3. Let F(t, S(t)) represent the price of a replicating portfolio h(t) whose value is V(t)...

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3. Let F(t, S(t)) represent the price of a replicating portfolio h(t) whose value is V(t) = h(t)TS(t). Assume that the stock price satisfies the Ito process dS(t) = (t, S(t))S(t)dt +(t,S(t))S(t)dW(t) Show that the arbitrage free price II(t) = F(t, S(t)) at time t [0,T], of a contingent claim (S(T)), can be obtained by solving the boundary value PDE 1 F(t, s)+srF(t, s) + - = 0 (t, s)Fss(t,s) rF(t, s) F(T,s) = (s) for all s R+. Here r is the risk free rate. Give a reason why the variable (t, S(t)) does not show up in the PDE. 3. Let F(t, S(t)) represent the price of a replicating portfolio h(t) whose value is V(t) = h(t)TS(t). Assume that the stock price satisfies the Ito process dS(t) = (t, S(t))S(t)dt +(t,S(t))S(t)dW(t) Show that the arbitrage free price II(t) = F(t, S(t)) at time t [0,T], of a contingent claim (S(T)), can be obtained by solving the boundary value PDE 1 F(t, s)+srF(t, s) + - = 0 (t, s)Fss(t,s) rF(t, s) F(T,s) = (s) for all s R+. Here r is the risk free rate. Give a reason why the variable (t, S(t)) does not show up in the PDE.