I recently had to derive the pricing formula for an American digital call option (or American binary/one touch call option). Although the final solution can be readily found in the internet, I was unable to find a full derivation of both, the first passage time of a drifted Brownian motion and the option value.
Although the pricing equation for the European and the American digital call only slightly differ by the discount factor of the first passage time, the resulting integral is a little more complicated to solve. The attached document provides the full derivation of both results.
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