Let r (interest rate), (volatility ), K (strike price ), T (time to expiration) and S(0) (present value of the stock) be positive constants. Suppose that the price of a stock at time t > 0 can be expressed as (see the attached picture for complete question)

Problem 3.2. Let r (interest rate), a (volatility ), K (strike price ), T (time to expiration) and 3(0) (present value of the stock) be positive constants. Suppose that the price of a stock at time t > 0 can be expressed as S(t) = 3(0) exp ((r' _ 1 2):: + m/tZ) 50' where Z N N(0, 1) is the standard normal with c.d.f. @(z). Show that: WT susm _ K)+] = 3(0) @(w) HT Ken.) 0%) Where _ i'T + %02T log(%) (,0 (EH/T 1 and 55+ = maa, O) = (|rs| + (B) for any real number rs