Question

You are trying to value a one-period call option on Twitter with a strike price of $32. The current risk-free rate is 10%. Next year, the price of Twtr will be $38 if the economy is good or $22 if the economy is bad. Unfortunately, you don’t know the current stock price for Twtr. However, you know that a put option on the stock with a strike price of $30 has a current price of $2.80. What is the value of the call option?

Answer 1

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $38 and a 50% chance of decreasing to $22.

The two possible stock prices are:

S+ = $38 and S– = $22. Therefore, since the exercise price is $32, the corresponding two possible call values are:

Cu= $6 and Cd= $0.

Step 2: Calculate the hedge ratio:

(Cu– Cd)/(uS0– dS0) = (6 – 0)/(38 – 22) = 6/16 = 0.375

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 – 2.67C0) = 32 – 2.67C0

and the certain end-of-year value is $22.

Step 4: Calculate the present value of $22 with a one-year interest rate of 10%:

$22/1.10 = $20

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

$32 – 2.67C0= $20

2.67C0 = $32 - $20

C0 = $12 / 2.67 = $4.50