Nonlinear Stochastic Model of Return matching to the data of New York and Vilnius Stock Exchanges
aa r X i v : . [ q -f i n . S T ] M a r Nonlinear Stochastic Model of Return matching to the dataof New York and Vilnius Stock Exchanges
V. Gontis, A. KononovičiusInstitute of Theoretical Physics and Astronomy of Vilnius University,[email protected]
Abstract
We scale and analyze the empirical data of return from New York and Vilnius stockexchanges matching it to the same nonlinear double stochastic model of return in financialmarket.
Volatility clustering, evaluated through slowly decaying auto-correlations, Hurst effect or /f β noise for absolute returns, is a characteristic property of most financial assets return time series[1]. Statistical analysis alone is not able to provide a definite answer for the presence or absence oflong range dependence phenomenon in stock returns or volatility, unless economic mechanisms areproposed to understand the origin of such phenomena [1, 2]. Whether results of statistical analysiscorrespond to long range dependence is a difficult question subject to an ongoing statistical debate[2, 3]. The agent based economic models [4, 5] as well as stochastic models [6, 7, 8, 9] exhibitinglong range dependence phenomenon in volatility or trading volume are of grate interest and remainan active topic of research.The properties of stochastic multiplicative point processes have been investigated analytically andnumerically and the formula for the power spectrum has been derived [10], later the model hasbeen related with the general form of the multiplicative stochastic differential equation [11, 12].The extensive empirical analysis of the financial market data, supporting the idea that the long-range volatility correlations arise from trading activity, provides valuable background for furtherdevelopment of the long-ranged memory stochastic models [13, 14]. The power law behaviour ofthe auto-regressive conditional duration process [15] based on the random multiplicative processand its special case the self-modulation process [16], exhibiting /f fluctuations, supported theidea of stochastic modelling with a power law probability density function (PDF) and long memory.A stochastic model of trading activity based on an stochastic differential equation (SDE) drivenPoisson-like process has been already presented in [8]. In the paper [9] we proposed a doublestochastic model, which generates time series of the return with two power law statistics, i.e., thePDF and the power spectral density of absolute return, reproducing the empirical data for theone-minute trading return in the NYSE. 1n this contribution we analyze empirical data from Vilnius Stock Exchange (VSE) in comparisonwith NYSE and stochastic model proposed in [9]. At the same time we demonstrate the scalingof statistical properties with longer time window of return. Recently we proposed the double stochastic model of return in financial market [9] based on thenonlinear SDE. The main advantage of proposed model is its ability to reproduce power spectraldensity of absolute return as well as long term PDF of return. In the model proposed we assumethat the empirical return r t can be written as instantaneous q -Gaussian fluctuations ξ with a slowlydiffusing parameter r and constant λ = 5 r t = ξ { r , λ } . (1)q-Gaussian distribution of can be written as follows: P r ,λ ( r ) = Γ( λ ) r √ π Γ( λ − ) r r + r ! λ/ , (2)The parameter r serves as a measure of instantaneous volatility of return fluctuations. See [9],for the empirical evidence of this assumption. Here r is defined in the selected time interval τ asa difference of logarithms of asset prices p : r ( t, τ ) = | ln[ p ( t + τ )] − ln[ p ( t )] | . (3)In this paper we consider dimensionless returns normalized by its dispersion calculated in thewhole length of realization. It is worth to notice that r ( τ ) is an additive variable, i.e., if τ = P i τ i ,then r ( τ ) = P i r ( τ i ) , or in the continuous limit the sum may be replaced by integration. We dopropose to model the measure of volatility r by the scaled continuous stochastic variable x , havinga meaning of average return per unit time interval. By the empirical analyses of high frequencytrading data on NYSE [9] we introduced relation: r ( t, τ ) = 1 + ¯ r τ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t s + τ s Z t s x ( s )d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4)where ¯ r is an empirical parameter and the average return per unit time interval x ( t s ) can bemodeled by the nonlinear SDE, written in a scaled dimensionless time t s = σ t t : d x = " η − λ − (cid:18) xx max (cid:19) (1 + x ) η − ( ǫ √ x + 1) x d t s + (1 + x ) η ǫ √ x + 1 d W s . (5) Here we abbreviate the official name NASDAQ OMX Vilnius Stock Exchange for the convenience η - exponent of multiplicativity, λ - powerlaw exponent of x long range PDF, ǫ - parameter dividing diffusion into two areas: stationary andexcited one, σ t - time scale adjustment parameter and x max - the upper limit of diffusion. Theterm (cid:16) xx max (cid:17) excludes divergence of x to the infinity. Seeking to discover the universal natureof financial markets we consider that all these parameters are universal for all stocks traded onvarious exchanges. In this paper we analyze empirical data from very different exchanges NewYork, one of the most developed with highly liquid stocks, and Vilnius, emerging one with stockstraded rarely.We solve Eq. (5) numerically introducing variable steps of dimensionless time t s,k +1 = t s,k + h k : h k = κ (cid:16) ǫ q x k + 1 (cid:17) (1 + x k ) η − , (6)where κ is precision parameter of numerical calculations, which should be less than 1. Then SDE,Eq. (5), can be replaced by iterative equation: x k +1 = κ " η − λ − (cid:18) xx max (cid:19) x k + κ q x k ζ k , (7)where ζ k is a normally distributed random variable with zero mean and unit variance. In paper [9] we analyzed the tick by tick trades of 24 stocks, ABT, ADM, BMY, C, CVX, DOW,FNM, GE, GM, HD, IBM, JNJ, JPM, KO, LLY, MMM, MO, MOT, MRK, SLE, PFE, T, WMT,XOM, traded on the NYSE for 27 months from January, 2005, recorded in the Trades and Quotesdatabase. The parameters of stochastic model presented in Section 2 were adjusted to the empiricaltick by tick one minute returns. An excellent agreement between empirical and model PDF andpower spectrum was achieved, see Fig. 3 in [9]. The same empirical data and model resultswith slightly changed values of parameters are given in Figure 1 (a,b). Noticeable difference intheoretical and empirical PDFs for small values of return r are related with the prevailing pricesof trades expressed in integer values of cents. We do not account for this discreteness in ourcontinuous description. In the empirical power spectrum one-day resonance - the largest spikewith higher harmonics - is present. This seasonality - an intraday activity pattern of the signal -is not included in the model either and this leads to the explicable difference from observed powerspectrum.Provided that we use scaled dimensionless equations derived while making very general assump-tions, we expect that proposed model should work for various assets traded on different exchangesas well as for various time scales τ . We analyze tick by tick trades of 4 stocks, APG1L, PTR1L,SRS1L, UKB1L, traded on VSE for 50 months since May, 2005, trading data was collected andprovided for us by VSE. Stocks traded on VSE in comparison with NYSE are less liquid – meaninter-trade time for analyzed stocks traded on VSE is 362 s, while for stocks traded on NYSE mean3nter-trade time equals 3.02 s. The difference in trading activity exceeds 100 times. This great dif-ference is related with comparatively small number of traders and comparatively small companiesparticipating in the emerging VSE market. Do these different markets have any statistical affinityis an essential question from the theoretical point of market modeling.First of all we start with returns for very small time scales τ = 60 s . For the VSE up to 95%of one minute trading time intervals elapse without any trade or price change. One can excludethese time intervals from the sequence calculating PDF of return. With such simple procedurecalculated PDF of VSE empirical return overlaps with PDF of NYSE empirical return (see Figure1 (a)). -8 -7 -6 -5 -4 -3 -2 -1 -2 -1 P(r) r (a) 10 -7 -6 -5 -4 -3 -2 S(f) f (b)10 -8 -7 -6 -5 -4 -3 -2 -1 -2 -1 P(r) r (c) 10 -7 -6 -5 -4 -3 -2 S(f) f (d)10 -8 -7 -6 -5 -4 -3 -2 -1 -2 -1 P(r) r (e) 10 -7 -6 -5 -4 -3 -2 S(f) f (f)
Figure 1: Comparison of empirical statistics of absolute returns traded on the NYSE (black thinlines) and VSE (light gray lines) with model statistics, Eqs. (1)-(7), (gray lines). Model parametersare as follows: λ = 5 ; σ t = 1 / · − s − ; λ = 3 . ; ǫ = 0 . ; η = 2 . ; ¯ r = 0 . ; x max = 1000 . PDFof normalized absolute returns is given on (a),(c),(e) and PSD on (b),(d),(f). (a) and (b) representsresults with τ = 60 s ; (c) and (d) τ = 600 s ; (e) and (f) τ = 1800 s . Empirical data from NYSE isaveraged over 24 stocks and empirical data from VSE is averaged over 4 stocks. τ = 60 s , and is related with low VSE market liquiditycontributing to the white noise appearance. The different length of trading sessions in financialmarkets causes different positions of resonant spikes. One can conclude that even so marginalmarket as VSE retains essential statistical features as developed market on NYSE. At the firstglance the statistical similarity should be even better for the higher values of return time scale τ .Further we investigate the behavior of returns on NYSE and VSE for increased values of τ = 600 s and τ = 1800 s with the specific interest to check whether proposed stochastic model scales inthe same way as empirical data. Apparently, as we can see in Figure 1 (d) and (f) PSDs ofabsolute returns on VSE and on NYSE overlap even better at larger time scale ( seconds and seconds). This serves as an additional argument for the very general origin of long rangememory properties observed in very different, liquidity-wise, markets. The nonlinear SDE is anapplicable model to cache up observed empirical properties. PDFs of absolute return observed inboth markets (see Figure 1 (c) and (e)) are practically identical, though we still have to ignorezero returns of VSE to arrive to the same normalization of PDF. We proposed a double stochastic process driven by the nonlinear scaled SDE Eq. (5) reproducingthe main statistical properties of the absolute return, observed in the financial markets. Sevenparameters of the model enable us to adjust it to the sophisticated power law statistics of variousstocks including long range behaviour. The scaled no dimensional form of equations gives anopportunity to deal with averaged statistics of various stocks and compare behaviour of differentmarkets. All parameters introduced are recoverable from the empirical data and are responsiblefor the specific statistical features of real markets. Seeking to discover the universal nature ofreturn statistics we analyse and compare extremely different markets in New York and Vilnius andadjust the model parameters to match statistics of both markets. The most promising result of thisresearch is discovered increasing coincidence of the model with empirical data from the New Yorkand Vilnius markets and between markets, when the time scale of return τ is growing. Observablespecific features of different markets could be a subject of another research based on the proposedmodel. For example, it is clear that parameter x max should be relevant to the maximum numberof active traders in the market and consequently should be specific for the every market. Furtheranalyses of empirical data and proposed model reasoning by agent behavior is ongoing. Acknowledgment
We would like to express gratitude towards NASDAQ OMX Vilnius Stock Exchange, which pro-vided empirical trade by trade data for our research.5 eferences [1] Willinger, W. Taqqu, M. and Teverovsky, V. 1999.
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