2 = max exp as 2-0. The Black-Scholes equation for a European put P(S. t) can be transformed into the diffusion equation aru ar2 using the transformations S = Ee*, t = T 21/o- and P = Eu(x, T)exp (k 1). (k +1) - where the constant k = 2r/o is determined by the volatility a and the risk free interest rate r. The initial condition is given by u = uo(2) at T = 0, where uo(2) 1). k (2) The boundary conditions are u(,T) + 0 as 2 + +0, (3) and (k 1). (k+1) u 2,T) exp (4) Let the problem (??)-(??) be solved numerically by a finite difference method. Consider a computational grid used for numerical solution of (??)-(??) in the domain [x-,x+] [0, Tmar = oT/2]. We set integers N- 0 (N- = N+) and M > 0 and define 8x *+/+ (8x = r-/N-) and 8T = T/M, where &r >0 and 87 > are the grid step size and time step size, respectively. The grid is then the discrete set of points (li, Tj) [x-,x+] x [0, 0T/2], where L; = idr for N-Sist and t; = jt for 0 Si SM. Students are required to independently investigate the topics of stability and convergence with regards to finite difference methods used to solve the initial boundary-value diffusion problem in order to answer the following questions: (a) What is a conditionally stable finite difference method? (b) Suppose that an explicit finite difference method is used for the numerical solution of the problem (??)-(??). The user starts computation on a grid with the grid step size 8x = 0.1 and the time step size defined as t = 0.88 2. (5) The user compares the result of the computation with the exact solution available to him and he decides to decrease the grid step size 8x by a factor of two to compute a more accurate numerical solution. The time step size is also redefined for the new value of dr according to (??). The user intends to repeat this procedure (i.e. halving the grid step size 8x and redefining the time step 8t accordingly) as many times as required for the numerical solution to converge to the exact solution within the prescribed tolerance. Will the numerical solution converge to the exact solution of the problem (??)-(??) as a result of the above-mentioned procedure? Explain your answer. (c) What is an unconditionally stable finite difference method? (d) Formulate an implicit finite difference method for the numerical solution of the problem (??)-(??). What are the appropriate discrete initial and boundary conditions for u(1,7)? Will the numerical solution obtained as a result of the computational procedure explained in (b) converge to the exact solution of the problem (??)-(??) when the implicit finite difference method is employed? Explain your answer. 2 = max exp as 2-0. The Black-Scholes equation for a European put P(S. t) can be transformed into the diffusion equation aru ar2 using the transformations S = Ee*, t = T 21/o- and P = Eu(x, T)exp (k 1). (k +1) - where the constant k = 2r/o is determined by the volatility a and the risk free interest rate r. The initial condition is given by u = uo(2) at T = 0, where uo(2) 1). k (2) The boundary conditions are u(,T) + 0 as 2 + +0, (3) and (k 1). (k+1) u 2,T) exp (4) Let the problem (??)-(??) be solved numerically by a finite difference method. Consider a computational grid used for numerical solution of (??)-(??) in the domain [x-,x+] [0, Tmar = oT/2]. We set integers N- 0 (N- = N+) and M > 0 and define 8x *+/+ (8x = r-/N-) and 8T = T/M, where &r >0 and 87 > are the grid step size and time step size, respectively. The grid is then the discrete set of points (li, Tj) [x-,x+] x [0, 0T/2], where L; = idr for N-Sist and t; = jt for 0 Si SM. Students are required to independently investigate the topics of stability and convergence with regards to finite difference methods used to solve the initial boundary-value diffusion problem in order to answer the following questions: (a) What is a conditionally stable finite difference method? (b) Suppose that an explicit finite difference method is used for the numerical solution of the problem (??)-(??). The user starts computation on a grid with the grid step size 8x = 0.1 and the time step size defined as t = 0.88 2. (5) The user compares the result of the computation with the exact solution available to him and he decides to decrease the grid step size 8x by a factor of two to compute a more accurate numerical solution. The time step size is also redefined for the new value of dr according to (??). The user intends to repeat this procedure (i.e. halving the grid step size 8x and redefining the time step 8t accordingly) as many times as required for the numerical solution to converge to the exact solution within the prescribed tolerance. Will the numerical solution converge to the exact solution of the problem (??)-(??) as a result of the above-mentioned procedure? Explain your answer. (c) What is an unconditionally stable finite difference method? (d) Formulate an implicit finite difference method for the numerical solution of the problem (??)-(??). What are the appropriate discrete initial and boundary conditions for u(1,7)? Will the numerical solution obtained as a result of the computational procedure explained in (b) converge to the exact solution of the problem (??)-(??) when the implicit finite difference method is employed? Explain your