Question

You are attempting to value a call option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it has a 50% chance of increasing to $120 and a 50% chance of decreasing to $80. The risk-free rate of interest is 10%. Based upon your assumptions, calculate your estimate of the the call option's value using the two-state stock price model. (

Answer 1

Answer

the call option's value using the two-state stock price model is 13.64

Step-by-step explanation:

First step:

We need to Calculate the option value at expiration based upon the assumption of a 50% chance of increasing to $120 and a 50% chance of decreasing to $80 first.

Then we know that the two possible stock prices are: S+ = $120 and also S- = $120.

Second step:

But the exercise price is $100,

Then we can say the corresponding two possible call values are: Cu = $20 and Cd = $0.

The hedge ratio can be calculated as follow

(Cu - Cd)/(uS0 - dS0) = (20 - 0)/(120- 80) = 0.50

Third step:

Form a riskless portfolio made up of one share of stock and two written calls. The cost of the riskless portfolio can be calculated as

: (S0 - 2C0) = 100 - 2C0

= 100- 2C0

Then the certain end-of-year value is $80

Fourth step:

We need to Calculate the present value of $80 with a one-year interest rate of 10%, which is

$80/1.10 = $72.72

Fifth step:

We need to equate the value of the hedged position to the present value of the certain payoff which is

$100 - 2C0 = $72.72

2C0 = $100 -$72.72

2C0= 27.28

C0 = 13.64

the call option's value using the two-state stock price model is 13.64