Mathematical Modeling in Finance with Stochastic Processes

4.2. MOMENT GENERATING FUNCTIONS 131

for all valuestfor which the integral converges.

Example.Thedegenerate probability distributionhas all the probability
concentrated at a single point. That is, ifXis a degenerate random variable
with the degenerate probability distribution, thenX=μwith probability 1
andXis any other value with probability 0. That is, the degenerate random
variable is a discrete random variable exhibiting certainty of outcome. The
moment generating function of the degenerate random variable is particularly
simple: ∑

xi=μ

exit=eμt.

If the moments of orderkexist for 0≤k≤k 0 , then the moment gen-
erating function is continuously differentiable up to orderk 0 att= 0. The
moments ofXcan be generated fromφX(t) by repeated differentiation:

φ′X=

d

dt

E

[

etX

]

=

d

dt

∫

x

etxfX(x)dx

=

∫

x

d

dt

etxfX(x)dx

=

∫

x

xetxfX(x)dx

=E

[

XetX

]

.

Then
φ′X(0) =E[X].
Likewise

φ′′X(t) =

d

dt

φ′X(t)

=

d

dt

∫

x

xetxfX(x)dx

=

∫

x

x

d

dt

etxfX(x)dx

=

∫

x

x^2 etxfX(x)dx

=E

[

X^2 etX

]

.