Daniel J. Duffy - Numerical Methods in Computational Finance

Not only do we have to approximate functions at mesh points, but we also have to come up with a scheme to approximate the derivative appearing in Equation (2.1). There are several possibilities, and they are based on divided differences . For example, the following divided differences approximate the first derivative of at the mesh point ;

(2.8)

The first two divided differences are called one-sided differences and give first-order accuracy to the derivative, while the last divided difference is called a centred approximation to the derivative. In fact, by using a Taylor's expansion (assuming sufficient smoothness of ), we can prove the following:

FIGURE 2.1Continuous and discrete spaces.

(2.9)

Note that the first two approximations use two consecutive mesh points while the last formula uses three consecutive mesh points.

We now decide on how to approximate Equation (2.1)using finite differences. To this end, we need to introduce two new concepts:

One-step and multistep methods

Explicit and implicit schemes.

A one-step method is a finite difference scheme that calculates the solution at time-level in terms of the solution at time-level . No information at levels , , or previous levels is needed in order to calculate the solution at level . A multistep method, on the other hand, is a difference scheme where the solution at level is determined by values at levels and possibly previous time levels. Multistep methods are more complicated than one-step methods, and we concentrate solely on the latter methods in this book.

An explicit difference scheme is one where the solution at time can be calculated from the information at level directly. No extra arithmetic is needed: for example, using division or matrix inversion. An implicit finite difference scheme is one in which the terms involving the approximate solution at level are grouped together and only then can the solution at this level be found. Obviously, implicit methods are more difficult to program than explicit methods because we must solve a system of equations at each time step.

We now introduce a number of important and useful difference schemes that approximate the solution of Equation (2.1). These schemes will pop up all over the place in later chapters. Understanding how the schemes work in a simpler context will help you appreciate them when we tackle partial differential equations based on the Black–Scholes model. They also help in our understanding of notation, jargon, and syntax.

The main schemes are:

Explicit Euler

Implicit Euler

Crank–Nicolson (or Box scheme)

The trapezoidal method.

The explicit Euler method is given by:

(2.10)

whereas the implicit Euler method is given by:

(2.11)

Notice the difference: in Equation (2.10)the solution at level can be directly calculated in terms of the solution at level n , while in Equation (2.11)we must rearrange terms in order to calculate the solution at level .

The next scheme is called the Crank–Nicolson or box scheme , and it can be seen as an average of explicit and implicit Euler schemes. It is given as (see notation in Equation (2.7)):

(2.12)

It is useful to know that the three schemes can be merged into one generic scheme as it were by introducing a parameter (the scheme is sometimes called the Theta method ):

(2.13)

and the special cases are given by:

(2.14)

The solution of Equation (2.13)is given by:

(2.15)