I had some thoughts while I was trying to sell alcohol to an under-age. I must've been crazy. alright, for a parabolic PDE of the Heston type , the ADE method indicates that for the first sweep we have ^ * *^ * *^ ^ * which requires the computation from the top to the bottom and from the right to the left. We start with solving the point at the upper-right corner (we define i goes from 1 to N and j goes from 1 to M) and then we go down before we move on to the left column. If we introduce the fictitious points, then the middle point can be determined by 8 surrounding points plus its own value at the previous time level. Ignore the 4 explicit points since they are not a problem. If we discretize the zero convexity condition and the same way those implicit and explicit terms are defined in the main equation, the ghost points and can be obsorbed by known interior points and itself. Then the "corner"ghost point U_(N+1,M+1)^(n+1) can also be eliminated. In order to kill off the last ghost point U_(N+1,M-1)^(n+1), we need to know the value of the point U_(N,M-1)^(n+1) which is to be solved next step. Therefore, we build a bi-diagonal matrix to solve for the rightmost boundary points implicitly. After getting the boundary values, we move to the left and the explicit life is always good. For the second sweep, we have* ^ ^* * ^* * ^Similarly, we start from the bottom to the top and from the right to the left. For the Heston model, we have Dirichlet at the bottom, ie. V=0 when S=0. We take natural conditions for vol=0. Fortunetely, the PDE for that condition doesn't come with second and mixed derivatives, so we have ()^ * * ^()*at the leftmost boundary(() means nothing there). Take the starting point U_(1,2)^(n+1) as an example. We can let the "corner" ghost point U_(0,1)^(n+1) be zero in consistency with the Dirichlet condition. Another ghost point U_(0,2)^(n+1) can be completely absorbed according to the natural condition. For the last ghost point U_(0,3)^(n+1), the above graph tells us that we need other 4 interior points. We can again solve the leftmost boundary values from the bottom to the top by constructing another bi-diagonal matrix. After that, the rest of the scheme would be totally explicit. Those are just my rough ideas. We probably need to define the initial ghost point values as a part of initial conditions. With directional computation we can avoid solving some crazy matrices. I think the ADE for the Heston model is workable but I have no clue of its robustness. Hope I made myself clear. Any comments and hints are very much welcome. The Latex editor is not working at this moment. Sorry for the sloppy notations. I will fix them later. gotta go sleep. see you guys.
Last edited by sammus
on March 10th, 2005, 11:00 pm, edited 1 time in total.