Call option

A call gives its holder the right to buy the underlying at the strike price. They will only use it when the underlying is worth more than the strike, so the payoff is the underlying price less the strike when that is positive and zero otherwise. The floor at zero is what makes an option an option: the holder cannot be forced into a loss beyond the price they paid for it.

That asymmetry has a consequence people find counterintuitive. Because the losing side is already truncated, anything that widens the range of possible outcomes makes the call more valuable, which is why volatility raises the price of a call even though a volatile share is not a better share. It raises the price of a put for the same reason on the other side.

In banking you rarely price a standalone option. You meet them embedded: the conversion right inside a convertible bond is a call on the issuer's shares held by the bondholder, warrants attached to a mezzanine tranche are calls held by the lender, an issuer's right to redeem a high yield bond early is a call it bought from the investors, and a management incentive pool is a set of calls on the equity of a portfolio company.

The naming trap is worth knowing. A European call may be exercised only at expiry and an American call at any point up to it, and neither name has anything to do with where the option trades or who issued it.