Daniel J. Duffy - Numerical Methods in Computational Finance

If we assume that the real parts of the eigenvalues are less than zero, we can conclude that the solution tends to the steady-state. Even though this solution is well-behaved the cause of numerical instabilities is the presence of quickly decaying transient components of the solution caused by the dominant eigenvalues of the matrix A in (2.45).

Let us take an example whose matrix has already been given in diagonal form:

The solution of this system is given by:

The solutions decay at different rates, and in the case of the explicit Euler method the inequality:

must be satisfied if the first component of the solution is to remain within the region of absolute stability. Unfortunately, choosing a time step of these proportions will be too small to allow for control over the round-off error in the second component.

In this case we fit the dominant eigenvalue. For variable coefficient systems and non-linear systems, we periodically compute the Jacobian matrix and carry out fitting on it.

The presence of different time scales in ODEs leads to a number of challenges when approximating them using the standard finite difference schemes. In particular, schemes such as explicit and implicit Euler, Crank–Nicolson, and predictor-corrector do not approximate these systems well, unless a prohibitively small time step is used. Let us take the example (Dahlquist and Björck (1974)):

with exact solution:

This is a stiff problem because of the different time scales in the solution. We carried out an experiment using the explicit Euler method, and we had to divide the interval into 1.2 million subintervals in order to achieve accuracy to three decimal places. The implicit Euler and Crank–Nicolson methods are not much better.

Robust ODE solvers for stiff system using the Boost C++ library odeint are discussed in Duffy (2018).

A special case of an initial value problem is when the number of dimensions n in an initial value problem is equal to 1. In this case we speak of a scalar problem, and it is useful to study these problems if one wishes to get some insights into how finite difference methods work. In this section we discuss some numerical properties of one-step finite difference schemes for the linear scalar problem:

where

The reader can check that the one-step methods ( Equations (2.10), (2.11)and (2.12)can all be cast as the general form recurrence relation :

where Then, using this formula and mathematical induction we can give an explicit solution at any time level as follows:

with:

for a mesh function . A special case is when the coefficients and are constant , that is:

Then the general solution is given by:

where we note that power of constant and .

In order to prove this, we need the formula for the sum of a series:

For a readable introduction to difference schemes, we refer the reader to Goldberg (1986).

In this chapter we gave an introduction to scalar ODEs and systems of ODEs. The main goal was to help the reader become acquainted with their mathematical and numerical foundations as well as become familiar with the associated notation. We recommend learning the main concepts in this chapter because many of them will be used and needed when we model one-factor and two-factor convection-diffusion-reaction PDEs such as the Black–Scholes equation, for example.

Contrary to popular thinking, there is more to ODEs than trying to find analytical solutions for them. Very few ODEs have analytical solutions, and we must resort to ODE solvers.

The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case .