Please provide complete answers with full calculations, derivations, explanations and proofs:

1. Black Scholes PDE: In a market in which the Black Scholes assumptions are satisfied, an asset makes the terminal payoff (S(T)) = S(T) > 1 where S(T) is the value of the underlying stock at time T. The asset's value if the stock price reaches zero is P(0,1) 0. (a) Show that the price of this power asset at 0 50, and maxosest S, 1 where S(T) is the value of the underlying stock at time T. The asset's value if the stock price reaches zero is P(0,1) 0. (a) Show that the price of this power asset at 0 50, and maxosest S,