Daniel J. Duffy - Numerical Methods in Computational Finance

In order to motivate the fitted scheme, consider the case of constant and . We wish to produce a difference scheme in such a way that the discrete solution is equal to the exact solution at the mesh points for this constant-coefficient case. We introduce a so-called fitting factor in the new scheme:

(2.24)

The motivation for finding the fitting factor is to demand that the exact solution of (2.1)(which is known) has the same values as the discrete solution of (2.24)at the mesh points.

Plugging the exact solution (2.22)into (2.24)and doing some simple arithmetic, we get the following representation for the fitting factor :

(2.25)

Having found the fitting factor for the constant coefficient case, we generalise to a scheme for the case (2.1)as follows:

(2.26)

In practice we work with a number of special cases:

(2.27)

In the final case is the hyperbolic cotangent function.

In later chapters we shall apply the fitting scheme to the one-factor and multifactor Black–Scholes equations, and we shall show that we get good approximations to the option price and its delta in all regions of space where is the underlying asset and is time (up to maturity ). This is in contrast to the Crank–Nicolson scheme where the spurious oscillations are seen, especially when the underlying is near the strike price or when the payoff function is discontinuous.

Real-life problems are very seldom linear. In general, we model applications using non-linear IVPs:

(2.28)

Here is a non-linear function in u in general. Of course, Equation (2.28)contains Equation (2.1)as a special case. However, it is not possible to come up with an exact solution for (2.28)in general, and we must resort to some numerical techniques. Approximating (2.28)poses challenges because the resulting difference schemes may also be non-linear, thus forcing us to solve the discrete system at each time level by Newton's method or some other non-linear solver. For example, consider applying the trapezoidal method to (2.28):

(2.29)

where is non-linear. Here see that the unknown term u is on both the left- and right-hand sides of the equation, and hence it is not possible to solve the problem explicitly in the way that we did for the linear case. However, not all is lost, and to this end we introduce the predictor-corrector method that consists of a set consisting of two difference schemes; the first equation uses the explicit Euler method to produce an intermediate solution called a predictor that is then used in what could be called a modified trapezoidal rule :

(2.30)

The predictor-corrector is used in practice; it can be used with non-linear systems and stochastic differential equations (SDE). We discuss this topic in Chapter 13.

We give an introduction to a technique that allows us to improve the accuracy of finite difference schemes. This is called Richardson extrapolation in general. We take a specific case to show the essence of the method, namely the implicit Euler method (2.11). We know that it is first-order accurate and that it has good stability properties. We now apply the method on meshes of size k and k /2, and we can show that the approximate solutions can represented as follows:

Then:

Thus, is a second-order approximation to the solution of (2.1).

The constant is independent of , and this is why we can eliminate it in the first equations to get a scheme that is second-order accurate. The same trick can be employed with the second-order Crank–Nicolson scheme to get a fourth-order accurate scheme as follows:

Then: