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Numerical Methods in Computational Finance. Daniel J. Duffy

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Numerical Methods in Computational Finance - Daniel J. Duffy

example, theorem 5.2.1 in Øksendal (1998) addresses these issues. We discuss SDEs in more detail in Chapter 13.

3.5 NUMERICAL METHODS FOR ODES

In this section we introduce a class of one-step methods to approximate the solution of ODE system (3.1).

The first step is to replace continuous time by discrete time. To this end, we divide the interval [0, T] into a number of subintervals. We define upper N plus 1 mesh points as follows:

0 equals t 0 less-than t 1 less-than ellipsis less-than t Subscript n Baseline less-than t Subscript n plus 1 Baseline less-than ellipsis less-than t Subscript upper N Baseline equals upper T period

In this case we define a set of subintervals left-parenthesis t Subscript n Baseline comma t Subscript n plus 1 Baseline right-parenthesis of size normal upper Delta t Subscript n Baseline identical-to t Subscript n plus 1 Baseline minus t Subscript n , 0 less-than-or-equal-to n less-than-or-equal-to upper N minus 1 .

In general, we speak of a non-uniform mesh when the sizes of the subintervals are not necessarily the same. However, in this book we consider in the main a class of finite difference schemes where the N subintervals have the same length (we then speak of a uniform mesh), namely normal upper Delta t equals upper T slash upper N . The variable h equals upper T slash upper N is also used to denote the uniform mesh size.

In general, we define y Subscript n to be the approximate solution at time t Subscript n and we write the functional dependence of y Subscript n plus 1 on t Subscript n Baseline comma y Subscript n Baseline and h by:

(3.21) y Subscript n plus 1 Baseline minus y Subscript n Baseline equals h normal upper Phi left-parenthesis t Subscript n Baseline comma y Subscript n Baseline comma h right-parenthesis

where normal upper Phi is called the increment function. For example, in the case of the explicit Euler method, this function is:

normal upper Phi left-parenthesis t comma y comma h right-parenthesis equals f left-parenthesis t comma y right-parenthesis period

The increment function represents the increment of the approximate solution. In general, the goal is to produce a formula for normal upper Phi that agrees with a certain exact relative increment with an error of upper O left-parenthesis h Superscript p Baseline right-parenthesis where p greater-than 1 without making it necessary to compute the derivative of f (Henrici (1962)). A special case of (3.21) is the fourth-order Runge–Kutta method:

(3.22) normal upper Phi left-parenthesis t comma y comma h right-parenthesis equals one sixth left-bracket k 1 plus 2 k 2 plus 2 k 3 plus k 4 right-bracket

where:

StartLayout 1st Row 1st Column k 1 2nd Column equals f left-parenthesis t comma y right-parenthesis 2nd Row 1st Column k 2 2nd Column equals f left-parenthesis t plus h slash 2 comma y plus one half italic h k 1 right-parenthesis 3rd Row 1st Column k 3 2nd Column equals f left-parenthesis t plus h slash 2 comma y plus one half italic h k 2 right-parenthesis 4th Row 1st Column k 4 2nd Column equals f left-parenthesis x plus h comma y plus italic h k 3 right-parenthesis period EndLayout

Other methods are:

Second-order Ralston method:

(3.23)

and Heun's (improved Euler) method:

(3.24)

This is a predictor-corrector method that also has applications to stochastic differential equations.

In general, explicit Runge–Kutta methods are unsuitable for stiff ODEs.

3.5.1 Code Samples in Python

We show hand-crafted code for the explicit Euler method as well as the schemes (3.22), (3.23) and (3.24). The main reason is to get hands-on experience with coding solvers for ODEs before using production solvers (for example, Python's or Boost C++ odeint libraries), especially for readers for whom numerical ODEs are new, for example readers with an economics or econometrics background. A good strategy is 1) get it working (write your own code and examples), then 2) get it right (use a library with the same examples) and then and only then get it optimised (use a library in a larger application).

The following code is for the methods:

Explicit Euler

Fourth-order Runge–Kutta (RK4)

Second-order Ralston.

Heun

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