Question

A fund manager is concerned about the performance of the market over the next year and plans to use one-year futures contracts on the S&P 500 to hedge the risk. The current index level is 1,200, and the one-year risk-free interest rate is 4% p.a. with continuous compounding. The current one-year futures price on a stock-index portfolio is 1,220. Assume that a dividend of $20 is expected after a year for a $1,200 investment in the market portfolio.

a) (4 points) Is the contract mispriced? Why? If yes, by how much is it overpriced (underpriced)?

b) (8 points) Identify an arbitrage opportunity such that you can obtain a riskless profit equal to the futures mispricing.

c) (8 points) Suppose that when you short sell the stocks in the market index, you do not receive any interest on the funds; instead, the broker receive it. Is there still an arbitrage opportunity now (assuming you don’t own the shares originally)? Explain the reason.

d) (10 points) Under the assumption of (c), i.e., you do not receive any interest on the funds if you short sell the stocks, whhat is the no-arbitrage range? That is, how high and how low can the futures price be such that there is no arbitrage opportunity?

Answer 1

Solution A) Fair Value of one year future i.e.

Fair Value = So*e^(rt) - D

So = Current Price = 1200

D= Dividend = 20

R = risk-free interest rate = 4%

T = Number of years = 1

Thus, fair Value of one year future = 1200*e^(0.04*1) - 20

= 1228.97

Solution A) Since the Contract is trading at $ 1220, which is is below ils Fair Value underpriced by $8.97 = (1228.97 -1200)

Solution B)

Ans b) Strategy to obtain risk less profit

- Steps

Risk Less to obtain Risk less profit :

1 )Buy futures @ $ 1220

2) Sell Short stocks@ 1220

3) Invest the proceeds (1200) at Continuous Compound Rate of 4%

After 1 year:

1) Receive Investment Proceeds = 1200*e^0.04

= 1200 * 1.0408

= 1248.97  

2) Square off the future at the current price, S1

Gain/loss from the transaction = S1 - 1220

3)B Stocks at Current price = -(S1+20)

Net Gain = 1248.97 + S1 - 1220 -S1 - 20

= 8.97

3) Since the Interest will not be received on Investing proceeds of short sold shares, hence, NO arbitrage opportunity will exits.

4) If Interest is not Received on Proceeds Investments

Fair Price of Future = 1200-20 = 1180

Any Price below $ 1180 will cause it to Undervalued  

On the other hand, if the Price is above $ 1180, we would still have to pay Interest for Borrowings to finance the Purchase of Stock.

Let the Price be Y.

Then, Today.

1 Short Future @ 1ooo

2) Buy stocks at 1200

(1248.97)

3) out of proceeds of borrowings we take today: Y

After 1 year:

1) pay loan along with interest = (1248.97)

2) Square off future with Current Price x= (X-Y)

3) Receive the dividend = +20

4) Short stock at the current price = X

Net gain/loss = -Y - 1228.97

For gain to be zero, Y = 1228.97

For prices above 1228.97, there would exist an arbitrage opportunity.