Question

You are attempting to value a call option with an exercise price of $109 and one year to expiration. The underlying stock pays no dividends, its current price is $109, and you believe it has a 50% chance of increasing to $139 and a 50% chance of decreasing to $79. The risk-free rate of interest is 10%. Calculate the call option's value using the two-state stock price model. (Do not round intermediate calculations and round your final answer to 2 decimal places.)

Answer 1

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $139 and a 50% chance of decreasing to $79.

The two possible stock prices are:

S+ = $139 and S– = $79. Therefore, since the exercise price is $109, the corresponding two possible call values are:

Cu= $30 and Cd= $0.

Step 2: Calculate the hedge ratio:

(Cu– Cd)/(uS0– dS0) = (30 – 0)/(139 – 79) = 30/60 = 0.5

Step 3: Form a riskless portfolio made up of one share of stock and two written calls. The cost of the riskless portfolio is:

(S0– 2C0) = 109 – 2C0

and the certain end-of-year value is $79.

Step 4: Calculate the present value of $79 with a one-year interest rate of 10%:

$79/1.10 = $71.82

Step 5: Set the value of the hedged position equal to the present value of the certain payoff:

$109 – 2C0= $71.82

2C0 = $109 - $71.82

C0 = $37.18 / 2 = $18.59

Notice that we never use the probabilities of a stock price increase or decrease. These are not needed to value the call option