The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 247 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 247

crete periods of time ∆ti, compounding is obtained by purely algebraic

formulas as follows:

Ct()i – Ct(i – ∆ti)

Ri∆ti = -----------------------------------------

Ct(i – ∆ti)

Solving for C(ti):

Ct()i = ( 1 + Ri∆ti)Ct(i – ∆ti)

By recursive substitution we obtain

Ct()i = ( 1 + Ri∆ti)( 1 + Ri – 1 ∆ti – 1 )...( 1 + R 1 ∆t 1 )Ct() 0

However, market interest rates are subject to rapid change. In the

limit of very short time intervals, the instantaneous rate r(t) would be

defined as the limit, if it exists, of the discrete interest rate:

Ct ( + ∆t)– Ct()

rt()= lim -----------------------------------------

t → ∆ (^0) ∆t Ct()
The above expression can be rewritten as a simple first-order differential
equation in C:
dC t
rt ()= ---------------
dt
()Ct
()
In a simple intuitive way, the above equation can be obtained consider-
ing that in the elementary time dt the bank account increments by the
amount dC = C(t)r(t)dt. In this equation, variables are separable. It
admits the family of solutions:

C = A exp( ∫ rt()td )

where A is the initial capital.

Linear Differential Equation

Linear differential equations are equations of the following type: