You are attempting to value a call option with an exercise price of $70 and one
ID: 2790541 • Letter: Y
Question
You are attempting to value a call option with an exercise price of $70 and one year to expiration. The underlying stock pays no dividends, its current price is $70, and you believe it has a 50% chance of increasing to $85 and a 50% chance of decreasing to $55. The risk-free rate of interest is 9%. Based upon your assumptions, calculate your estimate of the the call option's value using the two-state stock price model. (Do not round intermediate calculations. Round your answer to 2 decimal places.)Explanation / Answer
The value of the call option using the 2state model is $ 9.77. Working is provided below:
Working
First lets calculate the value of the call option at expiration. Value of Call:
C= Max(S-X, 0)
Where S= Stock Price
X=Strike Price
Su=85 Cu=Max (85-70,0)=15
Sd=55 Cd=Ma x(55-70,0)=0
Therefore, hedge ratio equals:
Cu-Cd/Su-Sd
=15-0/85-55
=15/30
=1/2
Therefore, to form a risk less portfolio if we buy one stock then we have to short 2 call options with the same strike that is 70 (hedge ratio is 1/2).
Hence the cost of our portfolio will S-2C = 70-2C
Now we would need to calculate the payoff for the risk less portfolio:
Payoff
S=85
S=55
Buy 1 share
85
55
Sell 2 Calls (X=70)
-30
0
Sum
55
55
Irrespective of what happens the riskless portfolio will have a payoff of $ 55 in 1 year. Now we need to take the present value of the payoff using discount rate 9%.
=55/1.09
=$ 50.4587
Therefore now the value of our call can be found as:
70-2C=50.4587
2C=19.5413
=9.77064=$9.77
Payoff
S=85
S=55
Buy 1 share
85
55
Sell 2 Calls (X=70)
-30
0
Sum
55
55
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