Value of a Call Option Stock in the Nantucket Corporation is currently selling for $25 per share. In one year, the price will be either $20 or $30. T-bills with one year to maturity are paying 10 percent. What is the value of a call option with a $20 exercise price? A $26 exercise price?

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Accepted Answer

With a $20 exercise price, the option can’t finish out of the money (it can finish “at the money” if the stock price is $20). We can replicate the value of the stock by investing the present value of $20 in T-bills and buying one call option. Buying the T-bill will cost $20/1.1 = $18.18.
If the stock ends up at $20, the call option will be worth zero and the T-bill will pay $20. If the stock ends up at $30, the T-bill will again pay $20, and the option will be worth $30 − 20 = $10, so the package will be worth $30. Because the T-bill–call option combination exactly duplicates the payoff on the stock, it has to be worth $25 or arbitrage is possible. Using the notation from the chapter, we can calculate the value of the call option:

[latex]S_{0} = C_{0} + E/(1 + R_{f})[/latex]
$25 = [latex]C_{0}[/latex] + $18.18
[latex]C_{0}[/latex] = $6.82

With the $26 exercise price, we start by investing the present value of the lower stock price in T-bills. This guarantees us $20 when the stock price is $20. If the stock price is $30, then the option is worth $30 − 26 = $4. We have $20 from our T-bill, so we need $10 from the options to match the stock. Because each option is worth $4 in this case, we need to buy $10/$4 = 2.5 call options. Notice that the difference in the possible stock prices (ΔS) is $10 and the difference in the possible option prices
(ΔC) is $4, so ΔS/ΔC = 2.5.
To complete the calculation, we note that the present value of the $20 plus 2.5 call options has to be $25 to prevent arbitrage, so:

$25 = 2.5 × [latex]C_{0}[/latex] + $20/1.1
[latex]C_{0}[/latex] = $6.82/2.5
= $2.73

Question 24.S-TP.1: Value of a Call Option Stock in the Nantucket Corporation is...

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