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%. Calculate the call option’s value using the two-state stock price model.

Short answer

$13.64

Step-by-step solution

Calculation of values at expiration

Two possible stock prices are: S+ = $120 and S– = $80.

Two possible call values are: Cu = $20 and Cd = $0 (as exercise price = $100)

Calculation of hedge ratio (H)

H = (Cu – Cd)/(uS0 – dS0)

= (20 – 0)/(120 – 80)

= 0.5

Now the cost of the riskless portfolio is: (S₀ – 2C₀) = 100 – 2C₀

End-of-year value is $80 (given)

Calculation of present value

Present value of $80 with a one-year interest rate of 10%: $80/1.1 = $72.73

The value of the hedged position equal to the present value of the certain payoff: The value of the call option (Co)

Current price - 2C₀ = Present value

$100 – 2C₀ = $72.73

C₀ = $13.64

Note that the probabilities of a stock price increase or decrease are not needed to value the call option.