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We are looking at two call options on the same stock, one with an exercise price of $20 and one with an exercise price of $30. The stock currently sells for $35. Its future price will be either $25 or $50. If the risk-free rate is 10 percent, what are the values of these call options?

Question

Don’t Call Us, We’ll Call You

We are looking at two call options on the same stock, one with an exercise price of $20 and one with an exercise price of $30. The stock currently sells for $35. Its future price will be either $25 or $50. If the risk-free rate is 10 percent, what are the values of these call options?

Accepted Answer

The first case (with the $20 exercise price) is not difficult because the option is sure to finish in the money. We know that the value is equal to the stock price less the present value of the exercise price:

[latex]C_{0} =S_{0}-E/(1+R_{f})[/latex]
= $35 − $20/1.1
= $16.82

In the second case, the exercise price is $30, so the option can finish out of the money. At expiration, the option is worth $0 if the stock is worth $25. The option is worth $50 − 30 = $20 if it finishes in the money.
As before, we start by investing the present value of the lowest stock price in the risk-free asset. This costs $25/1.1 = $22.73. At expiration, we have $25 from this investment.

If the stock price is $50, then we need an additional $25 to duplicate the stock payoff. Be-cause each option is worth $20 in this case, we need $25/$20 = 1.25 options. So, to prevent arbitrage, investing the present value of $25 in a risk-free asset and buying 1.25 call options must have the same value as the stock:

[latex]S_{0} = 1.25 imes C_{0}[/latex]+ $25/(1 + [latex]R_{f} [/latex])

$35 = 1.25 × [latex]C_{0}[/latex] + $25/(1 + .10)
= $9.82

Notice that this second option had to be worth less because it has the higher exercise price.

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