100 Frequently Asked Questions In Quantitative Finance
What is Itˆo’s Lemma?
Short Answer
Itˆo’s lemma is a theorem in stochastic calculus. It tells
you that if you have a random walk, iny,say,anda
function of that randomly walking variable, call itf(y,t),
then you can easily write an expression for the ran-
dom walk inf. A function of a random variable is itself
random in general.
Example
The obvious example concerns the random walk
dS=μSdt+σSdX
commonly used to model an equity price or exchange
rate,S. What is the stochastic differential equation for
the logarithm ofS,lnS?
The answer is
d(lnS)=
(
μ−^12 σ^2
)
dt+σdX.
Long Answer
Let’s begin by stating the theorem. Given a random
variableysatisfying the stochastic differential equation
dy=a(y,t)dt+b(y,t)dX,
wheredXis a Wiener process, and a functionf(y,t)
that is differentiable with respect totand twice differ-
entiable with respect toy,thenfsatisfies the following
stochastic differential equation
df=
(
∂f
∂t
+a(y,t)
∂f
∂y
+^12 b(y,t)^2
∂^2 f
∂y^2
)
dt+b(y,t)
∂f
∂y
dX.
Itˆo’s lemma is to stochastic variables what Taylor series
is to deterministic. You can think of it as a way of
expanding functions in a series indt, just like Taylor
series. If it helps to think of it this way then you must
remember the simple rules of thumb as follows.
