The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 252 Wednesday, February 4, 2004 12:51 PM

252 The Mathematics of Financial Modeling and Investment Management

f′(x+ ∆x)– f′()x

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

∆x

( ( ( ()

x

fx+ 2 ∆x)– fx+ ∆x) fx+ ∆x)– fx

---------------------------------------------------------– ----------------------------------------

∆x ∆x

= --------------------------------------------------------------------------------------------------------

∆x

fx( + 2 ∆x)– 2 fx( + ∆x)+ fx()

= ------------------------------------------------------------------------------

(∆x)

2

With this approximation, the original equation becomes

f′′()+ kf x

fx+ 2 ∆x)– 2 fx+ ∆x)+ fx

x ()≈------------------------------------------------------------------------------+ kf x

( ( ()

()= 0

(∆x)

2

fx( + 2 ∆x)– 2 fx( + ∆x)+ ( 1 + kx(∆ )^2 )fx()= 0

We can thus write the approximation scheme:

fx( + ∆x)= fx()+ ∆xf′() x

fx( + 2 ∆x)= 2 fx( + ∆x)– ( 1 + kx(∆ ) ()

2

)fx

Given the increment ∆xand the initial values f(0),f′(0), using the above

formulas we can recursively compute f(0 + ∆x), f(0 + 2∆x), and so on.

Exhibit 9.2 illustrates this computation.

In practice, the Euler approximation scheme is often not sufficiently

precise and more sophisticated approximation schemes are used. For

example, a widely used approximation scheme is the Runge-Kutta

method. We give an example of the Runge-Kutta method in the case of

the equation f′′+ f= 0 which is equivalent to the linear system:

x′= y

y′= –x

In this case the Runge-Kutta approximation scheme is the following:

k 1 = hy i()

h 1 = –hx i()