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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.

      2.3.1 Common Schemes

      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)

      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)

      2.3.2 Discrete Maximum Principle

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

      (2.17)Скачать книгу