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The price of a cash-or-nothing American binary put (resp. call) with strike <math>K<S</math> (resp. <math>K>S</math>) and time-to-expiry <math>T</math> is:
:<math> \frac{1}{2}e^{a\left(\xi-b\right)}\left\{ 1+\operatorname{sgn}(a)\operatorname{erf}\left(\frac{bT-a}{\sqrt{2T}}\right)+e^{2ab}\left[1-\operatorname{sgn}(a)\operatorname{erf}\left(\frac{bT+a}{\sqrt{2T}}\right)\right]\right\} \,</math>
where <math>\operatorname{erf}</math> denotes the [[error function]] and <math>\operatorname{sgn}</math> denotes the [[sign function]]. The above follows immediately from expressions for the Laplace transform of the distribution of the conditional first passage time of Brownian motion to a particular level.<ref>[
===Foreign exchange===
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