Question

The one-year futures price on a particular stock index portfolio is $1,825, the stock index is currently 1,800, the one-year risk-free interest rate is 3%, and the year-end dividend that will be paid on an $1,800 investment in the index portfolio is $25. a. By how much is the contract mispriced? b. Formulate a zero-net-investment arbitrage portfolio, and show how you can lock in riskless profits equal to the futures mispricing. c. Now assume that if you short-sell the stocks in the market index, the proceeds of the sale are kept with the broker and you do not receive any interest income on the funds. Is there still an arbitrage opportunity even though you do not own the shares in the index? How or why not? d. Given short-sale rules, what is the no-arbitrage band for the stock-futures price relationship? That is, given a stock index of 1,800, how high and how low can the futures price be without giving rise to arbitrage opportunities.

Answer 1

F_{1, actual} = $ 1,825; S₀ = $ 1,800; R_f = 3%; D = Year end dividend = $ 25

Part (a)

F_{1, no arbitrage} = S₀ x (1 + R_f) - D = 1,800 x (1 + 3%) - 25 = $ 1,829

Since, F_{1, actual} is not same as F_{1, no arbitrage} hence the prices are not in equilibrium. The contract is mis-priced by $ 1,829 - $ 1,825 = $ 4

Part (b)

Arbitrage portfolio:

• Buy the futures
• Short Sell the stock and
• Lend the proceeds of the short sale

Cash flow at t = 0 will be = Cost to buy the futures + proceed from short sell of the stock - Money lent = 0 + 1,800 - 1,800 = 0. Thus this is a zero net initial investment.

At the end of 1 year,

• Pay F_{1, actual} = 1,825 and take delivery of the stock
• Handover the physical stock, pay the dividend and close the short sell position
• Receive back the lent money along with interest = Amount lent x (1 + R_f) = 1,800 x (1 + 3%) = 1,854

Cash flows on maturity = -1,825 - 25 + 1,854 = $ 4

Thus we have made a risk free profit at the end of year 1, with a zero net initial investment. This is the arbitrage the question is asking about.

Part (c)

Now assume that if you short-sell the stocks in the market index, the proceeds of the sale are kept with the broker and you do not receive any interest income on the funds. In such a case,  there no longer exists an arbitrage opportunity.

Reasons: Cash flows on maturity = - 1,825 - 25 - 1,800 = - 50. Since cash flows are now negative, there is no longer an arbitrage opportunity.

Part (d)

Given short-sale rules, the no-arbitrage band for the stock-futures price relationship is:

S₀ - D < F < S₀ x (1 + R_f) - D

Hence, 1,800 - 25 < F < 1,800 x (1 + 3%) - 25

Hence, 1,775 < F < 1,829