Consider the Black-Scholes model for the price S(t) of a stock: dS(1) = S(1)dt + S(1) W(1), where rand o are constants, and W(1) E R is the Wiener process. Consider the 7-claim x = S(T)", where is a constant and I the expiry date. (a) (20 marks) By application of Ito's formula, show that E ($(t) satisfies a differential equation of the form u' (t) = ku(t), where k is a constant. (b) (14 marks) Derive the arbitrage-free price process II(t) of the claim x by solving the differential equation derived in Part (a). Consider the Black-Scholes model for the price S(t) of a stock: dS(1) = S(1)dt + S(1) W(1), where rand o are constants, and W(1) E R is the Wiener process. Consider the 7-claim x = S(T)", where is a constant and I the expiry date. (a) (20 marks) By application of Ito's formula, show that E ($(t) satisfies a differential equation of the form u' (t) = ku(t), where k is a constant. (b) (14 marks) Derive the arbitrage-free price process II(t) of the claim x by solving the differential equation derived in Part (a)