Numerical Methods in Computational Finance. Daniel J. Duffy

greater-than-or-equal-to 0 comma n equals 0 comma ellipsis comma upper N minus 1 comma w Superscript 0 Baseline greater-than-or-equal-to 0"/>

      implies that . Here, is a mesh function defined at the mesh points .

      Based on this definition, we see that the implicit Euler scheme is always positive while the explicit Euler scheme is positive if the term:

(2.18)

is positive. Thus, if the function achieves large values (and this happens in practice), we will have to make very small in order to produce good results. Even worse, if does not satisfy the constraint in (2.18) then the discrete solution looks nothing like the exact solution, and so-called spurious oscillations occur. This phenomenon occurs in other finite difference schemes, and we propose a number of remedies later in this book.

      Definition 2.2 A difference scheme is stable if its solution is based in much the same way as the solution of the continuous problem (2.1) (see Theorem 2.1), that is:

      (2.19)

      where:

      and:

Based on the fact that a scheme is stable and consistent (see Dahlquist and Björck (1974)), we can state in general that the error between the exact and discrete solutions is bounded by some polynomial power of the step-size :

      (2.20)

where is a constant that is independent of . For example, in the case of schemes (2.10), (2.11) and (2.12) we have:

      (2.21)

      Thus, we see that the Box method is second-order accurate and is better than the implicit Euler scheme, which is only first-order accurate.

2.4 SPECIAL SCHEMES

      We introduce exponentially fitted schemes that are used for boundary layer problems (for example, convection-dominated PDEs) and the extrapolation method to increase the accuracy of finite difference schemes. We shall see how to apply these techniques to more complex problems in later chapters. We also discuss predictor-corrector methods.

      2.4.1 Exponential Fitting

      We now introduce a special class of schemes with desirable properties. These are schemes that are suitable for problems with rapidly increasing or decreasing solutions. In the literature these are called stiff or singular perturbation problems (see Duffy (1980)). We can motivate these schemes in the present context. Let us take the problem (2.1) when is constant and is zero. The solution is given by a special case of (2.2), namely:

(2.22)

      If a is large then the derivatives of tend to increase; in fact, at , the derivatives are given by:

      (2.23)

      The physical interpretation of this fact is that a boundary layer exits near where is changing rapidly,