The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 251 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 251

EXHIBIT 9.1 Numerical Solutions of the Equation f ′= f with the Euler

Approximation for Different Step Sizes

This equation describes oscillatory motion, such as the elongation of a

pendulum or the displacement of a spring.

To approximate this equation we must approximate the second

derivative. This could be done, for example, by combining difference

quotients as follows:

fx( + ∆x)– fx()

f′()≈----------------------------------------

∆x

x

fx( + 2 ∆x)– fx( + ∆x)

f′(x+ ∆x) ≈---------------------------------------------------------

∆x