Question

You are attempting to value a call option with an exercise price of $108 and one year to expiration. The underlying stock pays no dividends, its current price is $108, and you believe it has a 50% chance of increasing to $133 and a 50% chance of decreasing to $83. The risk-free rate of interest is 9%. Calculate the call option’s value using the two-state stock price model.

Answer 1

Calculate the call-option value.
Given data:
Exercise price X = $108
Time to expiration T = 1 year
Current price of stock= $108
Value of stock when price goes up = $133
Value of stock when price goes down = $83
Risk-free interest rate= 9%
= 0.09
Calculate the hedge ratio.
Hedge ratio is the fraction of the difference between the range of option price and the range of stock price.
Symbolically,

Here, H is the hedge ratio,C​​​​​_{​​​​​​u} is the value of call option when stock price goes up,C_{​​​​​​d} is the value of call option when stock price goes down,uS_{​​​​​0} is the value of stock when price goes up and dS_{​​​​​0} is the value of stock when price falls. It is beneficial to exercise the call option when the price rises beyond the exercise price. For this reason C_{​​​​​​d} is taken as zero.

C_{​​​​​​U} = uS_{​​​​​0} - X

= $133 - $108

= $25

C_{​​​​​​d} = 0

Substitute $25 for Cu, 0 for Cd, $133 for uS0 and $83 for dS0 in equation (1) to find the hedge ratio H

H = (25-0)/(133-83)

H = 25/50

= 0.5
The portfolio to be risk-free should comprise of one share and two call options.

Cost of the portfolio = cost of the stock - cost of the two calls

= S - 2C

= 108 - 2C

The following table shows the payoff of the portfolio to be risk free.

Portfolio	S = $83	S = $133
Buy 1 share	83	133
Write 2 calls	0	-50
Total	83	83

Present value of the portfolio = 83/1.09 = 76.1468

The value is the difference between the exercise price and the value of option.

108 - 2C = 76.15

C = 15.925

Therefore the value of the option is 15.925