The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 255 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 255

EXHIBIT 9.4 Numerical Solution of the Equation f ′′ + f = 0 with the Runge-Kutta

Method

that is, knowledge of the future position of a system. However, they often

appear in static systems and when trying to determine what initial condi-

tions should be imposed to reach a given goal at a given date.

In the case of boundary conditions, one cannot write a direct recur-

sive scheme; it’s necessary to solve a system of equations. For instance, we

could introduce the derivative f′(x) = δ as an unknown quantity. The dif-

ference quotient that approximates the derivative becomes an unknown.

We can now write a system of linear equations in the following way:

fx(∆

∆x

 )^ = f^0 +δ^ ∆x

         

( ) 2 fx(∆^2

∆x

f 2 = ) – (
1 +k( ∆x ) )f (^0)
f(
3 ∆x) = 2 f (
2
) – (
1 +k( ∆x )
2 )fx ( )∆
.
.
.
2
f 1000 =
2 f (
999
∆x ) – (
1

• k ( ∆x ) ) f (
  998
  ∆x )