An immediate consequence of Eq. (13.26), is that (modulo a discount factor) it gives, for free, the BSM price of a digital (or binary) option, i.e., an option paying \(\$ 1\) if . Using the indicator function

we find

It is important to realize that, in this case, we must calculate \(d_{2}\) using the risk-neutral drift \(r\), as we are pricing the option.
Also note that the binary call features a discontinuous payoff at maturity, but thanks to the parabolic nature of the BSM equation, we find a continuous price as a function of \(S_{t}\). As one may expect, when approaching maturity, the derivative of the option price with respect to \(S_{t}\) will get steeper and steeper, as shown in Fig. 13.18. As we discuss in Section 13.8.3, this could make hedging difficult.

Data From Equation (13.26)

Data From Fig. 13.18

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