Positive correlation :
More experience, more salary
Where there’s a will, there’s a way.
The higher the Doy, the lower the temperature.
The more one works, the less free time one has.
The color of your shirt, the color of my shoes.
Coefﬁcient of Correlation
mpg Miles/(US) gallon
cyl Number of cylinders
disp Displacement (cu.in.)
hp Gross horsepower
drat Rear axle ratio
wt Weight (lb/1000)
qsec 1/4 mile time
am Transmission (0 = automatic, 1 = manual)
gear Number of forward gears
carb Number of carburetors
Can we forecast the return?
Imagine you have tossed a
normal coin ten times, last
ﬁve outcomes were all head.
Do you expect that the next
outcome would be head?
If there is no correlation, the
next outcome would be like
tossing a coin.
Seem hopeless T__T
H T T H
T H H H
1 2 3 4
5 6 7 8
9 10 11
Efﬁcient Market Hypothesis
A ﬁnancial economist and passionate defender of the efﬁcient markets hypothesis (EMH)
was walking down the street with a friend. The friend stops and says, "Look, there's a $20
bill on the ground."
The economist turns and says, "Boy, this must be our lucky day! Better pick that up quick
because the market is so efﬁcient it won't be there for very long. Finding a $20 bill lying
around happens so infrequently that it would be foolish to spend our time searching for
more of them. Certainly, after assigning a value to the time spent in the effort, an
'investment' in trying to ﬁnd money lying on the street just waiting to be picked up would be
a poor one. I am also certainly not aware of lots of people, if any, getting rich mining
beaches with metal detectors."
When he had ﬁnished they both look down and the $20 bill was gone!
Why squared return?
The variance of return is calculated from squared returns.
Why variance? What is it?
Variance is the degree of variation
High variance => high volatility=> high risk
Volatility is forecastable
High risk, high return (but return can be either + or - )
Major Stylized Facts for Return
I. The distribution of returns is not normal, it has a high
peak and fat tails.
II. There is almost no correlation between returns for
III. There is positive correlation between squared returns
on nearby days, likewise for absolute returns.
Time series models
General time series models:
MA : moving average
AR : autoregressive
ARMA : autoregressive moving average
Financial time series models:
EWMA : exponentially weighted moving average
(G)ARCH : (generalized) autoregressive conditional heteroskedastic
SV : stochastic volatility
Asset pricing model
Robert F. Engle Tim Bollerslev
Nobel Prize in Economic Sciences 2003
Generalized AutoRegressive Conditional
rt+1 = µ + t+1✏t+1
t+1 = ! + ↵(rt µ)2
where ✏t+1 ⇠ N(0, 1)
GARCH volatility forecasting
Using data up to 5 Nov 2015
Date Return SD
6 Nov 0 0.384
7 Nov 0 0.390
8 Nov 0 0.396
9 Nov 0 0.401
10 Nov 0 0.407
11 Nov 0 0.412
12 Nov 0 0.416
13 Nov 0 0.421
"When Professors Scholes
and Merton and I invested
in warrants, Professor
Merton lost the most
money. And I lost the least”
– Fischer Black –
Nobel Prize in Economic Sciences1997:
Fischer Black, Myron Scholes, and Robert
Challenging, isn’t it?
Financial time series is challenging as it is quite difﬁcult to
Multivariate time series is also of interest, but it is even more
difﬁcult to model multiple time series together.
Most ﬁnancial models were created some years ago, at the time
that less data were available.
Nowadays, we can access more and more data, that would be
a good opportunity to explore and create better models for