Question

Suppose that the risk-free interest rate term structure is flat 4% per annum with continuous compounding and that the dividend yield on a stock index is 1% per annum.

The index is standing at 9,200, and the futures price for a contract deliverable in eight months is 8,900.

Required:

a)      Today, what is the theoretical future price “F”?

b)      Today, what arbitrage opportunities does this create? Explain actions to carry it out and calculate the profit.

c)      Today, we enter into the future contract deliverable in eight months for an amount of 8,900 and after 2 months, we know that index trades at 8,200 (interest rate and dividend yield are still the same as two months ago).What is the value ”f” of the contract for a long position?

Answer 1

a)

Futures price as per the cost of carry model = Current futures price * e^ ( risk free rate - dividend rate )*n

= 9200 * [ ( 0.04-0.01)*8/12] = 9200 * e^0.02 = 9200*1.0202 = 9385.84

theoretical future price “F” = 9385.84

b)

The theoritical price is higher than the actual futures price. Hence we should short the index and go long the futures on index having maturity of 8 months. So index is sold at 9200 and then the same is bought at 8900 with the help of futures contract . So we have a arbitrage profit of 300

c )

We need to first calculate the present value of actual futures price at t=2 months

Present value = 8900 / 1/ e^[ risk free rate - dividend rate ) * n

= 8900 / e ^( 0.04 -0.01) * 6/12 = 8900 / e^0.015 = 8900/ 1.01511 = 8767.52

Value for the long futures position = 8200 - 8767.52 = -567.52