QUESTION 1Assume the Black-Scholes setting for an arbitrage-free financial market, i.e.dS(t) = rS(t)dt + σS(t)dW˜ (t) and S(0) = S0.Consider a European security with payoff ST2.(a)Find the arbitrage-free price of this security V0 at time 0.(b)Write down the arbitrage-free price of this security at time t, as a function of t and S.(c)Show that the function you have had in part (b) satisfies the Black-Scholes PDE and its terminal condition.QUESTION 2An American digital call struck at K has a payoff equal to 1 at time T if the stock price hits the level K at some time during the life of the option, i.e. until maturity T, and otherwise 0. Namely, it has the payoffVT = 1MTS≥K(a)Develop a formula that gives the arbitrage-free price of this security.(b)In the text there is pricing formula developed for American digital call. Develop also a formula for European digital call.(C)Suppose an asset follows geometric Brownian Motion and there are no interest rates. What can we say about the relative prices of out of the money American and European digital calls.