The Problem and Goal

This page is focused on approximating the PDEs when itโ€™s difficult to find the closed form solution. This is a study note summarized from the course MF-796 of Boston University Mathematical Finance Master Program.

The kind of PDEs we are interested in are the form of:

\[0 = A(t,x) + B(t,x) \frac{\partial f(t,x)}{\partial t} + C(t,x) \frac{\partial f(t,x)}{\partial x} + D(t,x) \frac{\partial^2 f(t,x)}{\partial x^2}\]

where

\[\frac{\partial f(t,x)}{\partial t} = \lim_{\delta \to 0} \frac{f(t+\delta ,x)-f(t,x)}{\delta}\] Dimensionality reduction:

  • \(\frac{\partial f(t,x)}{\partial t} = \frac{f(t+\delta ,x)-f(t,x)}{\delta} + O(\delta)\)
  • \(\frac{\partial f(t,x)}{\partial x} = \frac{f(t ,x+\Delta)-f(t,x)}{\Delta} + O(\Delta)\)
  • \(\frac{\partial^2 f(t,x)}{\partial x^2} = \frac{f(t ,x+\Delta)-2f(t,x)+f(t, x-\Delta)}{\Delta^2} + O(\Delta^2)\)

Now we can write:

\[0 = A(t,x) + B(t,x) \frac{\tilde{f}(t+\delta,x)}{\delta} + C(t,x) \frac{\tilde{f}(t,x+\Delta)-\tilde{f}(t,x)}{\Delta} + D(t,x) \frac{\tilde{f}(t,x+\Delta)-2\tilde{f}(t,x)+\tilde{f}(t,x-\Delta)}{\Delta^2}\] if \(\delta \approx 0\) and \(\Delta \approx 0\), then \(\tilde{f} \approx f\)

To solve this for every \(t\) and \(x\), we discretize time to \(N=T/\delta\) and space to \(M = X/\Delta\), the equation becomes:

\[0 = A(t_i,x_j) + B(t_i,x_j) \frac{\tilde{f}(t_{i+1}+,x_j)-\tilde{f}(t_{i}+,x_j)}{\delta} + C(t_i,x_j) \frac{\tilde{f}(t_i,x_{j+1})-\tilde{f}(t_i,x_j)}{\Delta} + D(t_i,x_j) \frac{\tilde{f}(t_i,x+{j+1})-2\tilde{f}(t_i,x_j)+\tilde{f}(t_i,x_{j-1})}{\Delta^2}\]

The boundary conditions:

  • \(\tilde{f}(t_N, x_j) = \tilde{f}(T,x_j) = f(T,x_j)\)

This approch of approximating the \(f\) by solving the discretized system of equations is called Finite Differences.

Alternative difference quotients:

  • Forward difference
    \(\frac{\partial f(t,x)}{\partial t} \approx \frac{f(t+\delta ,x)-f(t,x)}{\delta}\)
  • Backward difference
    \(\frac{\partial f(t,x)}{\partial t} \approx \frac{f(t,x)-f(t-\delta,x)}{\delta}\)
  • Central difference
    \(\frac{\partial f(t,x)}{\partial t} \approx \frac{f(t+\frac{\delta}{2},x)-f(t-\frac{\delta}{2},x)}{\delta}\)

Implicit Method

Derivation:

\[ \begin{aligned} 0 &= A(t_i,x_j) + B(t_i,x_j) \frac{\tilde{f}(t_{i+1}+,x_j)-\tilde{f}(t_{i},x_j)}{\delta} + C(t_i,x_j) \frac{\tilde{f}(t_i,x_{j+1})-\tilde{f}(t_i,x_j)}{\Delta} + \\ & \ \ \ \ \ D(t_i,x_j) \frac{\tilde{f}(t_i,x+{j+1})-2\tilde{f}(t_i,x_j)+\tilde{f}(t_i,x_{j-1})}{\Delta^2} \\ \tilde{f}(t_i,x_j) \ [\frac{B(t_i,x_j)}{\delta} + \frac{C(t_i,x_j)}{\Delta} + 2\frac{D(t_i,x_j)}{\Delta^2}] &= A(t_{i},x_j)+\\ & \ \ \ \ \ \tilde{f}(t_{i+1},x_j) \ [\frac{B(t_i, x_j)}{\delta}] + \\ & \ \ \ \ \ \tilde{f}(t_{i},x_{j+1})\ [\frac{C(t_i,x_{j})}{\Delta}+\frac{D(t_i,x_{j})}{\Delta^2}]+\\ & \ \ \ \ \ \tilde{f}(t_{i},x_{j-1}) \ [\frac{D(t_i,x_j)}{\Delta^2}] \\ P^{i,I} \cdot \tilde{f}^i&= Q^{i,I} \tilde{f}^{i+1} + \alpha^{i,I} \\ \tilde{f}^i &= (P^{i,I})^{-1} [Q^{i,I} \tilde{f}^{i+1} + \alpha^{i,I}] \\ \end{aligned} \]

Explicit Method

Crank-Nicolson Method