We would like to compute d(S T X) + , where S t follows the Geometric Brownian

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

d St St = = (r - q) dt+o (St, t) dZt.The function (ST − X)+ has a discontinuity at ST = X. Rossi (2002) proposed to approximate (ST − X)+ by the following function f (ST) whose first derivative is continuous, where

f(ST) = 0 (ST-X) 2 ST-X- 2 if ST < X if X  ST < X+. if ST if ST X +

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

f(ST) = f(So) + T T 1 fo f" (St) dSt + 5 + 2/2  o (St, t)s f" (S) dt. - 0 By taking the limit   0, explain

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