Digital Options: Digital (Or Binary) Options Can Only Have Two $\bullet$

Digital options: Digital (or binary) options can only have two payoffs at maturity. If the strike condition set in the option is met, the buyer will receive the full prespecified payoff. If not, the buyer receives no payoff. This is different from a traditional option where there exists an infinite number of payoffs. For example, we could have a digital option on the French CAC index, stating that the option buyer will get €200 if the CAC index is above 4,000 at expiration and zero otherwise. On this digital option, the buyer will get exactly 200 as soon as the CAC index is above the 4,000 level at expiration, whether it be 4,001, 4,100, or 5,000.
a. Draw the profit and loss curve at expiration as a function of the CAC index for these two options:
$\bullet$ Traditional call on the CAC index: Exercise price: 4,000; premium: 40.
$\bullet$ Digital call on the CAC index: Exercise price: 4,000; payoff if exercised: 200; premium: 40.
b. What are the relative advantages of the two options?
c. Assume that the volatility of the French stock market increases suddenly. Should the premium on the digital call increase more (or less) than the premium on the traditional call?