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Exact probability distribution function for multifractal random walk models of stocks
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July 2011 EPL, 95 (2011) 28007 doi: 10.1209/0295-5075/95/28007
www.epljournal.org
Exact probability distribution function for multifractal random walk models of stocks D. B. Saakian1,2,3(a) , A. Martirosyan4 , Chin-Kun Hu1 and Z. R. Struzik5 1
Institute of Physics, Academia Sinica - Nankang, Taipei 11529, Taiwan Yerevan Physics Institute - 2 Alikhanian Brothers St., Yerevan 375036, Armenia 3 Physics Division, National Center for Theoretical Sciences, National Taiwan University Taipei 106, Taiwan 4 Yerevan State University - Alex Manoogian 1, Yerevan 375025, Armenia 5 Graduate School of Education, The University of Tokyo - Bunkyo-ku, Tokyo 113-0033, Japan 2
received 8 November 2010; accepted in final form 2 June 2011 published online 4 July 2011 PACS PACS
89.65.Gh – Economics; econophysics, financial markets, business and management 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
Abstract – We investigate the multifractal random walk (MRW) model, popular in the modelling of stock fluctuations in the financial market. The exact probability distribution function (PDF) is derived by employing methods proposed in the derivation of correlation functions in string theory, including the analytical extension of Selberg integrals. We show that the recent results by Y. V. Fyodorov, P. Le Doussal and A. Rosso obtained with the logarithmic Random Energy Model (REM) model are sufficient to derive exact formulas for the PDF of the log returns in the MRW model. c EPLA, 2011 Copyright
In recent years, the fluctuation dynamics of stocks in financial markets have primarily been described in terms of multiplicative noise (multifractal random walk (MRW)) models [1–8] or coupled random walks [9]. The former models are widely used in financial engineering for risk analysis. The majority of these analyses consider mainly the scaling behaviour of the models. The tails of the distributions [10–17] are fitted to the experimental data, as the probability distribution function (PDF) itself is not known. We aim to reduce the MRW of ref. [3] to the statistical mechanics of a Random Energy Model (REM)-like model [18–20], and calculate the exact PDF. The availability of the exact PDF could dramatically improve the accuracy of the analysis of financial data. In the approach of refs. [1–3], the logarithm of returns ri of financial stock or an exponent such as the S&P 500 is observed with a minimal achievable time resolution τ . The logarithm of the return at the moment of time iτ is described by the following model: ri = xi exp[βyi ], (a) E-mail:
[email protected]
ri = 0,
(1)
where xi are independent random values from the normal distribution and β is the so-called intermittency parameter. If yi are constant, at a value y, the model describes the log-normal distribution of returns. More realistic models are those with stochastic volatility, where yi are random values [21]. For MRW models [3], yi are normally distributed random values, and yi yj = Cij , x2i = J 2 , i.e. the autocorrelation of x, is described by a constant J, and the correlations between two observations at times i and j are described by a matrix Cij . Logarithmic correlations are generally assumed in y: Cij = 2 ln dNij ,
dij N ; N≡
T τ
Cij = 0, dij N, ,
(2)
where dij τ is the time interval between two observations at times iτ and jτ , and N 1 is a parameter of the model, corresponding to T , the upper “correlation length” in the observed time series [9]. This model accurately describes experimental data of financial markets as shown, e.g., by Mandelbrot [22]. According to eq. (1) the observable ri (“logarithm of return”) decomposes into a product of two independent multipliers. The dynamical process described by eqs. (1)
28007-p1
D. B. Saakian et al. and (2) is referred to as the Bacry, Delour and Muzy MRW model [3]. The MRW model characterizes the behaviour of the logarithm of the returns in a certain “window of observation” time interval, to be referred to as the (time) scale s, entailing L resolution interval lengths, s ≡ Lτ as ZM RW (β, N, L) =
L
(3)
xi exp[βyi ].
(4)
i=1
We use N among the arguments of ZM RW , as the distribution of the random variables yi depends on the upper “correlation length”, through the variable N (eq. (2)). One can fit the available experimental data in order to obtain the PDF distribution of the returns using the MRW model [3] in eqs. (1)–(4), and a variety of numerical methods. In this letter, however, we show how this model can be solved analytically, to obtain an exact formula for the PDF for such a fit. In our derivation, we make use of the correspondence of the MRW model to the statistical mechanics of the logarithmic random energy model (REM) model [20]. Let us consider a sufficiently long time series of a duration T0 . Splitting the sequence into intervals of length s, we can calculate the PDF distribution ρ(Z) for the time series at the scale s: T0 /s L s ρ(β, N, L;Z) = δ Z− ri+(l−1)∗L , (5) T0 i=1
derive the solution for the MRW model. The MRW model can be considered as a random walk with the random time given by the MF model [6,22]. Indeed, here we consider a case of “simpler” motion corresponding to the internal time (described by the MF model). The motion spans a random walk by taking the form of “multifractal time”. The situation corresponds to the general principle of the description of complex systems [23]: using a simpler motion in some internal, unobserved space (pre-reality), and then mapping it randomly to the space of the observable (reality). Let us now provide explicit expressions of the probability distribution PM RW (ZM RW ). Consider the Fourier transform, with a parameter k, of ZM RW and the definition of the MF model (eq. (6)): k 2 J 2 2βyi exp [ikZM RW ] = exp − e 2 i 2 2 k J = exp − ZMF (2β) . (7) 2 The inverse Fourier transform of the distribution PM RW (β, N ; Z) of Z = ZM RW provides the following relationship: PM RW (β, N ; Z) = ∞ dh Z2 √ exp − PMF (2β, N ; h), 2hJ 2 2πhJ 2 0
(8)
where PMF (β, N, Z) is the distribution of Z = ZMF for the MF model by eq. (6). l=1 Thus, having obtained the probability distribution for the MF model, we can calculate the probability distribuwhere δ denotes the Dirac delta function. Here β, N, L tion for the MRW model. are the parameters of the distribution for Z, where Z is The function PMF (β, N ; h) depends on N as the logarithm of the return after Lτ period of time. In the arguments of the function, we separate the parameters PMF (β, N ; h) ≡ PˆMF (β; z), (9) and the function of the data by a semicolon. (The same convention is to be used throughout the text.) T0 /s is the z = h/Ze , number of intervals with the length L. We consider a case in which the number of such intervals is large. In this case where PˆMF (β; z) is defined above to serve as an intermeeq. (5), the empirical PDF, is expected to converge to diate notation. Equations (5) and (9) in ref. [20] with the the analytical formula we obtain. While calculating the substitution N = 1/ give the following expression for Ze : ρ(β, N, L; Z), it is important that we calculate the average 2 via non-overlapping intervals. As a consequence, we can e(1+β ) ln N . (10) Z = e identify the ρ(β, N, L; Z), calculated from the time series Γ(1 − β 2 ) with eq. (5), with the PDF of Z —the “partition function” in statistical physics models. The functions PMF and PˆMF are related through the reLet us first consider another sum instead of eq. (3), scaling of the arguments, defined in eqs. (9) and (10). where we omit the random variables xi : Thus, we can express the PDF for the MRW model as ZMF =
N
exp[βyi ].
(6)
PM RW (β, N ; Z) ≡ φβ (z), Z . z = Ze
i=1
(11)
This equation defines the multifractal (MF) model. The crux of our approach is in using the known results obtained We see that Ze defines the scale for the series sum ZM RW for the MF model by Fyodorov and Bauchaud [20], to in the argument of the analytical PDF φβ (z). The function 28007-p2
Exact probability distribution function for multifractal random walk models of stocks P Z Ze 1
P Z Ze 1.0
0.01
0.8
10– 4 0.6
–4
–2
10– 6
0.4
10– 8
0.2
10– 10
0
2
4
1.5 2.0
Z Ze
Fig. 1: (Colour online) The PDF of the MRW model by eq. (4) for 4β 2 = 0.4, 0.2, 0.1, 0 calculated using the exact formulas in eqs. (8)–(16). The higher values of β correspond to the higher values of distributions at the centre. The dashed line is the Gaussian distribution.
3.0
Z Ze 5.0 7.0 10.0 15.0 20.0
Fig. 2: (Colour online) The tails of the PDF for the MRW model at 4β 2 = 0.4, 0.2, 0.1, 0, obtained by the exact formulae, eqs. (8)–(16). The higher values of β correspond to the stronger tails of the distributions (higher lines in the plot). The dashed line is the Gaussian distribution.
In fig. 1, we provide the plots for the analytical expression φβ (z). We plot the PDF for Z/Ze at different values of the intermittency parameter β. The figures reflect a deviation from the Gaussian shape, and do not correspond to the Gaussian distribution one could naively expect from the central-limit theorem (CLT). The reader is referred to for z from eq. (9). ref. [14] for a characterization of the non-Gaussian properFyodorov et al. [20] derived the following distribution ties of the PDF. Our analytical result for the PDF shape for the series sum z of the MF model: shows fat tails, well documented across the financial liter ature. Indeed, according to the results by Fyodorov et al., 1 (13) in ref. [20], there is a scaling ds e−ht z 1−h (β), e−2t PˆMF (β; e−t ) = 2πi
φβ (z) is then calculated from eq. (8), by substituting PˆMF for PMF , obtained as shown below. Let us denote (12) z ≡ e−t
1 PˆMF (β; z)∼ ηMF , z 1 ηMF = 2 + 1. β
where the integral is carried out along a contour path parallel to the imaginary axis h = h0 + iw, h0 > 0. The generic moments z 1−s (β) are defined in ref. [20] for any complex s at a constant β, as z 1−h (β) = ×π
2 Γ(1 + β 2 (h − 1)) Aβ 2(h−1)(2+β (2h+1) Γ(1 − β 2 )1−h
This gives
3 3 G ( β + β1 + βh)Gβ ( 2β + βh)Gβ ( 2β + β2 + βh) 1−h β 2
Gβ ( β2 + β + βh)G2β ( β1 + βh)
(17)
1 , Z ηM RW 1 ηM RW = 2 + 1 2β
PM RW (β, N ;Z)∼ ,
(14)
(18)
for the scaling of the PDF tails of the MRW. We performed numerical evaluations of both distributions. We obtained the scaling exponent ηMF = 6 with 2% Gβ ( β1 + β)2 Gβ ( β2 + 2β) Aβ = . (15) accuracy for the distribution by eq. (17) with β 2 = 0.2 for 3β 1 3 3 Gβ ( 3β 2 + β )Gβ ( 2β + β)Gβ ( 2 + 2β ) z ≈ 500, while the same accuracy was obtained for ηM RW The function Gβ (x) has been defined in ref. [20] using the by eq. (18) for z ≈ 25. In fig. 2, we plot the tails of the PDF for MRW in log-log coordinates for different values results of Fateev [24]: of the intermittency parameter β. ∞ −Qt −xt In table 1, we provide the scaling exponent ηM RW ) (x − Q dt − e e 2 + ln Gβ (x) = fitted from the graphics PM RW (β; z) at different values −βt −t/β ln 2π t (1 − e )(1 − e ) 0 of z from the slope of the logarithm of the function vs. Q −x e−t Q − x)2 + 2 + . (16) logarithmic scale. To obtain greater than 40% accuracy 2 2 t for the fitting of the exponent ηM RW from the graphics of the function PM RW (β; z) we consider relatively small values of the function ∼ 10−5 . Therefore, while directly for Re(x) > 0, and Q = β + β1 . where
28007-p3
D. B. Saakian et al. Table 1: The fit of the scaling exponent ηM RW via the slope of the graphics in fig. 2 for the given value of z at 4β 2 = 0.2. The theoretical value for z = ∞ is 11.
z 5 10 15 20 PM RW (β; z) 8.6 ∗ 10−4 7 ∗ 10−6 2.1 ∗ 10−7 1.3 ∗ 10−8 ηM RW 6.03 8.2 9.2 10.2 fitting the experimental market data with our exact PDF formula, higher accuracy can be achieved in the regions near the centre of distribution compared with the tails of the distribution. Let us now consider the PDF of ZM RW (β, N, s) for the general scale s. A relation connecting the distributions at different scales can be derived from eqs. (2), (3) and reads ˆ L, L; Z), ρ(β, N, L; Z) = ρ(β, ln(N ) ˆ , β =β ln(L)
worth mentioning that for small intermittency, all the cascade processes are described approximately by the model in eqs. (1), (2), and such analytical solutions are generic. It is interesting to note that the full analytical solution of the most popular stock fluctuation model presented in this article requires reference to the results of string correlation functions. In this article we have investigated the MRW model formulated by Bacry, Delour and Muzy. The Markov switching multifractal model (MSM), suggested in [7], is now widely used for the market [27,28]. The MSM model closely resembles the MRW model, and there is a possibility that it can also be solved exactly using our method. ∗∗∗
(19)
This work was supported by Academia Sinica, National Science Council in Taiwan with Grant No. NSC 99-2911I-001-006, and National Center for Theoretical Sciences where N, L are defined via eqs. (2) and (3). We can use (Taipei Branch). this exact relation, eq. (19), to derive the distribution at the scale s = S, using the result at s = T . REFERENCES Now we may consider PDF φβˆ (Z/Zˆe ), where β2 , βˆ2 = v
v=
ln(S/τ ) . ln(T /τ )
(20)
Equations (10) and (20) produce 2
e(v+β ) ln N Zˆe = . Γ(1 − β 2 /v)
(21)
Equation (20) shows that the intermittency parameter β grows for the small scales s. This scale-dependency of the intermittency parameter βˆ2 may now serve to obtain a better estimate of the fat tail scaling in eq. (18). Thus, we derive an exact formula for the PDF of returns, eq. (8), using a recent advance in statistical physics [20], expressed in eqs. (9)–(16), achieved using the results of string theory in correlation functions [24]. To date, the exact solution has been available only for another, simpler model with stochastic volatility, that of Heston [25]. While the Heston model is successful in describing stock price fluctuations [21,26], the MRW model appears to be more adequate for modelling financial markets. We foresee that having an exact PDF will permit fitting the parameters of the MRW model better to real data, in particular in the case of small experimental data sets. This is in any case at least as reliable as estimating the asymptotics of the PDF. For this purpose, one can use our functions φβ (Z/Ze ), or consider Ze to be a free scale parameter and consider all possible choices of β and Ze to fit the experimental data. If we analyze the data at different scales S, we should fit the data using the functions φβ(S) (Z/Ze ), where β(S) is given by eq. (20). To date, an exact analytical expression has been available only for the logarithmic correlation in eq. (2). It is
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Exact probability distribution function for multifractal random walk models of stocks
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