A stock is currently priced at $64. The stock will either increase or decrease b
ID: 2649371 • Letter: A
Question
A stock is currently priced at $64. The stock will either increase or decrease by 20 percent over the next year. There is a call option on the stock with a strike price of $60 and one year until expiration.
If the risk-free rate is 4 percent, what is the risk-neutral value of the call option? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))
A stock is currently priced at $64. The stock will either increase or decrease by 20 percent over the next year. There is a call option on the stock with a strike price of $60 and one year until expiration.
Explanation / Answer
Case 1: The stock price will increase to 64 + 0.20 x 64 = 64 + 12.8 = $76.80
Case 2: The stock price will reduce to 64 - 0.20 x 64 = 64 - 12.8 = $51.80
The call option has a strike price of $60 and one year until expiration.
So the value of the call option at the end of the year:
case 1: 76.80 - 60 = $16.80, if the price of the stock is increased by 20% in one year
Case 2: Zero, if the price of the stock is decreased by 20% in one year
The main problem is we are not sure whether the price of the stock will rise or fall. So to headge the risk, if we purchase D stock and sell a call option today,
the portfolio can be written as DS - C where S is the stock price in two respective cases and C is the corresponding values of the call options.
Thus
in case 1: Portfolio value = DS - C = D x $ 76.80 - $16.80, and
in Case 2: Portfolio Value = DS - C = D x $51.80 - 0
The essence of Hedging is that, the portfolio value will remain same in any state of future conditions. Thus,
Dx $76.80 - $16.80 = D x $ 51.20
=> D x $25.60 = $16.80
=> D = $16.80 / $ 25.60 = 0.65625
Therefore for a risk neutral situation,
the investment = D x Current Price of the stock - Call Price = 0.65625 x $64 - C = $42 - C
The pay off in
case 1 : $76.80 x 0.65625 - $16.8 = $ 33.60
Case 2: $51.20 x 0.65625 - 0 = $ 33.60
Thus Present Value of $33.20 must be equal to the investment
$42 - C = 33.6 / (1+0.04)
=> C = $42 - $32.31 = $9.69
So Call Value = $ 9.69
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