Question

Suppose it is January 31 now. There is a stock index. The stock index level now stands at 3000. There is a futures contract on this stock index. The futures contract will mature in 5 months. The stock index pays dividends with dividend yield at 3% for the coming February and April and pays dividends with dividend yield of 4% for the coming March, May and June. The dividend yield is measured with continuously compounding. The risk-free rate with continuously compounding is 3.5%.

1. (a) What is the futures price for the futures contract on this index now? (2 Points)
2. (b) What arbitrage opportunity you have if the futures price now is 3050? ( 1 points)

Answer 1

a) February Dividend = $3000*(e^3%*(1/12) -1) =$7.51

Future value of February Dividend = $7.51*(e^3.5%(4/12)) = $7.60

March Dividend = $3000*(e^4%*(1/12) -1) =$10.02

Future value of March Dividend = $10.02*(e^3.5%(3/12)) = $10.10

April Dividend = $3000*(e^3%*(1/12) -1) =$7.51

Future value of April Dividend = $7.51*(e^3.5%(2/12)) = $7.55

May Dividend = $3000*(e^4%*(1/12) -1) =$10.02

Future value of May Dividend = $10.02*(e^3.5%(1/12)) = $10.05

June Dividend = $3000*(e^3%*(1/12) -1) =$7.51

Future value of June Dividend = $7.51*(e^3.5%(0/12)) = $7.51

Future Value of Stock Index = $3000*(e^3.5%(5/12)) - sum of FV of all dividends

= $3044.07 - ($7.60 + $10.10 + $7.55 + $10.05 + $7.51) = $3001.26

b) If the future price is $3050, there exists an arbitrage opportunity. To make use of this opportunity see below:

Now: Short 1 unit of future contract @$3050; Borrow $3000 @ 3.5%; Use borrowed money to buy stock index at $3000

Feb: Receive dividend of $7.51 and invest @3.5%

Mar: Receive dividend of $10.02 and invest @3.5%

Apr: Receive dividend of $7.51 and invest @3.5%

May: Receive dividend of $10.02 and invest @3.5%

June: Receive dividend of $7.51; Receive the dividend invested of $ 35.3; Deliver the stock at 3050; Pay back the borrowed amount with interest of $3044.07

Net Profit = 3050 -3044.07+7.51+35.3 =$48.74