Volume 9, Issue 2
Tricks of the Trade
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Path Integral Methods for Parabolic Partial Differential Equations with Examples from Computational Finance
7. Concluding Remarks
Although the examples presented in Section 6 were borrowed mostly from the realm of computational finance, it is important to realize that neither the general method developed in Sections 3 and 5, nor the Mathematica code used in Section 6 takes advantage of the special nature of the problems associated with financial applications in any way. The same method--and, indeed, the same computer code--can produce an approximate solution to a rather general one-dimensional parabolic PDE with boundary conditions imposed on some free or fixed boundary. Furthermore, this method does not require any manipulation whatsoever of the PDE.
As is well known, the "no free lunch" principle reigns in the world of computing, too, and the simplicity, the generality, and the precision with which we were able to solve some classical option valuation problems in the previous section did not come easily by any means. In terms of this metaphor, not only was the lunch not free, but it was, in fact, a rather expensive one. Indeed, an efficient integration procedure was absolutely crucial for the implementation of the method in Section 6. As explained in The Mathematica Book (see  p. 1071) the function Integrate "uses about 500 pages of Mathematica code and 600 pages of C code." The role played by the function ListInterpolation was just as crucial. Indeed, it allowed us to work on an exceptionally coarse grid in the state-space and then "restore" the values at all remaining points by a fairly sophisticated interpolation procedure. More importantly, we were able to feed NIntegrate with the object produced by ListInterpolation. There are intrinsic obstacles to such "shortcuts" in essentially all modifications of the finite difference method.
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