From Renewal Theory To High-Frequency Finance

Transcription

1 From Renewal Theory To High-Frequency Finance Mauro Politi * Enrico Scalas ** * Universitá degli studi di Milano Dipartimento di Fisica, via Celoria Milano ITALY mauro.politi@unimi.it ** Universitá degli Studi del Piemonte Orientale Dipartimento di Scienze e Tecnologie Avanzate, via Bellini 25/G Alessandria ITALY enrico.scalas@mfn.unipmn.it ABSTRACT. Based on the continuous-time random walk (CTRW) formalism for high-frequency financial data, we present some preliminary recent results on the following issues: We analyze the structure of waiting times between consecutive trades and fit them with Tsallis q -exponentials and Weibull functions. We define stochastic integrals on CTRWs and we study the (non-markovian) case of nonexponentially distributed waiting times. KEYWORDS: Renewal Theory, Compound Poisson Processes, Continuous-Time Random Walks, Stochastic Calculus, Econophysics 1 re soumission à MASHS 27- Brest, France, le 28 April 27.

2 2 1 re soumission à MASHS 27- Brest, France. 1. Introduction In the last two decades, the use of automatic systems in stock markets has made many tick-by-tick series available and has determined an increasing interest in the study of high-frequency finance. In 1998, Engle and Russel introduced the so-called Autoregressive Conditional Duration (ACD) models emphasizing the importance of intertrade durations [ENG 98, ENG ]. Before that, in 1982, Engle had introduced generalized autoregressive heteroskedastic models taking into account phenomena such as volatility claustering [ENG 82, BOL 86]. Mandelbrot and, later, Praetz suggested the use of mixtures of normal distributions to account for the leptokurtic character of empirical return distributions [MAN 67, PRA 72]. These theoretical developments paved the way for the study of high-frequency financial time-series. In particular, in the Econophysics literature, it is now common to find applications and studies on high-frequency finance. Among the most used tools, there are Continuous Time Random Walks (CTRWs), introduced by Montroll and Weiss in Physics for the study of anomalous diffusion processes [MON 65]. In this paper, we present a short review of the CTRW model and we discuss preliminary results on the following issues: the characterization of waiting-time survival functions and the definition of stochastic integrals on CTRWs. 2. Continuous-Time-Random-Walk models Let us start from the observation of real data. The now classical Black and Scholes assumptions describe the time evolution of a share price S(t) as a geometric Wiener process. However, the geometric Wiener process as well as its self-similar generalizations [MAN 3] cannot be used for tick-by-tick data, as the scaling properties of these processes break down. This is shown in Figure 1; if we observe S(t) under a very short time window (some minutes), it becomes a step-function. Scaling is present only in the hydrodynamic limit, i.e. when the prices are observed from a long distance, as described in refs. [SCA, SCA 4a] The simplest model describing the tick-by-tick behaviour is a random walk defined as follows [PRE 67, SCA 6]. Let us call t i, with i = 1...N, the random instants in which a jump in the price S(t) occurs. Moreover, let us define the log-return x(t) = ln(s(t)/s(t )) for the origin of continuous trading in the Stock Market, t ; without loss of generality we can assume t =. Now, we define two random variables: the waiting times or durations τ i = t i t i 1 and the tick-by-tick log-returns ξ i = ln(s(t i )/S(t i 1 )). Let ϕ(τ, ξ) be the joint probability density function. The two marginal densities can be written as ψ(τ) = ϕ(τ, ξ)dξ [1]

3 High-Frequency Finance 3 Figure 1. Dependence of the time series shape on the scale of observation. For a small time window scaling breaks down and we observe a step function. (General Electric prices traded at NYSE, October 1999) λ(ξ) = ϕ(τ, ξ)dτ and are easily empirically accessible. If ϕ(τ, ξ) = ψ(τ)λ(ξ), we call this process uncoupled CTRW and coupled CTRW otherwise. With these definitions in mind, we can write for the log-return process x(t): n(t) x(t) = ξ j [2] with x() =, where n(t) it is equal to the number of events (jumps) with t i < t: n(t) = max(n; t n < t). By probabilistic arguments [MON 65, SCA, SCA 4a], it is possible to derive the following integral equation giving the probability density, p(x, t), for the log-return to be in position x at time t, conditioned on the fact that it was in position x = at time t = : p(x, t) = δ(x)ψ(t) + t + j=1 ϕ(x x, t t )p(x, t )dt dx [3] where δ(x) is Dirac s delta function and Ψ(τ) is the survival function (or survival probability, or complementary cumulative distribution function) defined as: Ψ(τ) = 1 τ ψ(τ )dτ = + τ ψ(τ )dτ. [4]

4 4 1 re soumission à MASHS 27- Brest, France. It is important to notice that a CTRW is Markovian if and only if ψ(τ) is an exponential distribution: ψ(τ) = µexp( µτ). In this case, the log-price fluctuations follow a Compound Poisson process and one can write the following closed form for p(x, t): p(x, t) = (µt) k e µt λ n (x) [5] k! k= where λ n (x) represents the n-fold convolution of the λ(x) functions. For a general ψ(τ) it is possible to write: p(x, t) = P (k, t)λ n (x) [6] k= where P (k, t) is the probability to have k jumps between and t. It is important to note that, in the exponential case, this probability is independent on the starting point. In fact, generally speaking, the probability to have k events between t and t + t is different from the one to have the same number of jumps between t and t + t. As possible to understand by this brief introduction, the time structure of CTRWs is strictly related with the arguments of renewal theory [FEL 66]. 3. Waiting-time structure A priori, one could argue that there is no strong reason for independent market investors to place buy and sell orders in a time-correlated way. This argument would lead one to expect a Poisson process for interorder durations. If price formation were a simple thinning of the bid-ask process, then exponential waiting times should be expected between consecutive trades as well [COX 79]. But, in the last few years, some independent empirical studies had shown that for a wide different kind of financial products (stocks, futures and forex trades) the empirical survival function for intertrade durations Ψ(τ) is not exponential (see references in [SCA 4b]). This has to do both with the market price formation mechanisms and with the double-auction process. The empirical results on the survival probability set limits on possible statistical market models for price formation. An interesting result has been obtained by Lillo and Farmer, who find that the signs of orders in the London Stock Exchange obey a long-memory process [LIL 4] as well as by Bouchaud and co-workers [BOU 4]. Further studies on market microstructure will clarify this point. For shares it is possible to find a well defined behavior. As shown in Fig. 2, the survival function interpolates between an exponential behavior and a power law while τ is increasing. In order to understand the reason for this anomalous shape, it is useful to consider the intertrade process as a mixture of Poisson processes [SCA 7, JEW 82, MCN 8]. In this way, Ψ(τ) can be written as a weighted sum of exponential distributions: Ψ(τ) = g(µ)e µτ dµ with g(µ)dµ = 1. [7]

5 High-Frequency Finance 5 Semi Log plot of empirical Survival function 1 UTX Oct 99 Exponential law Log Log plot of Empirical survival function UTX Oct 99 Power law curve (exponent 2) Figure 2. Empirical survival function vs. exponential distribution and vs. power law. It is important to notice that the behavior of the data is "far" from the exponential as well as from the power law. Studying the spectrum g(µ) gives us some pieces of information on the process memory as well as its non-stationarity. The inverse problem defined in 7 can be solved by numerical real-inversion of a Laplace transform for noisy data. A discussion on this problem can be found in [POL 7]. It is also possible to get information by trying to fit the survival probabilities with selected families of curves. In the literature it is possible to find references using either Mittag-Leffler [GOR 97] or Weibull distributions 1 [IVA 4]: ψ(τ) w(τ; a, b) = ax b 1 e axb Ψ(τ) = e axb [8] Ψ(τ) E β ( ατ β ), E β (z) := n= z n Γ(βn + 1). [9] We are currently fitting two different distribution to each stock intertrade durations belonging to the DJIA for the month of October, We choose to revisit the old studies on the Weibull distributions and to use a new family of function. Indeed, in recent years, some functions arising in the framework of Tsallis non-extensive statistical mechanics [TSA 88] have been used to fit financial distributions, such as log-returns [BOR 2] and volumes [des 6]. For intertrade durations, we use the q-exponential functions defined as e x q (1 + (1 q)x) 1 1 q [1] ψ(τ) (e µx q ) q = (1 + (1 q)( µ q x)) q 1 q [11] Ψ(τ) = e µx q. Two fitting methods have been used. The first one is a kind of maximum-likelihood algorithm and the distributions parameters that minimize some objective function are selected. In the second one, we match the mean value and the second moment of the 1. There are different equivalent forms to parametrize the Weibull distribution.

6 6 1 re soumission à MASHS 27- Brest, France. empirical distribution to the theoretical predictions. A set of two equations is derived and solved for the two parameters of either the q-exponential or the Weibull distribution. The first method only gives a good description of the distribution shape whereas the second one also takes into account the information on the low-order moments. For both methods we use goodness of fit tests for one-tail distributions. The first method is not able to discriminate between the two families of curves. With the second method, we can see a significant difference between the Weibull and the q-exponential fits. In Fig.3 it is possible to see results based on the second method. A problem in the inter BA Oct 99 Exponential Weibull q Exponential XON Oct 99 Exponential Weibull q Exponential Figure 3. Example of a non-parametric fit with Weibull, q-exponential and exponential distribution. (Boeing and Exxon traded at NYSE, October 1999) pretation of the results is that the families have two parameters. One of them can be simply seen as a time-scale parameter (a for the Weibull and µ q for the q-exponential), whereas the other one does not have any simple meaning. However, we can study its distribution on different stocks and find that its values have little dispersion around the average. We can also see some systematic dependence between parameter estimates and the number N of trades registered for every share. In Fig.4 the values of q and β are plotted vs. N as obtained in the second method. For further discussion on the Values of q parameter vs. the number of jumps Value of Weibull s β parameter vs. the number of jumps Values of q for the fits Mean Value q=1.151 (std=.66) x Value of β for the fits Mean Value β =.837 (std=.73) x 1 4 Figure 4. Parameter q (left) for the q-exponential and β (right) for the Weibull as a function of the number of trades.

7 High-Frequency Finance 7 problem we point to the following papers [IVA 4, EIS 6, POL ]. 4. Stochastic calculus on CTRWs The classical Black and Scholes theory assumes that, in a risk neutral setting, the price S(t) obeys a geometric Brownian motion described by the following diffusive stochastic differential equation (SDE): ds(t) = rs(t)dt + σs(t)d W t, [12] where σ is the volatility, r is the risk free interest rate and dw t is a Wiener process defined by the properties: W =, a.s.; W t is a.s. continuous; W t has independent and stationary increments with distribution W t W s N(, t s). As previously discussed, share prices behave differently. An behavior of the underlying more similar real high-frequency-processes can be found in [MER 76]. It is essentially an uncoupled CTRW with exponentially distributed durations and normally distibuted log-returns called Normal Compound Poisson Process (NCPP). Option pricing with respect to pure jump processes is discussed in the recent book by Cont and Tankov [CON 3]. A translation of Merton s ideas appeared in a nice paper for physicists [ZYG 3] that triggered our interest on the issue. Indeed, it is possible to define the rules of stochastic calculus over the NCPPs and this is possible with some restrictions on the properties of jump distributions, such as the finiteness of all its moments: + ξ n λ(ξ)dξ <. [13] We have derived rules for the definition of stochastic calculus on a more general class of uncoupled CTRWs, when ψ(τ) is non-exponential but has a finite first moment [POL ]. Let us recall that the integral over the Wiener process J t = t H(W s )dw s [14] is not precisely determined in a conventional sense, unless the additional rule of choosing the intermediate points during construction of approximate sums is specified. Ito s choice is J t = lim H(W si )[W i+1 W i ] [15] i and results, due to the statistical independence of increments for the Wiener process, in the martingale property of the integral: J t =. As a consequence, ordinary calculus rules do not apply to Ito s integrals. In fact, if, according to Ito, one has I t =, for I t = t G (W s )d W s, [16]

8 8 1 re soumission à MASHS 27- Brest, France. we cannot identify I t with G = G(W t ) G(). There exists a definition of stochastic integral due to Stratonovich, which is free of this inconvenience, i.e., it is consistent with the ordinary rule dg(w s ) = G (W s )dw s (differential written without the symbol ): J t = lim H(W (si+1 +s i )/2)[W si+1 W si ]. [17] i This corresponds to the evaluation of the integrand in the middle of successive time intervals. The practical importance of Stratonovich integrals is due to the fact that they follow the ordinary calculus rules. This applies also to SDEs and the Stratonovich SDE dy t = [f(y t ) Dg(y t )g (y t )]dt + g(y t )dw t, [18] where D is a normalization constant W 2 t = 2Dt, is equivalent to the following Ito SDE (equivalent means that they have the same solutions): It is now clear that to define stochastic integrals Q t = dy t = f(y t )dt + g(y t )d W t. [19] t G (x)dx t R t = t G (x)d x t [2] and some analogous SDEs on a CTRW x t = x(t) = n(t) i=1 ξ i, the average dg(x) has a key role. Using the independence and the stationarity of the increments and the Stratonovich definition, it is possible to write: dg(x) = G(x + dx) G(x) = dk G ) dx k (dxk k! and [ZYG 3] has shown that for compound Poisson process with ψ(τ) = µ exp( µτ) one gets: (dx k ) = µdt + k=1 [21] ξ k λ(ξ)dξ + o[(dt) 2 ]. [22] In our work, we show how to define the stochastic calculus over a generic CTRW with non-exponential distribution ψ(τ) with a finite mean value: + τψ(τ)dτ <. [23] Indeed, it is possible to rewrite equation 22 by replacing µ with a function h(t) called renewal density function (dx k ) = h(t)dt + ξ k λ(ξ)dξ + o[(dt) 2 ], [24] where h(t) is defined as the first derivative of the renewal function H(t): H(t) = E[n(t)]. [25]

9 High-Frequency Finance µ 1 =1/1 µ 2 =1/3 a 1 =.5 a 2 =.5.6 h(t) seconds Figure 5. h(t) function for a simple mixture of exponentials. It shows a typical exponential decay of the memory. H(t) is the solution of the integral equation H(t) = 1 Ψ(t) + t H(t s)ψ(s)ds, [26] called renewal equation. h(t)dt can be interpreted as the expected number of events between t and t + dt. For a complete presentation of renewal theory, one can consult [COX 7, FEL 66]. Computing h(t) is not an easy task. As an example, let us consider a simple mixture of two exponential distributions: with a 1 + a 2 = 1. We find: ψ(τ) = a 1 µ 1 e µ 1τ + a 2 µ 2 e µ 2τ, [27] h(t) = µ 1 µ 2 a 1 µ 2 + a 2 µ 1 + e (a1µ2+a2µ1)t (a 1 µ 2 + a 2 µ 1 ) 2 (a 1a 2 2µ a 2 1a 2 µ 3 2 [28] (2a 1 a 2 2 a 2 1a 2 )µ 2 1µ 2 (2a 2 1a 2 a 1 a 2 2)µ 1 µ 2 2). [29] In Fig.5 it is sketched for the values a 1 =.5, a 2 =.5 µ 1 =1/1 and µ 2 =1/3. As shown, this kind of processes have an exponentially decaying memory.

10 1 1 re soumission à MASHS 27- Brest, France. 5. Conclusions We have briefly discussed some preliminary results obtained on the application of the CTRW model to the analysis of intertrade durations as well as generalizations of stochastic calculus. It turns out that both q-exponentials and Weibull distributions can be used for fitting the empirical survival function, even if q-exponentials give better results in the tail region when parameters are obtained by estimates of the first two moments. As for stochastic calculus, equation 24 can be used to compute Ito s compensators and will be applied to intraday option pricing. Future work on the application of CTRWs to high-frequency finance will be focused on the relationship with popular ARCH-GARCH models. 6. References [BOL 86] BOLLERSLEV T., Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, vol. 31, 1986, p [BOR 2] BORLAND L., Option Pricing Formulas Based on a Non-Gaussian Stock Price Model, Phisical Review Letters, vol. 22, 22, [BOU 4] BOUCHAUD J., GEFEN Y., POTTERS M., M.WYART, Fluctuations and Response in Financial Markets: The subtle nature of random price changes, Quantitative Finance, vol. 4, 24, p [CON 3] CONT R., TANKOV P., Financial modelling with Jump Processes, Chapman & Hall/Crc Financial Mathematics Series, 23. [COX 7] COX D., Renewal Theory, Methuen & Co., 197. [COX 79] COX D., ISHAM V., Point Processes, Chapman and Hall, London, [des 6] DESOUZA J., MOYANO L. G., QUEIROS S. M., On statistical properties of traded volume in financial markets, The European Physical Journal B, vol. 5, 26, p [EIS 6] EISLER Z., KERTÉSZ J., Size matters: some stylized facts of the stock market revisited, The European Physical Journal B, vol. 51, 26, p [ENG 82] ENGLE R. F., Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation, Econometrica, vol. 5, 1982, p [ENG 98] ENGLE R. F., RUSSELL J. R., Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data, Econometrica, vol. 5, num. 66, 1998, p [ENG ] ENGLE R. F., The Econometrics of Ultra-High-Frequency Data, Econometrica, vol. 1, num. 68, 2, p [FEL 66] FELLER W., An Introduction to Probability Theory and Its Applications, John Wiley & Sons, New York, [GOR 97] GORENFLO R., MAINARDI F., Fractional calculus: integral and differential equations of fractional order, A. Carpinteri and F. Mainardi (Eds), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlang, Wien and New York, 1997, Page 223.

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