Download Modelling intradaily durations for the direction of the price changes

Modelling intradaily durations
for the direction of the price changes
Giovanni De Luca∗
Silvia Golia†
1
Introduction
The methodologies for the analysis of financial time series has recently extended
for the availability of the so-called tick-by-tick or ultra-high frequency data,
i.e. data which are stored when there is the occurrence of the market event
one is interested to. Because of their nature of irregularly spaced data, the
time between them (duration) has to be treated as a random variable. For the
modelling of durations the class of Autoregressive Conditional Duration (ACD)
models has been introduced. It allows one to predict next durations. However,
the joint modelling of durations and a market event can be more fruitful. In
this work the direction of a price change is analyzed using ultra-high frequency
data.
2
The class of ACD models: short vs long memory
The class of ACD models has been introduced in Engle and Russell (1998). Let
n be the number of events observed at random times ti , i = 1, . . . , n, and let
xi = ti − ti−1 be the duration between the (i − 1)-th and the i-th event. The
main assumption of these models is that all the temporal dependence in the
durations is captured by the mean function. The expected i-th duration, given
the past, is written as a function of the past durations, i.e.
E(xi |xi−1 , xi−2 , . . . , x1 ) = g(xi−1 , xi−2 , . . . , x1 ; θ),
(1)
where θ is the parameter vector characterizing g. The ACD class of models consists of the parameterization of these expected durations and can be expressed
∗ DESI - Sezione Statistica, Università di Verona, via dell’Artigliere, 19 - 37129 Verona
(e-mail: [email protected])
† Dipartimento di Metodi Quantitativi, Università di Brescia, C. da S.Chiara, 50 - 25122
Brescia (e-mail: [email protected])
1
as
xi
ψi
= ψi i , {i } ∼ i.i.d. with E(i ) = 1
= g(xi−1 , xi−2 , . . . , x1 ; θ).
(2)
Equation (2) means that xi , given the past, are independent and identically distributed and f (xi |xi−1 , xi−2 , . . . , x1 ; θ) = f (xi |ψi ; θ). In the basic formulation
of the ACD(p,q) process (Engle and Russell, 1998) the conditional expected
durations ψi are expressed as
ψi = ω +
q
X
αj xi−j +
j=1
p
X
βj ψi−j ,
j=1
Pp
Pq
where ω > 0, αj , βj ≥ 0 and j=1 αj + j=1 βj < 1 in order to guarantee the
positivity of the durations and the stationarity of the process respectively. This
model takes into account only the short dependence in the expected durations.
If a longer dependence is allowed for, ψi must have a different specification.
A possible one is given by the Fractional Integrated Autoregressive Conditional
Duration process (FIACD) (Jasiak, 1999):
ψi
= ω + β(L)ψi + [1 − β(L) − [1 − φ(L)](1 − L)d ]xi
= ω + β(L)ψi + Λ(L)xi ,
(3)
where β(L) = β1 L + β2 L2 + . . . + βp Lp , φ(L) = φ1 L + φ2 L2 + . . . + φq Lq
and all the roots of 1 − β(L) and 1 − φ(L) lie outside the unit circle, Λ(L) =
λ1 L + λ2 L2 + λ3 L3 + . . . is a polynomial of infinite order, ω > 0 and 0 ≤ d ≤ 1.
The FIACD is a long memory process with an hyperbolic rate of decay in the
autocorrelation function. The long range dependence may span several tradings
days and should be accounted for in models and forecasts of market activity.
Some constraints on the parameters have to be imposed in order to guarantee
the positivity of the expected durations ψi : βi ≥ 0 for i = 1, 2, . . . , p and λk ≥ 0
for k = 1, 2, . . . For the latter no general rule to be applied to the original
parameters exists. The conditions must be analitically derived every time.
It is interesting to observe that for d = 0, the FIACD process is led back to an
ACD; for d = 1 one obtains an integrated process. For 0 < d ≤ 1 the expansion
of (1 − L)d evaluated in L = 1 is equal to zero, and the sum of all the coefficients
is equal to 1; this means that the first unconditional moment of the duration
is infinite and the FIACD process is not weakly stationary. Nevertheless Jasiak
(1999) shows that the FIACD(p,d,q) class of processes is strictly stationary and
ergodic for 0 ≤ d ≤ 1.
The most popular specification of such a process is the FIACD(1,d,1) process.
In this case it is easy to derive the parameters of the polynomial Λ(L) as function
of the parameters β, φ and d.
Let πk = (−1)k [ d(d−1)(d−2)...(d−k+1)
] be the terms of the expansion of (1 − L)d ,
k!
then:
λ1 = φ − β + d, λk = φπk−1 − πk
k = 2, 3, . . . .
2
The constraints on the parameters β and φ which guarantee the positivity of
the durations are the following:
0 ≤ β ≤ φ + d,
φ≤
1−d
.
2
For the FIACD(2,d,0) process, which will be used in Section 4, the parameters
and the constraints are the following:
λ1 = d − β1 ,
0 ≤ β1 ≤ d,
3
λ2 = −π2 − β2 , λk = −πk
d(1 − d)
0 ≤ β2 ≤
.
2
k = 3, 4, . . .
The prediction of the direction of the price
Bauwens and Giot (2000) proposed a first model for forecasting the direction of
the price of a financial asset. They considered a joint model for duration (xi )
and price change direction (yi ) at the end of the duration xi . yi is a binary
random variable, i.e.
+1 if price increases
yi =
−1 if price decreases.
Different models are obtained according to the assumption on the hazard function of xi , h(xi ).
A first simple possibility is to assume a hazard function constant with respect
to the durations xi , but variable with respect to yi . The interpretation is the
following: a different duration is expected according to what happened to the
price process (increase or decrease) while the duration process does not matter.
The hazard function h(xi |yi ) = λ(yi ) implies xi |yi ∼ Exp(λ(yi )).
A more refined model allows the durations to have a role in the prediction of
the direction of stock prices, including a more complex structure in the hazard
function, which is now
h(xi |yi ) = ψi−1 (yi ),
so that
xi |yi ∼ Exp(ψi−1 (yi )).
(4)
An ACD-type model is being assumed for the duration xi , with the salient feature that the conditional expected duration, E [xi |xi−1 , xi−2 , . . . , x1 ] = ψi (yi ),
depends on another process, the price process.
The model in (4) is characterized by parameters depending on the price
process and is denoted as exponential Asymmetric ACD (AACD). The simplest
representation implies
+
ψi if yi = +1
ψi =
ψi− if yi = −1
3
where
+
+
ψi+ = g(ψi−1
, . . . , ψi−p
, xi−1 , . . . , xi−q ; θ + )
and
−
−
ψi− = g(ψi−1
, . . . , ψi−p
, xi−1 , . . . , xi−q ; θ − ).
Furthermore, it is possible to specify a dependence on the price change of
+
+
the previous period, allowing ψi+ = g(ψi−1
, . . . , ψi−p
, xi−1 , . . . , xi−q ; θ + (yi−1 ))
−
−
−
and ψi− = g(ψi−1
, . . . , ψi−p
, xi−1 , . . . , xi−q ; θ (yi−1 )).
In the framework of competing risks models, the joint density function is
written as:
I i
1−Ii
1
xi
1
xi
f (xi , yi |yi−1 ) =
exp − +
exp − −
(5)
ψi+
ψi
ψi−
ψi
where Ii = 1 if yi = +1 and Ii = 0 if yi = −1, i.e. as the product of the
density function of xi characterized by the expression of ψi corresponding to
the state (yi ) which occurred, and the survival function of xi characterized by
ψi corresponding to the state which did not occur. The density function (5) is
used for the maximization of the likelihood function with respect to (θ + , θ − ).
Integrating out (5) with respect to the duration xi , it is possible to get the
probability
Ii 1−Ii
1
ψi+
p(yi |yi−1 ) =
4
1
ψi+
1
ψi−
+ ψ1−
i
.
(6)
The application
The prices of Italian stock Generali, available from Borsa Italiana Spa has been
analyzed om the period from 3rd through 19th of April 2000. The number of
available durations is 19696. The minimum duration is 1 second, the maximum
710 seconds, i.e. 11 minutes and 50 seconds. The average duration between
successive changes in prices is 19 seconds with standard deviation 27.16.
Before performing any analysis, the daily seasonal component s(t) has been
estimated using a cubic spline with knots set on each hour. The adjusted series
has been obtained removing the daily sesonal factor in the following way: xadj
=
i
xi
.
s(ti−1 )
The adjusted time series has been initially estimated using a log-AACD(2,2)
model1 proposed by Bauwens and Giot (1997) which is a variant of the AACD
model that does not require the positivity of the coefficients. In order to evaluate
the in-sample performance of this model, the probability of an increase of the
price (i.e. the probability (6)) has been computed at each time ti . When it is
n
P
P
P
1 ψ+
i
= exp
ω1 +
n
P2
ψi− = exp
ω3 +
j=1
2
j=1
α1j i−j
α3j i−j
Ii−1 + ω2 +
Ii−1 + ω4 +
4
P2
j=1
2
j=1
α2j i−j
α4j i−j
(1 − Ii−1 ) +
(1 − Ii−1 ) +
P2
j=1
2
j=1
+
βj+ log ψi−j
−
βj− log ψi−j
o
o
more (less) than 0.5, an increase (decrease) of the price is predicted. In 84.46%
(84.13%) of cases an increase (decrease) of the price is correctly predicted.
Even if the performance obtained with the short-range dependence LogAACD model is satisfactory, an Asymmetric FIACD(2,d,0) model2 , with a truncation point of the polynomial of infinite order Λ(L) set equal to 1000, has been
applied to the series because of the hyperbolic decay of the autocorrelation
function. The prior expected improvement did not come up. The percentages
of correct predictions of the movements of the prices has settled down around
53%, implying that this model does not provide a better forecast than the short
memory version.
5
Conclusions
In the latest years the availability of the so-called tick-by-tick data has pushed
statisticians towards new methodologies for getting more information on financial tradings. In this paper a class of joint models on the direction of price
changes and the duration between two consecutive price changes is considered
and a long memory approach is also proposed. The Log-AACD and the AFIACD models are applied to the Generali time series, showing the weakness of
the long memory model. Further analysis is needed in order to understand the
real potentialities and limits of such models.
References
BAUWENS, L. and GIOT, P. (1997): The logarithmic ACD model: an application to the bid-ask quote process of three NYSE stocks. Core Discussion
Paper 9789, Université Catholique de Louvain, Belgium.
BAUWENS, L. and GIOT, P. (2000): Asymmetric ACD models: introducing
price information in ACD models with a two state transition model. Core
Discussion Paper 9844, Université Catholique de Louvain, Belgium.
ENGLE, R. and RUSSELL, J. (1998): Autoregressive conditional duration;
a new model for irregularly spaced transaction data. Econometrica, 66,
1127–1162.
JASIAK, J. (1999): Persistence in intertrade durations.Manuscript, York University.
2 ψ+
i
ψi− =
=
ω1 +
ω3 +
P∞
P∞
j=1
j=1
λ1j xi−j
λ3j xi−j
Ii−1 + ω2 +
Ii−1 + ω4 +
P∞
P∞
j=1
5
j=1
λ2j xi−j
λ4j xi−j
(1 − Ii−1 ) +
(1 − Ii−1 ) +
P2
P2
j=1
j=1
+
βj+ ψi−j
−
βj− ψi−j