Stat 280 Lab 99
Friday, December 03, 2004
This lab is about performing some exploratory data analysis for common stocks and
stock indexes. First, a few things must be understood. Stocks and many other physical
or economic systems are too poorly-understood to be modeled deterministically. Instead
we model them in time or space by an appropriate stochastic process.
For stocks, the most common model is the log-normal diffusion model, known as
geometric Brownian motion (GBM). Not wishing to confuse, the equation for the
evolution in time of this model is () (0) .
ii
t
ii
Xt X e
µσν
′t
∆
+∆
= Although this model has
consistently been shown to not truly represent the markets, in many respects it is “close
enough.” The nice thing about GBM is that is parameterized just by two population
parameters, its growth and variance, or (,)
θµ
σ
′
=
, where 2
2
i
ii
σ
µµ
′=−
In our analysis, we usually refer to the time ticks t
∆
as 1-day increments. The process is
modeled in terms of price CHANGES (to get to normality!), i.e. 1t
t
t
X
R
X
+
=, from which
we take the natural log to get . This r is modeled as Gaussian (Normal). For
instance, is the price of the stock today is 100 and yesterday it was 95, then
R=100/95=1.053 (note that the percent increase is R-1, or 5.3%), and its “mathematical”
return r=ln(1.053)=.0516, which is CLOSE to the “percent” return but not exactly equal
to it. Some factoids are that , for small r, and
ln( )rR=
pct
rr≤pct
rr%11
r
rR e
=
−= −.
However, we are mostly concerned with the basic definition ln( )rR
=
Normal.
Once one has estimates for the population parameters, it is easy to simulate stock path
progression and other things. The standard estimators for 2
and
µ
σ
are 2
2
ˆR
S
t
σ
=∆ and
2
ˆ2
r
t
σ
µ
=+
∆, with =1/252 (1 day). The variance is especially easy to calculate, it’s
just the sample variance of the daily returns r . For daily data,
t∆
_
ˆ252
annual r daily
S
σ
=,
recalling 2
σ
σ
=. The growth rate
2
ˆ21
daily
daily
r
σ
µ
=
+.
For a typical stock or market index we might get ˆˆ
( , ) (.082,.165)
gbm gbm
GBM GBM
µ
σ
=
,
showing an annual growth rate of 8.2% and annual “volatility” of 16.5%. So for a stock
with annual volatility of 100%, we might expect that in 95% of the cases the price of the
stock could double within a year.
