
2 1 Option Valuation and the Volatility Smile
a basic building-block in modeling financial variables, and extensively used to de-
scribe most periodic changes of random prices, for example, daily returns of stock
prices. Brownian motion was first introduced by the Scottish botanist R. Brown in
1827 to document random movements of small particles in fluid. The French math-
ematician L. Bachelier (1900) applied Brownian motion in his lately honored PhD
thesis “The Theory of Speculation” to analyze the price behavior of stocks and op-
tions. But first in 1905, A. Einstein delivered a formal mathematical analysis of
Brownian motion and derived a corresponding partial differential equation. After-
wards Brownian motion became an important tool in physics. In honor of N. Wiener
who made a great contribution to studying Brownian motion from the perspective
of stochastic theory, a standard Brownian motion is also called a Wiener process.
Let (
Ω
,F ,(F
t
)
t0
,Q)
1
be a complete probability space equipped with a prob-
ability measure Q. F
t
is
σ
-algebra at time t and may be interpreted as a collection
of all information up to time t.
2
A standard Brownian motion is defined as follows.
Definition 1.1.1. Brownian motion: A standard Brownian motion {W(t),t 0} is
a stochastic process satisfying the following conditions,
1. for 0 t
1
t
2
··· t
n
< ∞,W(t
1
),W(t
2
) −W(t
1
),··· ,W(t
n
) −W(t
n−1
) are
stochastically independent;
2. for every t > s 0,W(t) −W(s) is normally distributed with a mean of zero and
a variance of t −s,
W(t) −W(s) ∼ N(0,t −s);
3. for each fixed
ω
∈
Ω
,W(
ω
,t) is continuous in t;
4. W(0)=0 almost surely.
Condition 1 says that non-overlapping increments of W(t) are uncorrelated. Con-
dition 2 states that any increment W (t)−W(s) is distributed according to a Gaussian
law with a mean of zero and a variance of t −s. As a Brownian motion W (t) is de-
fined on F
t
, the probability measure Q is then a Gaussian law. In other words, the
dispersion of a standard Brownian motion is proportional to time length. A further
implication of condition 2 is that a standard Brownian motion is homogeneous in
the sense that the distribution of any increment depends only on time length t −s
and not on time point s. Condition 3 explains that a standard Brownian motion has
continuous paths over time. The last condition just requires that a standard Brown-
ian motion starts at zero almost surely, and implies the expected value of any W(t)
is equal to zero. Summing up, a standard Brownian motion belongs to a class of the
processes with stationary independent increments.
Since a standard Brownian motion is normally distributed and defined on the
entire real line, it may not be applied appropriately to model price levels that are
usually strictly positive. A realistic concept for price movements is a geometric
Brownian motion, an exponential function of a Brownian motion.
1
For more detailed descriptions on probability space,
σ
-algebra and filtration, please refer to other
mathematical textbooks, for example, Øksendal (2003).
2
Throughout this book, for notation simplicity, we will not always express a conditional expecta-
tion by explicitly adding F
t
.