We would like to compute d(S_T −X)⁺, where S_t follows the Geometric Brownian process

The function (S_T − X)⁺ has a discontinuity at S_T = X. Rossi (2002) proposed to approximate (S_T − X)⁺ by the following function f (S_T) whose first derivative is continuous, where

Here, є is a small positive quantity. By applying Ito’s lemma, show that

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d St St = = (r - q) dt + o (St, t) dZt.