Help with heat equation

Hello I have the following question

I am trying to understand how to do from the heat equation part Can anyone help?

Answer 1

[Image Analyst]

Try

sigma = 0.2;
r = 0.05;
S0 = 10 : 10 : 100;
K = 50
T = 2
xmax = 10
xmin = -10
N = [10, 100, 200, 400, 800];
M = N
for k = 1 : length(N)
    thisM = M(k);
    thisN = N(k);
    blsprice(................
    absoluteError = .........
    plot(S0, absoluteError, 'b*-');
    hold on;
    grid on;
end

I don't know what "the Call Option" or the heat equation or blsprice() is so I can't help with that, but at least this is a start. Presumably blsprice() uses those parameters as inputs. I'm also not sure how to calculate the absolute error.

[Janvier Solaris]

Thank you for your reply

This is what I have so far:

S0= 10:10:100;
K = 50;
r = .05; 
T = 2; 
sigma = .2;

callV0 = blsprice(S0, K, r, T, sigma)

M = [ 10 100 200 400 800];

N = M;

xmin = -10;

xmax = 10;

dx = (xmax-xmin)/(M+1);                     

dt = T/N;                          

r= dt/(dx*dx);                          

v = zeros(M,N);                        

x = linspace(xmin,xmax, M)                  

v(:,1) = max(S0*exp((r-sigma**2/2)*T+sigma*x)-K,0)  

%v(:,1) = max(exp(x)-1,0);              

         for n = 2:N                      

           %lines below are initial and boundary conditions
             v(1,n) = v(1, 1);            
              v(M,n) = v(M,1);

              v(2:M-1,n) = (1-2*r)*v(2:M-1,n-1)+r*v(1:M-2,n-1)+r*v(3:M,n-1);
         end

error = abs(callV0 - V)

surf(v)

The issue and where I am getting stuck are my boundary conditions. I am assuming xmin and xmax are the boundary conditions but I don't know how to incorporate them where I have initial conditions

[Image Analyst]

You can't do

for n = 2:N

because N is an array. You have to think what you really want. You need to extract the values of M and N from the array and use them somehow.

Even if you did

for n = N

which is allowed, n would take on the value of N(1) first, which is 10. So then you say

v(1, 10) = v(1,1);

well, what about the rest of the array? What about columns 2 through 9?

You also shouldn't do

v = zeros(M,N);

because they're arrays. You can if you use thisM and thisN and put that inside the loop as I showed you. Take another crack at it.

[Janvier Solaris]

Hi @Image Analyst, I was able to modify my code as such which works similarly to your modifications;

    S0 = 10:10:100;
  K = 50;
  r = .05;
  T = 2;
  sigma = .2;
  price = blsprice(S0,K,r,T,sigma)

amax = 10;
amin = -10;
M = [10 100 200 400 800];
N = [10 100 200 400 800];
for k = 1:5

      dx = (amax - amin)/(M(k)-1);

      dt = T/(N(k)-1);                         

      v = zeros(M(k), N(k));                       

      r= dt/dx^2;                          

      x = linspace(amin,amax,M(k));
      v(:,1) = max(exp(x)-1,0);               

         for n = 2:N(k)                      

           %2 lines below are initial and boundary conditions
             v(1,n) = v(1, 1);            
              v(N(k),n) = v(M(k),1);

              v(2:M(k)-1,n) = (1-2*r)*v(2:M(k)-1,n-1)+r*v(1:M(k)-2,n-1)+r*v(3:M(k),n-1);

         end

         approx = v(2:M(k)-1,n)
  end
      error = abs(approx - price)  
     B= table(approx')
     G=table(price')

surf(v)

the loop grabs every item within the array as thisN and thisM would have .

Now the only issue I am getting is that I am getting the error, the limits are too large and it is not plotting for my 3d plot. when I change the array to for k = 1: 3 then it works but not for 400 and higher would you know anything about "limits are too large" error?