Download Predicting Returns and Volatilities with Ultra

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1
EFFICIENT MARKET
HYPOTHESIS
In its simplest form asserts that excess
returns are unpredictable - possibly
even by agents with special information
Is this true for long horizons?
It is probably not true at short horizons
Microstructure theory discusses the
transition to efficiency
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Why Don’t Informed Traders
Make Easy Profits?
Only by trading can they profit
If others watch their trades, prices will
move to reduce the profit
When informed traders are buying,
sellers will require higher prices until
the advantage is gone.
Trades carry information about prices
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3
TRANSITION TO EFFICIENCY
Glosten-Milgrom(1985), Easley and O’Hara(1987),
Easley and O’Hara(1992), Copeland and Galai(1983)
and Kyle(1985)
Two indistinguishable classes of traders - informed
and uninformed
When there is good news, informed traders will buy
while the rest will be buyers and sellers.
When there are more buyers than sellers, there is
some probability that this is due to information
traders – hence prices are increased by sophisticated
market makers.
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4
CONSEQUENCES
Informed traders make temporary
excess profits at the expense of
uninformed traders.
The higher the proportion of informed
traders, the
faster prices adjust to trades,
wider is the bid ask spread and
lower are the profits per informed trader.
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5
Easley and O’Hara(1992)
Three possible events- Good news, Bad
news and no news
Three possible actions by traders- Buy,
Sell, No Trade
Same updating strategy is used
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BEGINNING OF DAY
P(INFORMATION)=
P(GOOD NEWS)=
P(AGENT IS INFORMED)=
P(UNINFORMED WILL BE BUYER)=
P(UNINFORMED WILL TRADE)=
END OF DAY
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Easley Kiefer and O’Hara
Empirically estimated these probabilities
Econometrics involves simply matching
the proportions of buys, sells and nontrades to those observed.
Does not use (or need) prices,
quantities or sequencing of trades
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50.3
50.2
50.1
50.0
49.9
10
20
30
40
50
EVA
60
70
80
90
100
EVB
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50.3
50.2
50.1
50.0
49.9
10
20
30
40
50
EVA
60
70
80
90
100
EVB
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ASKING QUOTES WITH VARIOUS FRACTIONS
OF INFORMED TRADERS
50.30
50.25
50.20
50.15
50.10
50.05
50.00
2
4
6
ASK1
ASK_EKO
8
10
ASK2
ASK3
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14
ASK4
11
ASK QUOTES AFTER A SEQUENCE OF BUYS
WITH INTERVENING NONTRADES
50.30
50.25
50.20
50.15
50.10
50.05
50.00
2
4
6
8
EVA
EVAN
EVA2N
10
12
EVA3N
EVA4N
EVA5N
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12
INFORMED TRADERS
What is an informed trader?
Information
Information
Information
Information
about
about
about
about
true value
fundamentals
quantities
who is informed
Temporary profits from trading but
ultimately will be incorporated into
prices
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HOW FAST IS THIS
TRANSITION?
Could be decades in emerging markets
Could be seconds in big liquid markets
Speed depends on market
characteristics and on the ability of the
market to distinguish between informed
and uninformed traders
Transparency is a factor
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14
HOW CAN THE MARKET DETECT
INFORMED TRADERS?
When traders are informed, they are
more likely to be in a hurry(short
durations)
When traders are informed, they prefer
to trade large volumes.
When bid ask spreads are wide, it is
likely that the proportion of informed
traders is high as market makers
protect themselves
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15
EMPIRICAL EVIDENCE
Engle, Robert and Jeff Russell,(1998) “Autoregressive
Conditional Duration: A New Model for Irregularly Spaced Data,
Econometrica
Engle, Robert,(2000), “The Econometrics of Ultra-High
Frequency Data”, Econometrica
Dufour and Engle(2000), “Time and the Price Impact of a
Trade”, Journal of Finance, forthcoming
Engle and Lunde, “Trades and Quotes - A Bivariate Point
Process”
Russell and Engle, “Econometric analysis of discrete-valued,
irregularly-spaced, financial transactions data”
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APPROACH
Model the time to the next price change
as a random duration
This is a model of volatility (its inverse)
Model is a point process with
dependence and deterministic diurnal
effects
NEW ECONOMETRICS REQUIRED
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PRICE PATH
Time
Price Duration
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Econometric Tools
Data are irregularly spaced in time
The timing of trades is informative
Will use Engle and Russell(1998)
Autoregressive Conditional Duration
(ACD)
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THE CONDITIONAL
INTENSITY PROCESS
The conditional intensity is the
probability that the next event occurs at
time t+t given past arrival times and
the number of events.
 (t , N(t ); t1,..., t N( t ) ) 
lim
t  0
P( N(t  t )  N(t ) N(t ), t1,..., t N( t ) )
t
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THE ACD MODEL
The statistical specification is:
 i .
 ii .
 i  E  x i t i 1 ,...,t1    i t i 1 ,...,t1 ; 
x i   i i
where xi is the duration=ti-ti-1,  is the
conditional duration and
is an i.i.d.
random variable with non-negative support

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TYPES OF ACD MODELS
Specifications of the conditional
duration:
 i    xi 1   i 1
 i     j xi  j    j i  j
 
 i    xi , y i , z i 
Specifications of the disturbances
Exponential
Weibul
Generalized Gamma
Non-parametric
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MAXIMUM LIKELIHOOD
ESTIMATION
For the exponential disturbance

xi 
L   log  i   
i 
i 
which is so closely related to GARCH
that often theorems and software
designed for GARCH can be used for
ACD. It is a QML estimator.
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MODELING PRICE
DURATIONS
WITH IBM PRICE DURATION DATA
ESTIMATE ACD(2,2)
ADD IN PREDETERMINED VARIABLES
REPRESENTING STATE OF THE MARKET
Key predictors are transactions/time,
volume/transaction, spread
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Model 1
Parameter

Model 2
.2107
(6.14)
.3027
(18.22)
1
.0457
(2.60)
.0507
(2.24)
2
.1731
(5.94)
.1578
(5.19)
1
.0769
(1.00)
.1646
(1.61)
2
.5609
(8.07)
.4600
(5.16)
-.0440
(-12.65)
-.0359
(-13.40)
#Trans/Sec
Spread
Volume/Trans
-.0782
(-15.68)
-.0041
(-4.58)
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EMPIRICAL EVIDENCE
Engle, Robert and Jeff Russell,(1998) “Autoregressive
Conditional Duration: A New Model for Irregularly Spaced Data,
Econometrica
Engle, Robert,(2000), “The Econometrics of Ultra-High
Frequency Data”, Econometrica
Dufour and Engle(2000), “Time and the Price Impact of a
Trade”, Journal of Finance, forthcoming
Engle and Lunde, “Trades and Quotes - A Bivariate Point
Process”
Russell and Engle, “Econometric analysis of discrete-valued,
irregularly-spaced, financial transactions data”
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STATISTICAL MODELS
There are two kinds of random
variables:
Arrival Times of events such as trades
Characteristics of events called Marks
which further describe the events
Let x denote the time between trades
called durations and y be a vector of
marks
{( xi , yi ),i  1,...N }
Data:
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A MARKED POINT PROCESS
Joint density conditional on the past:


( xi , yi ) Fi 1 ~ f ( xi , yi xi 1 , yi 1 ; i )
can always be written:
f ( x i , y i x i 1 , y i 1 ;  i ) 
g ( x i x i 1 , y i 1 ;1i )q ( y i x i , x i 1 , y i 1 ;  2i )
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MODELING VOLATILITY WITH
TRANSACTION DATA
Model the change in midquote from one
transaction to the next conditional on the
duration.
Build GARCH model of volatility per unit of
calendar time conditional on the duration.
Find that short durations and wide spreads
predict higher volatilities in the future
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GARCH(1,1)
VARIABLE
Coef
Std.Err Z-Stat
GARCH&ECON
Coef
Std.Err Z-Stat
MEAN
DURS
-0.008
AR(1)
0.279
MA(1)
-0.656
0.004 -1.892 -0.007
0.002 -4.027
0.023
0.022
12.29
0.186
8.507
0.019 -33.86 -0.570
0.016 -35.70
VARIANCE
C
0.988
0.092
10.74 -0.111
0.047 -2.358
ARCH(1)
0.245
0.020
12.33
0.250
0.013
18.73
GARCH(1)
0.622
0.025
24.70
0.158
0.014
11.71
0.587
0.028
21.27
1/DUR
DUR/EXPDUR
-0.040
0.005 -7.992
LONGVOL(-1)
0.096
0.011
8.801
SPREAD(-1)>>
0.736
0.065
11.29
SIZE>10000
0.193
0.119
1.624
1/EXPDUR
LOGLIK
-112246.3
-107406.4
LB(15)
93.092
0.000
40.810
0.000
LB2(15)
30.422
0.004
169.12
0.000
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30
EMPIRICAL EVIDENCE
Engle, Robert and Jeff Russell,(1998) “Autoregressive
Conditional Duration: A New Model for Irregularly Spaced Data,
Econometrica
Engle, Robert,(2000), “The Econometrics of Ultra-High
Frequency Data”, Econometrica
Dufour and Engle(2000), “Time and the Price Impact of a
Trade”, Journal of Finance, forthcoming
Engle and Lunde, “Trades and Quotes - A Bivariate Point
Process”
Russell and Engle, “Econometric analysis of discrete-valued,
irregularly-spaced, financial transactions data”
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31
APPROACH
Extend Hasbrouck’s Vector
Autoregressive measurement of price
impact of trades
Measure effect of time between trades
on price impact
Use ACD to model stochastic process of
trade arrivals
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Cumulative percentage quote revision after an
unexpected buy
0.08
0.06
0.04
0.02
1/17/91
12/24/90
0
1
3
5
7
9
11
13
15
17
19
21
Transaction Time (t)
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Cumulative percentagequote revisionafter an
unexpected buy
0.08
1/17/91
0.06
0.04
12/24/90
0.02
20:50
18:45
16:40
14:35
12:30
10:25
08:20
06:15
04:10
02:05
0:00
0
Calendar time(min:sec)
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SUMMARY
The price impacts, the spreads, the speed of
quote revisions, and the volatility all respond
to information variables
TRANSITION IS FASTER WHEN THERE IS
INFORMATION ARRIVING
Econometric measures of information
high shares per trade
short duration between trades
sustained wide spreads
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35
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36
EMPIRICAL EVIDENCE
Engle, Robert and Jeff Russell,(1998) “Autoregressive
Conditional Duration: A New Model for Irregularly Spaced Data,
Econometrica
Engle, Robert,(2000), “The Econometrics of Ultra-High
Frequency Data”, Econometrica
Dufour and Engle(2000), “Time and the Price Impact of a
Trade”, Journal of Finance, forthcoming
Engle and Lunde, “Trades and Quotes - A Bivariate Point
Process”
Russell and Engle, “Econometric analysis of discrete-valued,
irregularly-spaced, financial transactions data”
http://weber.ucsd.edu/~mbacci/engle/
37
Jeffrey R. Russell
Robert F. Engle
University of Chicago
University of California, San Diego
Graduate School of Business
http://gsbwww.uchicago.edu/fac/jeffrey.russell/research/
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38
IBM
Transaction Price
105.4
105.3
105.2
105.1
105
104.9
104.8
0
2
4
6
8
10
12
14
Time (Minutes)
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Goal: Develop an econometric model for discrete-valued,
irregularly-spaced time series data.
Method: Propose a class of models for the joint distribution
of the arrival times of the data and the associated price changes.
Questions: Are returns predictable in the short or long run?
How long is the long run? What factors influence this
adjustment rate?
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Hausman,Lo and MacKinlay
Estimate Ordered Probit Model,JFE(1992)
States are different price processes
Independent variables
Time between trades
Bid Ask Spread
Volume
SP500 futures returns over 5 minutes
Buy-Sell indicator
Lagged dependent variable
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A Little Notation
Let ti be the arrival time of the ith transaction where
t0<t1<t2…
A sequence of strictly increasing random variables is
called a simple point process.
N(t) denotes the associated counting process.
Let pi denote the price associated with the ith transaction
and let yi=pi-pi-1 denote the price change associated with
the ith transaction.
Since the price changes are discrete we define yi to take
k unique values.
That is yi is a multinomial random variable.
The bivariate process (yi,ti), is called a marked point process.
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We take the following conditional joint distribution of the
arrival time ti and the mark yi as the general object of interest:

f yi , ti y i 1 ,t i 1

where y i 1   yi 1 , yi  2 ,...  and t i 1  ti 1 , ti 2 ,...
In the spirit of Engle (2000) we decompose the joint distribution
into the product of the conditional and the marginal distribution:

 


f yi , ti y i 1 ,t i 1  g yi y i 1 ,t i  q ti y i 1 ,t i 1

 
 


?
ACD
Engle and Russell (1998)
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SPECIFYING THE
PROBABILITY STRUCTURE
Let x i be a kx1 vector which has a 1 in only
one place indicating the current state
Let  i be the conditional probability of all the
states in period i.
A standard Markov chain assumes
 i  Px i 1
Instead we want modifiers of P
 i  P (x i 1 , i 1 , z i ,ti 1,ti )x i 1
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RESTRICTIONS
For P to be a transition matrix
It must have non negative elements
All columns must sum to one
To impose these constraints,
parameterize P as an inverse logistic
function of its determinants
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THE PARAMETERIZATION
For each time period t, express the
probability of state i relative to a
base state k as:
log  i ,t /  k ,t   Ai x t 1  bi ,
for
i  1,..., k  1
Which implies that:
Pij 
exp  Aij  bi 
k 1
1   exp  Aim  bm 
m 1
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Rewriting the k-1 log functions as h() this can be written in simple
form as:
(2)
h( )  Ax  b
where A is an unrestricted (k-1)x(k-1) matrix, b is an unrestricted
(k-1)x1 vector and x is a the (k-1)x1 state vector.
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MORE GENERALLY
Let matrices have time subscripts and
allow other lagged variables:
h t   At xt 1  Bt t 1  Ct h t 1   Dt zt
The ACM likelihood is simply a
multinomial for each observation
conditional on the past
LACM (x ;  )   xt ' log(t )
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THE FULL LIKELIHOOD
The sum of the ACD and ACM log
likelihood is

t 
L ( x , ;  , )   xt ' log(t )   log( t )  
t 

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Even more generally, we define the Autoregressive Conditional
Multinomial (ACM) model as:
h i    At , j xi  j   i  j    Bt , j xi  j   Ct , j h i  j   GZi
p
q
r
j 1
j 1
j 1
Where h : ( K 1)  ( K 1) is the inverse logistic function.
Zi might contain ti, a constant term, a deterministic function
of time, or perhaps other weakly exogenous variables.
We call this an ACM(p,q,r) model.
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The data:
58,944 transactions of IBM stock over the 3 months of Nov.
1990 - Jan. 1991 on the consolidated market. (TORQ)
98.6% of the price changes took one of 5 different values.
70
60
P ercent
50
40
30
20
10
0
-1
0
1
P rice C ha ng e
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We therefore
consider a 5
state model
defined as
1,0,0,0 if p < -.125
i

0,1,0,0 if - .125  p < 0
i



xi  0,0,0,0 if p i = 0


0,0,1,0 if 0 < p i  .125


0,0,0,1 if p i > .125

It is interesting to consider the sample cross correlogram of
the state vector xi.
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Sample cross correlations of x
lag = 1










       
        
     
 

       
up 2
up 1
down 1
down 2
up 2
up 1
down 1
down 2
2
6
    
   


   








7










3
    
   


   


    
4






8
       
        
     
 

       
11
12
13
    
   


   


    
    
   


   


    
    
   


   


    
  
   
  

  
5
    
   


   


    
9
10
    
   


   








    
   


   








14
15
    
   


   


    
http://weber.ucsd.edu/~mbacci/engle/
    
   


   


    
53
Parameters are estimated using the joint distribution of arrival
times and price changes.

 


f yi , ti y i 1 ,t i 1  g yi y i 1 ,t i  q ti y i 1 ,t i 1

 
 


ACM
ACD
Initially, we consider simple parameterizations in which
the information set for the joint likelihood consists of the
filtration of past arrival times and past price changes.
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54
ACM(p,q,r) specification:
h i      A V
p
1/ 2
j i j
j 2
x
i j
  i  j    B j xi  j   C j h i  j  
q
r
j 1
j 1
ln(  i ) g1   i g 2  ( i  i ) g3   i g 4
Where  i  ti  ti 1 and gj are symmetric.
ACD(s,t) Engle and Russell (1998) specifies the conditional
probability of the ith event arrival at time ti+ by
 
  I i 1   0   where  i  E  | ti 1 , ti  2, ..., xi 1 , xi  2 ,...
 i  i 
1
v
w
 i j t
ln  i       j
   j ln  i  j     j xi  j    j i2 j
j 1
j 1
j 1
 i  j j 1
s
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55
Conditional Variance of Price Changes as a Function of Expected Duration
0.008
0.007
0.006
0.004
0.003
0.002
0.001
4.
9
4.
6
4.
3
4
3.
7
3.
4
3.
1
2.
8
2.
5
2.
2
1.
9
1.
6
1.
3
1
0.
7
0.
4
0
0.
1
Volatility
0.005
Expected Duration
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56
Simulations
We perform simulations with spreads, volume, and transaction
rates all set to their median value and examine the long run
price impact of two consecutive trades that push the price
down 1 ticks each.
We then perform simulations with spreads, volume and
transaction rates set to their 95 percentile values, one at a
time, for the initial two trades and then reset them to their
median values for the remainder of the simulation.
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57
Price impact of 2 consecutive trades each pushing the price
down by 1 tick.
0
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
-0.05
Dollars
-0.1
-0.15
-0.2
-0.25
-0.3
Transaction
Median
High Transaction Rate
Large Volume
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Wide Spread
58
0
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
-0.01
Dollars
-0.02
-0.03
-0.04
-0.05
-0.06
Transaction
High Transaction Rate
Large Volume
Wide Spread
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59
Conclusions
1. Both the realized and the expected duration impact the
distribution of the price changes for the data studied.
2. Transaction rates tend to be lower when price are falling.
3. Transaction rates tend to be higher when volatility is higher.
4. Simulations suggest that the long run price impact of a
trade can be very sensitive to the volume but is less
sensitive to the spread and the transaction rates.
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60