Daniel J. Duffy - Numerical Methods in Computational Finance

This equation is useful because it can be mapped to C++ code and will be used by other schemes by defining the appropriate value of the parameter .

Finally, the trapezoidal method is similar to Crank–Nicolson, but it takes a slightly different averaging mechanism:

(2.16)

Having developed some difference schemes, we would like to have a way of determining if the discrete solution is a good approximation to the exact solution in some sense. Although we do not deal with this issue in great detail, we do look at stability and convergence issues.

Definition 2.1The one-step difference scheme of the form (2.13)is said to be positive if:

(2.17)

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 2.2A 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.

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.

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, and it has been shown that classical finite difference schemes fail to give acceptable answers when is large (typically values between 1000 and 10000). We get so-called spurious oscillations , and this problem is also encountered when solving one-factor and multifactor Black–Scholes equations using finite difference methods. We have resolved this problem using so-called exponentially fitted schemes . We motivate the scheme in the present context, and later chapters describe how to apply it to more complicated cases.