Question 1: = The Black-Scholes equation for a European put P(S,t) can be trans- formed into the diffusion equation au (1) ar2 using the transformations S = Ee, t =T 27/02 and P = Eu(x, T)exp (k 1). (k+1) 7 2 4 (-47 (+7):), 0 where the constant k = 2r/o is determined by the volatility o and the risk free interest rate r. The initial condition is given by u = uo(2) at T = 0, where Uo(x) = max exp (5(x 1)a) exp $($(x + 1)v),0). k (2) The boundary conditions are u(1,T) + 0 as r + +0, (3) and u(,7) exp (x - 1) = + (+ 12 )) - k. 2 k + as -oo. (4) 4 Let the problem (1)-(4) be solved numerically by a finite difference method. A computational grid used for numerical solution of (1)-(4) is generated in the domain [z ,x+] [O, Tmax] with the grid step size 8x and time step size . 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 when the diffu- sion problem is solved? (b) Let an explicit finite difference method be used for numerical solution of the problem (1)-(4). The user starts computation on a grid with the grid step size 8x = 0.2 and the time step size defined as St = 0.481. (5) a The user compares the result of the computation with the exact solution available to him and he decides to decrease the grid step size or by a factor of two to compute a more accurate numerical solution. The time step size is also redefined for the new value of 8x according to (5). 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 (1)-(4) as a result of the above- mentioned procedure? Explain your answer. (c) What is an unconditionally stable finite difference method when the diffusion problem is solved? (d) Formulate an implicit finite difference method for numerical solution of the problem (1)-(4). 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 (1)-(4) when the implicit finite difference method is employed? Explain your answer. Question 1: = The Black-Scholes equation for a European put P(S,t) can be trans- formed into the diffusion equation au (1) ar2 using the transformations S = Ee, t =T 27/02 and P = Eu(x, T)exp (k 1). (k+1) 7 2 4 (-47 (+7):), 0 where the constant k = 2r/o is determined by the volatility o and the risk free interest rate r. The initial condition is given by u = uo(2) at T = 0, where Uo(x) = max exp (5(x 1)a) exp $($(x + 1)v),0). k (2) The boundary conditions are u(1,T) + 0 as r + +0, (3) and u(,7) exp (x - 1) = + (+ 12 )) - k. 2 k + as -oo. (4) 4 Let the problem (1)-(4) be solved numerically by a finite difference method. A computational grid used for numerical solution of (1)-(4) is generated in the domain [z ,x+] [O, Tmax] with the grid step size 8x and time step size . 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 when the diffu- sion problem is solved? (b) Let an explicit finite difference method be used for numerical solution of the problem (1)-(4). The user starts computation on a grid with the grid step size 8x = 0.2 and the time step size defined as St = 0.481. (5) a The user compares the result of the computation with the exact solution available to him and he decides to decrease the grid step size or by a factor of two to compute a more accurate numerical solution. The time step size is also redefined for the new value of 8x according to (5). 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 (1)-(4) as a result of the above- mentioned procedure? Explain your answer. (c) What is an unconditionally stable finite difference method when the diffusion problem is solved? (d) Formulate an implicit finite difference method for numerical solution of the problem (1)-(4). 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 (1)-(4) when the implicit finite difference method is employed? Explain your
