Consider two types of barrier options with barrier price X , (1) an up-and-out call option, and (2) a down- and-out put option. The up-and-out option begins with S < X , and expires immediately and with zero payoff if the price S rises to X . The down-and-out option begins with S > X and expires immediately and with zero payoff if the price S falls to X . ( a ) For the up-and-out call option, suppose that X < E . What must be true about the price of such an barrier call option? Explain. ( b ) For the down-and-out put option, suppose that X > E , What must be true about the price of such a barrier put option? Explain. ( c ) Now suppose that X < E for the down-and-out put option. Given that for a plain van illa option, we have the boundary conditions, P ( S ( T ) , T ) = max( E S ( T ) , 0) , P ( S ( t ) , t ) 0 as S ( t ) , P (0 , t ) = e r ( T t ) E , derive the three boundary conditions for the down-and-out p ut option. Assume the option to be european. (Hint: In 9.3.1, we derived the boundary conditions for the knock-out call.) ( d ) Transform the boundary conditions for the down-and-out pu t option which you derived in part (c) from ( S, t ) coordinates to ( x, ) coordinates using the transformations which convert the B lack-Scholes PDE to the parabolic PDE: u/ = 2 u/x 2 . As in the notes, define x B = ln[ X/E ]. You may find that 5.4 and 9.3.1 of the notes are helpful. Write P DO for V so that, for example, P BO ( S ( t ) , t ) = E v ( x, ). First transform the conditions in terms of v ( x, ), then convert to u ( x, ).