Question 21.5: Two European call options with a strike price of $50 are wri...
Two European call options with a strike price of $50 are written on two different stocks. Suppose that tomorrow , the low-volatility stock will have a price of $50 for certain. The high-volatility stock will be worth either $60 or $40, with each price having equal probability. If the exercise date of both options is tomorrow, which option will be worth more today?
PLAN
The value of the options will depend on the value of the stocks at expiration. The value of the options at expiration will be the stock price tomorrow minus $50 if the stock price is greater than $50, and $0 otherwise.
EXECUTE
The low-volatility stock will be worth $50 for certain, so its option will be worth $0 for certain. The highvolatility stock will be worth either $40 or $60, so its option will pay off either $0 or $60 – $50 = $10.
Because options have no chance of a negative payoff, the one that has a 50% chance of a positive payoff has to be worth more than the option on the low-volatility stock (with no chance of a positive payoff).
EVALUATE
Because volatility increases the chance that an option will pay off, the options have very different values even though the expected value of both stocks tomorrow is $50—the low-volatility stock will be worth this amount for sure, and the high-volatility stock also has an expected value of \$40(\frac{1}{2} )+\$60(\frac{1}{2} )=\$50.