Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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