4. 20 Points: Suppose a stock price ST is lognormal under the risk-neutral measure P~, specifically, we get that for any 0t0. Show that Pt the price of the derivative security at time t with payoff f(s)=sp is given by Pt=(St)pe(22p+r)(p1)(Tt). In other words, show that (St)pe(2e2p+r)(p1)(Tt)=E~t[er(Tt)(St)pep((r222)(Tt)+TtZ)]. 4. 20 Points: Suppose a stock price ST is lognormal under the risk-neutral measure P~, specifically, we get that for any 0t0. Show that Pt the price of the derivative security at time t with payoff f(s)=sp is given by Pt=(St)pe(22p+r)(p1)(Tt). In other words, show that (St)pe(2e2p+r)(p1)(Tt)=E~t[er(Tt)(St)pep((r222)(Tt)+TtZ)]