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%.

1. What is the value of the risk-free portfolio at time zero? (1 mark)

Answer 1

The given scenario can be represented as follows

From the given information we can see the price will either increase to $ 16 or decline to $ 14.

We can create a risk-free portfolio buy buying 1 stock today at a price of $ 15 and taking short position on delta (∆) number of call option with a strike price of $ 14.5 and maturity period of 1 month.

Since the portfolio is risk free then the value of portfolio would remain same for all price changes.

After a month assume the price is $ 16.5 the the value of portfolio will be

Similarly when price is $ 14 then the

These two values must be equal

The portfolio will be risk free when we hold 1 stock and short 0.8(∆) number of stock.

Plugin in this value in the following equation

Now convert it to present value

Value of portfolio today = $ 11.1441

Note: Delta is known as the optimal hedge ratio.

I have considered call option thus it will be exercised when stock price is greater than strike price. When price is equal to 16.5 then option will be exercised and payoff to long position holder will be fu = Max(16.5-14.5,0) = Max(2,0) =2. In case of price $ 14 the option will remain unexcericed thus value of fd = 0. Further the portfolio had short position on option thus it is subtracted from the value of portfolio.

The below part is not asked but I have calculated it for reference.

Now calculating the value of portfolio at time t=0

The value of portfolio bat time t = 0 is

V = 15 - f

Equate it to the discounted value

Value of option = $ 3.86 / share.