·
Wiener
Processes
and
Ito's
etntna
Any variable whose value changes over time in an uncertain
way
is
said to follow a
stochastic process.
Stochastic processes can be classified as
discrete time
or
continuous
time.
A discretetime stochastic process
is
one where the value
of
the variable can
change only
at
certain fixed points in time, whereas a continuoustime stochastic
process
is
one where changes can take place at any time. Stochastic processes can also
be classified as
continuous variable
or
discrete variable.
In a continuousvariable process,
the underlying variable can take any value within a certain range, whereas in a discrete
variable process, only certain discrete values are possible.
This chapter develops a continuousvariable, continuoustime stochastic process for
stock prices. Learning about this process
is
the first step to understanding the pricing
of
options and
other more complicated derivatives.
It
should be noted that,
in
practice,
we
do not observe stock prices following continuousvariable, continuous
time processes. Stock prices are restricted to discrete values (e.g., multiples
of
a cent)
and changes can be observed only .when the exchange
is
open. Nevertheless, the
continuousvariable, continuoustime process proves to be a useful model for many
purposes.
Many people
feel
that continuoustime stochastic processes are so complicated that
they should be left entirely to "rocket scientists". This
is
not so. The biggest hurdle to
understanding these processes
is
the notation. Here
we
present a stepbystep approach
aimed at getting the reader over this hurdle.
We
also explain an important result known
as
Ito's lemma
that
is
central to the pricing
of
derivatives.
12.1
THE
MARKOV
PROPERTY
A
Markov process
is
a particular type
of
stochastic process where only the present value
of
a variable
is
relevant for predicting the future. The past history
of
the variable and
the
way
that the present has emerged from the past are irrelevant.
Stock prices are usually assumed to follow a Markov process. Suppose that the
price
of
IBM stock
is
$100
now.
If
the stock price follows a Markov process, our
predictions for the future should be unaffected by the price one week ago, one month
263
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264
CHAPTER
12
ago, or one year ago. The only relevant piece
of
information
is
that the price
is
now
$100.
1
Predictions for the future are uncertain and must be expressed in terms
of
probability distributions. The Markov property implies that the probability distribu
tion
of
the price at any particular future time
is
not
dependent on the particular path
followed by the price in the past.
The Markov property
of
stock prices
is
consistent with the weak form
of
market
efficiency. This states that the present price
of
a stock impounds all the information
contained in a record
of
past prices.
If
the weak form
of
market efficiency
were
not
true,
technical analysts could make aboveaverage returns
by
interpreting charts
of
the past
history
of
stock prices. There
is
very little evidence that they are in fact able to do this.
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 Spring '11
 DonBlasius
 Math, Normal Distribution, Probability theory, K. Ito

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