to be paid

= 1358.5 - 986.62 = 371.88