4. The value W(R, t) of an option on an underlying asset with price R at time t satisfies the Black Scholes PDE aw . . +rS TW 40 (2) for 0 Vio? iii) The price V of the bull put option spread at time t is V-V-V. Write down the Black-Scholes PDE for the combined bull put option spread, i.e., the equation which is satisfied by V. (User for the risk-free interest rate, o for the volatility, t for time and S for the price of the underlying at time t.) iv) Carefully explain what the "initial" conditions are for this problem. v) Carefully explain what the boundary conditions are for this problem. vi) Let Smax 0 be the maximal price of the underlying used in the finite difference scheme and t [0,7]. Set up a grid (St,) = (nAS, jAt) for n 0,1,2,N+1 and j = 0, 1, 2,..., J, where AS = Smax N+1 T and J Let VV (St) denote the approximate solution. At the time point t; and underlying price S,, give a) central difference approximations of O(AS) for the partial deriva- tives av aV and b) the forward difference approximation of O(At) for the partial deriva- tive at e) We want to use a finite difference method for computing an approxi- mate solution to equation Black Scholes equation for the bull put option spread. A) Show that the finite difference equation has the form V+an+B V + YnV+ = nn-1 and find the coefficients an, B and n n+1 B) Write down the 'initial' and boundary conditions for V. C) Is this method an implicit method or an explicit method? (3)