Derivation of the closed-form formulas for European options Let DT = max(0, ST – K) be the payoff of a European call with strike K and maturity T. The following questions provide a step-by-step derivation of the closed-form formulas for c0 and p in the lognormal model.

(a) Show that  where  and  otherwise.

(b) Show that ST has the same distribution as 

(c) Show that Y > K if and only if 

(d) Using the change of variable , show that .

(e) Establish the closed-form formulas for c0 and p0.

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