Question

The current futures price of a stock is $15 per share. One month later, when the futures option expires, the futures price could have risen to $16.5 per share or declined to $14 per share. The strike price is $14.5. The risk-free rate is 6%.

What is the cost of futures contract at time zero

Answer 1

Let the number of shares be x.

Portfolio value = (stock price*number of shares) - one futures contract value

where futures contract value = max(market price at expiry - strike price, 0)

Portfolio value if price rises = (16.5*x) - max(16.5-14.5, 0) = 16.5x - 2

Portfolio value if price falls = (14*x) - max(14-14.5, 0) = 14x

Under no arbitrage condition, the portfolio value, whether price rises or falls after one month, has to be the same.

16.5x - 2 = 14x

x = 2/(16.5-14) = 0.8

So, portfolio value at expiry is 14x = 14*0.8 = 11.20

Present Value (PV) of portfolio value at T = 0 = portfolio value at expiry/(1+ riskfree rate/12)

= 11.20/(1+6%/12) = 11.1443

Let the cost of the futures contract be f.

Then, at T = 0,

portfolio value = 15x - f

15x - f = 11.1443

f = (15*0.8) -11.1443

f = 0.8557 (Cost of the futures contract)