Question

You are attempting to value a call option with an exercise price of $75 and one year to expiration. The underlying stock pays no dividends, its current price is $75, and you believe it has a 50% chance of increasing to $95 and a 50% chance of decreasing to $55. 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. (Do not round intermediate calculations. Round your 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 $95 and a 50% chance of decreasing to $55. The two possible stock prices are: S+= $95 and S–= $55. Therefore, since the exercise price is $75, the corresponding two possible call values are: C_u= $20 and C_d= $0.

Step 2: Calculate the hedge ratio: (C_u– C_d)/(uS₀– dS0) = (20 – 0)/(95 – 55) = 0.50

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) = 75 – 2C₀and the certain end-of-year value is $55.

Step 4: Calculate the present value of $55 with a one-year interest rate of 10%:

$55/1.10 = $50.00

Step 5: Set the value of the hedged position equal to the present value of the certain payoff: $75 – 2C₀ = $50.00

Step 6: Solve for the value of the call: C₀ = $12.50

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