T opic#1:Modelling stock
prices
Financial Risk Management2014-15
January2015
FRM c Dennis PHILIP2015
1Modelling stock prices2
1Modelling stock prices
Modelling the evolution of stock prices is
about introducing a process that will ex-杭州灵隐寺求什么最灵
plain the random movements in prices.This
可乐怎么画randomness is explained in the(weak form)
知难而退的意思
E¢cient Market Hypothesis(EMH)that can
be summarized in two assumptions:
1.Past history is re‡ected in prent price
2.Markets respond immediately to any如何补血
new information about the ast
This means that the past“path”followed
by the stock price is irrelevant(indepen-
dent)to where the stock price value will be
in the future.All that matters is today’s
price.
普通会计一个月多少钱Only the current value of the stock price
variable is relevant for predicting the future.
The above is the de…nition of“Markov Process”.
In a Markov process,future movements in a
variable depend only on where we are,not
the history of how we got where we are.
FRM c Dennis PHILIP2015
1Modelling stock prices3 So we are going to de…ne a process for how
stock price changes over time.
If ast price is S.Suppo price changes
to S+dS in a small time interval(say dt).
Then we can decompo returns dS into de-
terministic/anticipated part and a ran-
dom part where prices changed due to some
external unanticipated news.
dS
= dt+ dw
S
The randomness in the random part is ex-
plained by a Brownian Motion process(also
known as Wiener process)and scaled by the
volatility of returns.
We can introduce time subscripts and re-
arrange to get
dS t= S t dt+ S t dw t
This process is called the Geometric Brown-
ian Motion.
FRM c Dennis PHILIP2015
徐徐升起1Modelling stock prices4 Why have we ud Brownian Motion process
to explain randomness?
–In practice,we e that stock prices be-
have,atleast for long stretches of time,
like random walks with small and fre-
quent jumps
–BM…ts the characteristics of the share
price.Imagine a heavy particle(share
price)that is jarred around by lighter
particles(trades).Trades a¤ect the
中国人体艺术摄影price movement.
–In statistics,random walk,being the
simplest form,have limiting distribu-
tions and since BM is a limit of the
random walk,we can easily understand
the statistics of BM(u of Central
Limit Theorem,CLT)
Next we e,what is this w(and in turn what is dw)?
Brownian motion is a particular type of Markov
process.It is also referred to as the“Wiener
process”.
FRM c Dennis PHILIP2015
1Modelling stock prices5 Consider a variable w following a Wiener
process who change in value during one
year is a normal distribution with mean0
and variance of1
(w1 w0)="1 N(0;1)
Another way to e it is:w1=w0+"1(last
period value+new information/shock)
The change in w during the cond year
is again a normal distribution with mean
0and variance of1
(w2 w1)="2 N(0;1)
Another way to e it is:w2=w1+"2
Since the variable w follows a Markov process,
the two normal distributions are indepen-
dent.
Adding i.ally distributed"0s will
result in a normal distribution.And,the
mean is the sum of the means and the vari-
ance is the sum of the variances.
(w2 w0)="1+"2 N(0;2)
FRM c Dennis PHILIP2015
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