Daniel J. Duffy - Numerical Methods in Computational Finance

Lemma 2.1 (Positivity).Let the operator be defined in Equation (2.1), and let be a well-behaved function satisfying the inequalities:

Then the following result holds true:

Roughly speaking, this lemma states that you cannot get a negative solution from positive input.

You can verify it by examining Equation (2.2)because all terms are positive.

The following result gives bounds on the growth of .

Theorem 2.1Let be the solution of Equation (2.1). Then:

This result states that the value of the solution is bounded by the input data. In other words, it is a well-posed problem .

We wish to replicate these properties in our difference schemes for Equation (2.1).

For completeness, we show the steps to be executed in order to produce the result in Equation (2.2).

(2.3)

Then from Equation (2.1)we see:

or:

Integrating this equation between and gives:

This style of mathematical analysis will be used in other contexts in this book, for example when transforming convection-diffusion-reaction equations (in particular, the Black–Scholes equation) to adjoint form .

The IVP Equation (2.1)is a model for all the linear time-dependent differential equations that we encounter in this book. We no longer think in terms of scalar problems in which the functions in Equation (2.1)are scalar-valued, but we can view an ODE at different levels of abstraction. To this end, we focus on the generic homogeneous ODE with solution :

(2.4)

This equation subsumes several special cases:

1 The variable is a square matrix, and then Equation (2.4)represents a system of ODEs. This is a very important area of research having many applications in science, engineering, and finance.

2 The variable is an ordinary or partial differential operator, and then Equation (2.4)represents an ODE in a Hilbert or Banach space.

3 The variable is a tridiagonal or block tridiagonal matrix that originates from a semi-discretisation in space of a time-dependent partial differential equation (PDE) using the Method of Lines (MOL) as discussed in Chapter 20.

4 The formal solution of (2.4)is:(2.5) In other words, we express the solution in terms of the exponential function of a matrix or of a differential operator. In the former case, there are many ways to compute the exponential of a matrix (see Moler and Van Loan (2003)).

5 The solution of Equation (2.4)can be simplified by matrix or operator splitting of the operator : (2.6)For example, we can split a matrix into two simpler matrices, or we can split an operator into its convection and diffusion components. In other words, we solve Equation (2.4)as a sequence of simpler problems in (2.6). These topics will be discussed in Chapters 18, 22, and 23.

6 The initial value problem (2.1)was originally used as a model test of finite difference methods in (Dahlquist (1956)). The resulting results and insights are helpful when dealing more complex IVPs.

Finally, this chapter and Chapter 3are recommended for readers who may not be familiar with ODE theory and ODE numerics. It is a prerequisite before moving to partial differential equations.

We now discuss finding an approximate solution to Equation (2.1)using the finite difference method . We introduce several popular schemes as well as defining standardised notation.

The interval or range where the solution of Equation (2.1)is defined is . When approximating the solution using finite difference equations, we use a discrete set of points in where the discrete solution will be calculated. To this end, we divide into equal intervals of length , where is a positive number called the step size . (We also use the symbol to denote the step size in many cases.) We number these discrete points as shown in Figure 2.1. In general all coefficients and discrete functions will be defined at these mesh points only. We adopt the following notation:

(2.7)