Application of the Beck model to stock markets: Value-at-Risk and portfolio risk assessment

Physica A 387 (2008) 1225–1246 www.elsevier.com/locate/physa

Application of the Beck model to stock markets: Value-at-Risk and portfolio risk assessment M. Kozaki, A.-H. Sato ∗ Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Received 22 February 2007; received in revised form 17 August 2007 Available online 13 October 2007

Abstract We apply the Beck model, developed for turbulent systems that exhibit scaling properties, to stock markets. Our study reveals that the Beck model elucidates the properties of stock market returns and is applicable to practical use such as the Value-at-Risk estimation and the portfolio analysis. We perform empirical analysis with daily/intraday data of the S&P500 index return and find that the volatility fluctuation of real markets is well-consistent with the assumptions of the Beck model: The volatility fluctuates at a much larger time scale than the return itself and the inverse of variance, or “inverse temperature”, β obeys 0-distribution. As predicted by the Beck model, the distribution of returns is well-fitted by q-Gaussian distribution of Tsallis statistics. The evaluation method of Value-at-Risk (VaR), one of the most significant indicators in risk management, is studied for q-Gaussian distribution. Our proposed method enables the VaR evaluation in consideration of tail risk, which is underestimated by the variance–covariance method. A framework of portfolio risk assessment under the existence of tail risk is considered. We propose a multi-asset model with a single volatility fluctuation shared by all assets, named the single β model, and empirically examine the agreement between the model and an imaginary portfolio with Dow Jones indices. It turns out that the single β model gives good approximation to portfolios composed of the assets with non-Gaussian and correlated returns. c 2007 Elsevier B.V. All rights reserved.

PACS: 89.65.Gh; 02.50.Ey; 05.10.Gg Keywords: Econophysics; Stock markets; Value-at-Risk; Portfolio risk assessment; Tsallis statistics

1. Introduction Nowadays, the improvement of risk management systems becomes a crucial issue for financial institutions. The fundamental feature of risk management is an estimation of risk and control of its amount within a limited risk buffer. The assets exposed to risk, such as stocks, bonds, and loans are called risk assets. Risk assets sometimes fluctuate unpredictably but such fluctuations are mutually not independent, but correlated. The amount of risk corresponding to risk assets is measured by statistical methods such as Value-at-Risk explained below. Based on the estimated risk, financial institutions have to reserve a capital with which the loss would be compensated when the risk is realized. As a capital, or a risk buffer, is limited, control of the total amount of risk and the allocation strategy of the risk buffer ∗ Corresponding author.

E-mail address: [email protected] (A.-H. Sato). c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.10.023

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between risk assets to maximize earnings become important problems. In order to handle such problems the portfolio theory offers an established framework. Value-at-Risk (VaR) is one of the risk estimation methods widely employed in the practice of risk management [1, 2]. It was propagated by the BIS regulation, where it was adopted as the standard method for evaluating market risk [3]. The brief definition of VaR is described as follows: a VaR with confidence level c for the period of t days means that the losses larger than the amount of the VaR occurs with probability 100(1−c)% during t days. The current standards of VaR measurement are the variance–covariance method and the Monte Carlo method. Variance–covariance method (VC) evaluates a VaR with the assumption of Gaussian distributed returns. Its name comes from the fact that a variance–covariance matrix of assets plays an important role in this method. Based on the portfolio theory, it allows us intuitive analysis and easy calculation of risk, which has enabled VC to propagate early in practice. The VC has a problem, however, that it ignores fluctuations of volatility (heteroskedasticity) which is widely observed in actual markets and resulting fat-tailness of return distributions and thus likely to underestimate the amount of risk. Monte Carlo method (MC) supposes that the return process obeys a certain time series model such as GARCH model [4] and numerically evaluate a VaR with a Monte Carlo simulation. Since it is a numerical method, MC puts few restrictions on models and theoretically intractable models can be easily evaluated. However, the heavy load of simulation is the shortcoming of MC. Provided that one analyzes the effect of changing weightings of a portfolio on VaR, it may cost enormous number of simulations and may be impossible. To summarize, VC has a fault of underestimating risk, while MC spends much computational resources and has difficulty in the analytical aspect. Hence a method managing both the characteristics of markets such as fat-tailness and the theoretical analysis is needed. In this paper, we propose a risk evaluating method which preserves the tractable framework of VC method and takes the characteristics of markets into consideration, by the use of Tsallis statistics in statistical physics and its dynamical foundation model, the Beck model. The Tsallis statistics is a generalization of ordinary Boltzmann–Gibbs statistics and is introduced by Tsallis with the motivation of providing a framework of statistical mechanics for farfrom-equilibrium systems which often exhibit scaling properties [5]. It is successfully applied to various areas such as fully developed turbulence [6,7], pure-electron plasma [8], cosmic rays [9], economics [10], option pricing [11], and others [12]. The Beck model is introduced by Beck, as a dynamical foundation of Tsallis statistics [13,14]. Although it is originally intended to describe mechanical systems such as turbulence, its basic idea of fluctuating temperature is well-consistent with the heteroskedasticity of markets. Thus, in recent studies, it is employed to elucidate price fluctuations in markets [15,16]. In Section 2, we explain the framework of Tsallis statistics and the Beck model to the extent of requirement of our applications. In Section 3, we perform empirical analysis of daily and high frequency data of S&P500 index to investigate the applicability of the Beck model. In Sections 4 and 5, we deal with two applications of Tsallis statistics and the Beck model: A VaR evaluation method and portfolio risk assessment. In Section 6, we discuss a few points about our applications and Section 7 is devoted to conclusion. 2. Theoretical preparations 2.1. Tsallis statistics Tsallis statistics or nonextensive statistical mechanics is a generalization of ordinary Boltzmann–Gibbs statistical mechanics to describe statistical behavior of complex systems. Here we briefly introduce a generalized canonical distribution and its typical realization, q-Gaussian distribution [17]. Tsallis introduced a generalized entropy functional [5]   Z 1 Sq [ p] = k 1 − p(x)q dx , q > 1, (1) q −1 where q is a system dependent parameter and k a positive constant. The parameter q is often called Tsallis’ q-index. In the limit of q → 1, this functional converges into the ordinary Boltzmann–Gibbs entropy Z S[ p] = k p(x) ln p(x)dx. (2)

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Let us consider the maximization of Sq with conditions Z p(x)dx = 1, Z P (q) (x)U (x)dx = Uq ,

1227

(3) (4)

R where P (q) (x) = p(x)q / p(x)q dx is called escort probabilities [18], U (x) is a given potential and Uq a given internal energy. In the case of parabolic potential U (x) = x 2 discussed below, Uq becomes a generalized variance σq2 [19]. σq2 is equivalent to “temperature” T0 , introduced later in Section 2.2, which corresponds to volatility or the intensity of market fluctuations. One obtains the following probability distribution function by applying the Lagrange multiplier method and the variational principle to Eq. (1) under constraints Eqs. (3) and (4), p(x) =

1

n

˜ Z q (β)

o− 1 ˜ − 1)U (x) q−1 , 1 + β(q

(5)

where Z q is called a generalized partition function, β˜ denotes the inverse temperature 1/T˜ of the system under consideration. This is a generalized canonical distribution in terms of Tsallis statistics. Particularly, in the case of U (x) = x 2 , the canonical distribution becomes p(x) =

1 ˜ Z q (β)

n

˜ − 1)x 2 1 + β(q

 − 1 0 ˜ = β(q ˜ − 1) 2 Z q (β)

o−

1 q−1

,

  1 2

  1 0 q−1 − 12   , 1 0 q−1

(6)

where 0(x) is a Gamma function defined as Z ∞ 0(x) = e−t t x−1 dt.

(7)

(8)

0

This specific canonical distribution is called a q-Gaussian distribution, as it converges into a Gaussian distribution in the limit of q → 1. 2.2. Beck model The Beck model was originally introduced by Beck as a model of fully developed turbulent flows [13,14]. It explains the reason why turbulent flows exhibit fat-tailness in its distribution and are well-described by Tsallis statistics, due to fluctuation of noise intensity or temperature. It gives a dynamical foundation of Tsallis statistics. Besides turbulence, the Beck model is successfully applied to financial markets in several studies [15,16]. Below we formalize the Beck model for our purpose of study. Primarily, we introduce the one-variable Beck model and show the appearance of q-Gaussian caused by the fluctuation of temperature or, in financial terms, volatility. Secondarily, we define the Beck model extended to multivariable case. The introduced model has a generalized form of multivariable model originally discussed by Beck [14]. 2.2.1. One variable model Let us consider the following Langevin equation for one variable, u˙ = −γ u + σ L(t),

(9)

where L(t) is a Gaussian white noise with hL(t)i = 0 and hL(t1 )L(t2 )i = 2δ(t1 − t2 ), γ is a friction constant, and σ the strength of L(t), respectively. Beck considered that the case σ fluctuates in time so that the inverse temperature β = γ /σ 2 of the system obeys a stochastic differential equation β˙ = a1 (β, t) + a2 (β, t)W (t),

(10)

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where W (t) is a Gaussian white noise with hW (t)i = 0 and hW (t1 )W (t2 )i = 2δ(t1 − t2 ), which is statistically independent of the fluctuation of u. Rather than specifying the forms of a1 and a2 , we assume the following properties on β fluctuation; (1) the time scale of β fluctuation is longer than that of u that the system can temporarily reach local equilibrium before β significantly varies, (2) β has a stationary distribution function. Upon these assumptions, we can approximately evaluate the distribution function of u in equilibrium. In local equilibrium where β can be taken as constant, u is distributed by r   β β 2 exp − u . (11) p(u|β) = 2π 2 If one observes u in longer time scale in which β fluctuates, the distribution of u unconditioned to β is given by Z p(u) = p(u|β) f (β)dβ. (12) In this paper, we assume the stationary distribution function of β to be the 0-distribution  α   1 α αβ β α−1 exp − , α > 2, β0 > 0. f (β) = 0(α) β0 β0

(13)

The average and the variance of β are given by hβi = β0 ,

D

E β2 β 2 − hβi2 = 0 . α

(14)

In this case, Eq. (12) is calculated as p(u) =

0(α + 12 ) 0(α)0( 12 )



β0 α

1  2

β0 2 1+ u 2α

−α− 1 2

,

(15)

which is the form of q-Gaussian distribution derived from Eq. (5) (U (u) = u 2 ), − 1  ˜ − 1)u 2 q−1 . p(u) ∝ 1 + β(q

(16)

The assumption that inverse temperature β obeys the 0-distribution means that the distribution of temperature T = 1/β is the inverse 0-distribution,   1 αT0 α −(α+1) f T (T ) = (αT0 ) T exp − , (17) 0(α) T where hT i = T0 ,

D

E T 2 − hT i2 =

T02 . α−1

(18)

2.2.2. Multivariable model We generalize the Beck model to the multivariable case [14]. Consider N variables u 1 , . . . , u N . The coupled system of stochastic differential equations of u i is assumed to be u˙ i = −γ

N X

Jij u j + σi L i (t),

i = 1, . . . , N ,

(19)

j=1

where Jij are coupling constants, L i are mutually independent Gaussian white noises with hL i (t)i = 0 and

L i (t1 )L j (t2 ) = 2δij δ(t1 − t2 ), and σi denote the strength of L i (t), respectively. We assume that J = (Jij ) is a symmetric positive definite matrix, i.e. all eigenvalues of J are positive and Jij = Jji for all i, j. In the same way

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as the one-variable case, we consider βi = γ /σi2 (i = 1, . . . , N ) to slowly fluctuate in time. The local equilibrium distribution of u 1 , . . . , u N is given by [20]   1 N 1 (20) p(u|β1 , . . . , β N ) = (2π )− 2 |Σ |− 2 exp − u> Σ −1 u , 2   1 1 D˜ 11 . . . D˜ 1N  λ1 + λ1  λ1 + λ N   > . .  V , . . Σ = 2V  (21) . .    1 1 D˜ N 1 . . . D˜ N N λ N + λ1 λN + λN where λ1 , . . . , λ N are eigenvalues of J , V is the corresponding eigenvector matrix:   λ1   .. J = V ΛV > , Λ= , . λN and D˜ indicates 

σ12 γ   > D˜ = V   

(22)

 ..

.

 −1  β1   > V = V   2 σN  γ

 ..

. β N−1

  V.

(23)

If we describe the stationary distribution of βi as f (β1 , . . . , β N ), we can evaluate the marginal distribution of u by Z Z p(u) = · · · p(u|β1 , . . . , β N ) f (β1 , . . . , β N )dβ1 . . . dβ N . (24) The specific form of Eq. (24) in our applications are introduced in Section 5. 3. Data analysis We performed empirical data analysis in order to check the adequacy of applying the Beck model to stock markets. We analyzed log returns of S&P500 index in various time scales from 5 min to a day. The intraday data of S&P500 index cover the periods from Aug. 1997 to Dec. 2006 which contain 182,760 data points with 5 min resolution and the daily closing prices of S&P500 index cover the periods of Jan. 1950 to Sept. 2006 which consist of 14,263 data points. The intraday data are offered by RC Research [21] and the daily data are obtained at Yahoo! finance [22]. In the following, before explaining our results, we refer to parameter estimation methods that we employed in the analysis. Then we verify whether β, which corresponds to the inverse of the volatility, satisfies the Beck model assumption. Finally, we present the agreement of returns distributions with q-Gaussian, as expected by the Beck model. 3.1. Parameter estimation methods Here we explain parameter estimation methods used in the following data analysis. In addition to simple estimation methods such as method of moments, we exploit Maximum Likelihood Estimation (MLE) and Generalized Method of Moments (GMM), especially to estimate parameters of a q-Gaussian distribution. 3.1.1. Maximum Likelihood Estimation Maximum Likelihood Estimation (MLE), one of the popular methods to estimate parameters of a distribution, suggests that the most efficient estimator of parameters is the one which maximizes the (log-)likelihood function of a distribution.

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Let X 1 , . . . , X N be independently and identically distributed random variables from p(x; θ), where θ = (θ1 , . . . , θ K ) is a set of parameters identifying a shape of the distribution. The joint distribution of them is given by p(x1 , . . . , x N ; θ) =

N Y

p(xi ; θ).

(25)

i=1

The log-likelihood function parameterized by θ is the logarithm of Eq. (25), ln L(θ; x1 , . . . , x N ) =

N X

ln p(xi ; θ).

(26)

i=1

To maximize the function, parameters should satisfy ∂ ln L(θ; x1 , . . . , x N ) = 0, ∂θk

k = 1, . . . , K .

(27)

The solutions of these equations give the estimator by MLE. In our concrete case, as the probability distribution function of a q-Gaussian distribution is given by o− 1 1 n q−1 1 + β(q − 1)u 2 , Z q (β)     1 1 1 0 0 − 1 2 q−1 2   Z q (β) = (β(q − 1))− 2 , 1 0 q−1 p(u; q, β) =

(28)

(29)

the joint distribution of samples can be written as p(u 1 , . . . , u N ; q, β) =

N Y i=1

o− 1 1 n q−1 1 + β(q − 1)u i2 . Z q (β)

(30)

Taking the logarithm of Eq. (30) yields the log-likelihood function ln L(q, β; u 1 , . . . , u N ) = −N ln Z (q, β) −

N o n 1 X ln 1 + β(q − 1)u i2 . q − 1 i=1

(31)

For the sake of convenience, we introduce a transformation of parameters α = 1/(q − 1), λ = β(q − 1) and estimate them instead of q, β. Then, the log-likelihood function changes into ln L(q, β; u 1 , . . . , u N ) = N F(α, λ) − α

N X

  ln 1 + λu i2 ,

(32)

i=1

where F(α, λ) =

    1 1 1 ln λ + ln 0(α) − ln 0 α − − ln 0 2 2 2

(33)

is the formal free energy function of q-Gaussian. The equations to be solved for MLE are N   ∂ ∂F X ln L(α, λ) = N − ln 1 + λu i2 = 0, ∂α ∂α i=1

(34)

N X ∂ ∂F ui ln L(α, λ) = N − 2α = 0, 2 ∂λ ∂λ i=1 1 + λu i

(35)

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where   1 ∂F , = Ψ (α) − Ψ α − ∂α 2

∂F 1 = , ∂λ 2λ

(36)

and Ψ (x) is a polygamma function defined by Ψ (x) = d ln 0(x)/dx. Since the simultaneous equations (34) and (35) cannot be solved analytically, we performed numerical calculation with a quasi-Newton method to obtain an estimator. 3.1.2. Generalized Method of Moments Generalized Method of Moments (GMM) is mainly propagated by Hansen [23] and is widely used in the field of econometrics. While the ordinary method of moments makes use of up to K th moments to estimate K parameters, GMM utilizes higher-order moments than K th and chooses parameters which minimize the cost functions about moments to be satisfied with the estimated parameters. The formulation of GMM is as follows. Define a vector function   X i − hX i iθ   .. h(θ, X i ) =  (37) .   . X ir − X ir θ If N samples x1 , . . . , x N are given, we can compose a sample average of h, g N (θ) =

N 1 X h(θ, xi ). N i=1

(38)

Obviously, it satisfies hg N (θ0 )i = 0

(39)

at the true parameters θ 0 . The estimator of GMM is a set of parameters which minimizes a cost function J (θ) = g N (θ)> W g N (θ),

(40)

where W is a certain weighting matrix. The choice of W is rather arbitrary within a condition of positive definiteness, so we have chosen W = I through our analysis (another, superior choice of W is shown, e.g., in Ref. [23]). To apply GMM to q-Gaussian distributions, in order to avoid vanishing of odd moments, we slightly modified GMM: We do not use moments but absolute moments. So we consider   |X i | − h|X i |iθ   .. h(θ, X i ) =  (41)  .

 r r |X i | − |X i | θ instead of Eq. (37). The absolute moments of a q-Gaussian distribution can be evaluated as     1 0 n+1 0 q−1 − n+1

n n 2 2  , n = 1, 2, . . . .    |u| = (β(q − 1))− 2 1 0 21 0 q−1 − 21

(42)

By performing parameter estimations of q-Gaussian in practice, it is revealed that MLE is likely to overestimate the value of q, so that the variance of an estimated distribution diverges, which is unrealistic. Therefore, in the following data analysis, we estimate parameters from data with GMM except when evaluation of a likelihood function is needed. 3.2. Statistical properties of volatility We investigated whether the assumptions of the Beck model on β, namely slow fluctuation and 0-distribution, agree with the actual market data. Primarily, we examined the time scale of β fluctuation, or relaxation time. Note that, in the following, we treat “temperature” T = 1/β instead of β and regard the relaxation time of T as an alternative to that of β.

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Fig. 1. The autocorrelation coefficients of u 2 and T .

As β, or equally T , is an implicit variable, we cannot calculate its relaxation time directly. Thus we make use of the following relation between the autocorrelation of squared return u 2 and that of T : ρ (2) (τ ) = e−2γ τ +

1 hT (0)T (τ )i  (1 − e−2γ τ ),

3 T (0)2

(43)

where γ denotes the relaxation constant of the return. The derivation of this relation is shown in Appendix A. Since ρ (2) (τ ) and γ can be numerically calculated with actual time series, we can estimate the autocorrelation coefficient of T (t) as  3 ρˆ (2) (τ ) − e−2γ τ hT (0)T (τ )i = ρT (τ ) := , (44) hT (0)i 1 − e−2γ τ where ρˆ (2) (τ ) denotes experimentally calculated ρ (2) (τ ). Fig. 1 shows the autocorrelation coefficients of u 2 and T obtained as above. We determined γ = 100, because an observation of the autocorrelation of u revealed u to be almost δ-correlated, namely, γ can be seen to be sufficiently large. Note that the tails of ρT (t) in Fig. 1 do not asymptotically decline to zero because the mean of T is obviously greater than zero. In order to be consistent with the ordinary definition of the autocorrelation, we have to reduce u 2 and T by its mean value as ρ¯

(2)



2 u(0)2 u(τ )2 − u(0)2 (τ ) = 2 ,



u(0)4 − u(0)2

ρ¯T (τ ) =

hT (0)T (τ )i − hT (0)i2 . 

T (0)2 − hT (0)i2

(45)

(46)

The relation between reduced autocorrelation functions is given by ρ¯ (2) (τ ) = e−2γ τ + C T ρ¯T (τ )(1 − e−2γ τ ), where a coefficient C T is defined as

2 T − hT i2 CT =  . 3 T 2 − hT i2

(47)

(48)

Fig. 2 shows ρ¯T (τ ) estimated by the use of Eq. (47). One can find that the autocorrelation ρ¯T (τ ) declines to 0.2 within 20–60 days. It suggests that the relaxation time of T is around the order of months (one month ≈ 20 business days) and that separation of time constants is possible as the Beck model assumed. In addition, Fig. 2 indicates that the autocorrelation of T , that is a representative of volatility, exhibits power-law decay at the tails, which is reported as a long time tail or a long time memory in several studies [24–26].

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Fig. 2. The mean-reduced autocorrelation coefficient of T . A power-law decay τ −α with α = 0.336 is drawn with a dashed line for reference (R 2 = 0.486, χ 2 = 3.92 × 10−3 ).

Fig. 3. The empirical distribution f T (T ) of S&P500 returns with window length L = 40. Dashed line indicates the fitted inverse 0-distribution with α = 1.27, T0 = 0.420 (R 2 = 0.0105, χ 2 = 6.19 × 10−4 ). Inset: linear–linear representation of the same data.

Secondarily, as the time scale of T fluctuation is obtained, we checked the distribution of T . We separated whole time series of returns into terms of length L and calculated temporal T in each subseries as Ti =

1 L

(i+1)L X

¯ 2, (u(k) − u)

(49)

k=i L+1

where u¯ is the mean of returns. L is selected in the range of 20–60 days based on the order of the relaxation time of T. As shown in Fig. 3, the empirical distribution of Ti with window length L = 40 is well-fitted by inverse 0distribution. Note that it exhibits power-law decay at the tail as inverse 0-distribution declines with f T (T ) ∝ T −(α+1) , for T  1. 3.3. Empirical distributions of returns In Section 3.2, we demonstrated the agreement of properties of empirical volatility of S&P500 with the assumptions of the Beck model. This result means that the stock market returns are expected to obey q-Gaussian distributions. In order to verify this expectation, we calculated empirical distributions of returns and estimated q-Gaussian parameters ˜ As shown in Fig. 4, the return distributions of S&P500 exhibit fat-tailness and are well-fitted by q-Gaussian in q, β. the broad time scale of minutes to day. If the Beck model is suitable for stock markets, the following is also expected; if we observe a return distribution within a term of various lengths N , we will find that in small N the distribution is well-fitted by Gaussian while it is better fitted by q-Gaussian as N increases.

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Fig. 4. The empirical distributions of S&P500 returns. The frequency of data is 5, 10, 30 min and 1 day from bottom to top. Each set of points is vertically shifted for clarity of presentation and is plotted with the estimated q-Gaussian distribution.

Fig. 5. The values of AIC scaled by observing term length N for Tsallis canonical distribution (q-Gaussian) and Gaussian fitted to S&P500 returns.

The fitness of a distribution to given data can be measured by Akaike Information Criterion (AIC) Refs. [27,28]. AIC is defined as ˆ + 2k, AIC = −2 ln L(θ)

(50)

where L is the likelihood function of the evaluating distribution function, θˆ is its maximum likelihood estimator and k denotes the number of parameters. The smaller value of AIC implies the better fitness of the distribution considered for the given data. We evaluated the values of AIC for the q-Gaussian and the ordinary Gaussian distribution applied to returns of S&P500. We divided the data into nonoverlapping terms of length N and calculated Eq. (50) for each term. Averaging them over the entire data gives the mean value of AIC for a certain N . We repeated this procedure with different values of N . As shown in Fig. 5, it is confirmed that q-Gaussian exceeds Gaussian when N is longer than 40 days, which supports the validity of applying the Beck model and is consistent with the time scale of β fluctuation. 3.4. Relations between parameters By comparing Eq. (15) with Eq. (16), the relations 1 1 =α+ q −1 2

⇔

q =1+

β0 2α

⇔

T0 =

(q − 1)β˜ =

1 , α + 1/2

T˜ , 3−q

(51) (52)

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Fig. 6. The relations between parameters of distributions of returns and β; (a) the relation of β0 and q, β˜ described by Eq. (52), (b) the relation of α and q described by Eq. (51).

are obtained between the distribution of returns and that of “inverse temperature” β observed over sufficiently long time. In order to investigate whether the relations above are satisfied in actual markets, we performed the following examination. Similar to the examination on the T -distribution, we separated the data into nonoverlapping terms of length N = k L, where L is chosen from among the appropriate time scale of the β fluctuation, e.g., L = 40 and N sufficiently large to regard returns as q-Gaussian distributed, e.g., N = 480. For each separated term, we estimated the parameters q, β˜ of a returns distribution and created k samples of β out of k L returns by which we determined α, β0 of a β-distribution. Here we plotted the obtained couples of the distribution parameters in Fig. 6 along Eqs. (51) and (52). It seems that both the relations are almost satisfied. 4. Application I: Value-at-Risk 4.1. Theoretical definition of VaR In Section 1, we intuitively explained the basic idea of Value-at-Risk (VaR). Before we analyze VaR under the Beck model, we introduce the theoretical definition of VaR [2]. Let y be a return of an asset for periods of t days and µ be the corresponding expected return. Suppose that u = y −µ is distributed by p(u). Then, the VaR with a confidence level c for t periods is defined as the value satisfying Z −VaR p(u)du = 1 − c. (53) P(u < −VaR) = −∞

The confidence level c is generally chosen from the range of 95%–99.5%. The length of periods t is typically a day to several days for a monitoring purpose and is a quarter to one year for a management purpose. In case of the variance–covariance method, where u is assumed to be Gaussian distributed, the VaR is simply expressed as VaR = σt z(c),

(54)

where σt is the standard deviation of u in t days and z(c) is a 100c percentile of standard Gaussian distribution which can be found in elementary textbooks of statistics. Therefore, to evaluate VaR in this case, just estimate the standard deviation σt by historical samples of u, and multiplying z(c) immediately yields the answer. On the other hand, in the case of the Beck model, or whatever model apart from Gaussian distributions, one has to calculate Eq. (53) directly to evaluate VaR in general. However, we developed a VaR evaluation method for the Beck model which adjusts VaR of Gaussian distributions, Eq. (53), to q-Gaussian. 4.2. Contribution of tail risk to VaR As far as returns are sampled from a Gaussian distribution, there is a direct proportion between volatility and VaR. However, as actual market returns obey a fatter-tailed distribution than Gaussian, the downside risk might increase

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Fig. 7. The VaR as a function of q for confidence levels c = 0.95, 0.99, 0.995. The variance is scaled to 1. Table 1 The approximated VaR by Eq. (56) c 95.0% 99.0% 99.5%

VaR1.0

R(v)

1.65 2.33 2.58

−0.03 0.30 0.45

VaRq 1.1

1.2

1.3

1.64 2.40 2.70

1.64 2.47 2.82

1.63 2.54 2.93

compared to Eq. (54) despite the same variance. If we represent such increments of VaR as tail risk, soon the problem of how to quantify the amount of tail risk arises. As the answer to this problem, we propose q-index of Tsallis statistics which inevitably appears in the Beck model as an indicator of tail risk, and quantitatively evaluate the effect of tail risk represented by q on VaR. Although the definite VaR for a q-Gaussian distribution can be numerically calculated by the direct evaluation of Eq. (53), to enable the intuitive analysis, we developed a convenient method to approximately evaluate the VaR: Based on the value of VaR at q = 1, namely, in case of Gaussian distribution, we performed the linear approximation of VaR as a function of q. We demonstrate this method in Fig. 7. There numerically calculated VaRs of q-Gaussian with unit variance, for various confidence levels as a function of q and corresponding tangents at q = 1 is shown. It is seen that, for usual confidence levels and in the neighborhood of q = 1, the tangents give a good approximation to exact VaR. We obtained exact form of a slope of a tangent at q = 1 as   3 v3 dv = − v , (55) dq q=1 8 3 the proof of which is shown in Appendix B. By Eq. (55), in the case of q & 1, one can approximately evaluate the VaR with tail risk as VaRq = VaR1.0 (1 + R(VaR1.0 )1q) ,

(56)

v2

where 1q = q − 1 and R(v) = 38 ( 3 − 1) is a coefficient of VaR increments per tail risk. We reported increments of VaR obtained by the linear approximation Eq. (56) in Table 1. The VaR increasing ratio to tail risk is, for example, 3% per 0.1 increment of q for the confidence level 99%. 4.3. Performance of VaR with tail risk When tail risk is taken into consideration, one has to estimate the amount of tail risk, i.e., q-index in addition to the variance. Hence it would be worth to evaluate the influence of the error accompanied with a q-estimation to VaR. To this end, we utilize the following facts [29]. Suppose that θˆ is an unbiased estimator of parameters θ of a

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ˆ i.e., the covariance matrix of θˆ is evaluated by distribution p(x; θ) with N samples. Then, the estimation error of θ, the Cram´er–Rao inequality, h i G(θ)−1 , (57) V θˆ ≥ N where the inequality is in the sense of positive semidefiniteness, G(θ) denotes the Fisher information matrix of θ. Additionally, if θˆ is the most likelihood estimator, it is asymptotically distributed by N (θ 0 , G −1 /N ) for large N , where θ 0 is a set of true parameters. The Fisher information matrix of a q-Gaussian distribution for the parameters α and λ is obtained in the following. In general, the Fisher information matrix of p is defined as   ∂ ln p ∂ ln p G ij = . (58) ∂θi ∂θ j Applying an identity h∂ ln p/∂θi i = 0, one obtains an equivalent expression of Eq. (58),  2  ∂ ln p G ij = − . ∂θi ∂θ j

(59)

Moreover, let us describe the distribution function as p(x; θ) = e−Φ (x;θ )+F(θ ) . Then it is convenient to calculate the Fisher information matrix by  2  ∂ Φ ∂2 F − G ij = . ∂θi ∂θ j ∂θi ∂θ j

(60)

(61)

Consider the expression of p with a different set of parameters η. In this case, by the chain rule of partial differential, both the Fisher information matrices for η and θ have a conversion rule G 0kl =

X

G ij

i, j

∂θi ∂θ j , ∂ηk ∂ηl

(62)

or in the matrix expression, G 0 = J > G J,

(63)

where J denotes the Jacobian matrix, Jij = ∂θi /∂η j . If one describes the q-Gaussian distribution with the expression of Eq. (60), 1 ln λ + ln 0(α) − ln 0 (α − 1/2) − ln 0 (1/2) , 2   Φ(x; α, λ) = α ln 1 + λx 2 . F(α, λ) =

(64) (65)

Hence the Fisher information matrix for the parameters (α, λ) is evaluated as     1 1 0 0 Ψ α − − Ψ (α)   2 2αλ  2   (66) G(α, λ) =   . 1 3 1 2− 2αλ 2λ α+1 q  Now we consider a parameter transformation σ = u 2 , δ = σ/α. The Fisher information matrix for the transformed parameters is given by G(δ, σ ) = J > G(α, λ)J,

(67)

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Fig. 8. The 99% VaR estimators with error bounds. The error bars denote 95% confidence intervals for 1000 samples. The estimator error bounds denoted by “Gauss” and “Tsallis” are numerically calculated ones, while the other is the theoretical error bounds obtained by Eq. (68).

where J = ∂(α, λ)/∂(δ, σ ) is the Jacobian matrix. Then, by Eq. (56), the variance of the VaR estimation error is evaluated by h i h i     d q = z c2 V σˆ + 2z c2 R(z c )Cov σˆ , δˆ + z c2 R(z c )2 V δˆ V VaR (68) ˆ σˆ ) ∼ N ((δ0 , σ0 ), G −1 /N ). with asymptotic distribution of estimator (δ, In Fig. 8, we compared 95% confidence intervals of VaR estimators for a q-Gaussian distribution evaluated by Eq. (68) with error bounds numerically calculated by Monte Carlo simulation. A comparison between the estimation with Gaussian assumption and the estimation with q-Gaussian is also made in Fig. 8, which shows us considerable underestimation of the Gaussian assumption. 5. Application II: Portfolio risk assessment 5.1. Framework of the variance–covariance method First, we explain a risk assessment framework of a portfolio based on the variance–covariance method before we introduce an extended framework by the Beck model. Let y1 , . . . , y N be returns of N assets and x1 , . . . , x N be their component weightings. The return of a portfolio is given by w=

N X

N X

xi yi ,

i=1

xi = 1.

(69)

i=1

Consider y = (y1 , . . . , y N )> obeying a multivariate Gaussian distribution   N 1 1 p(y) = (2π )− 2 |C|− 2 exp − (y − µ)> C −1 (y − µ) , 2

(70)

where µ denotes the expected return of y. The covariance of y is given by

 Cij = (yi − µi )(y j − µ j ) .

(71)

Since each return of assets is Gaussian distributed and the variance of w is given by D E X w 2 − hwi2 = xi Cij x j ,

(72)

i, j

the distribution of portfolio return w can be written as r   (w − µw )2 1 exp − , p(w) = 2π x> Cx 2x> Cx

(73)

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where µw = x> µ. Therefore, the risk of portfolio in this case is expressed by component weightings and covariance matrix of asset returns. For example, the optimal problem for the asset allocation is described by minimize x> Cx subject to x> µ = µw , N X xi = 1, i=1

xi > 0,

i = 1, . . . , N .

The VaR of a portfolio, given a set of weightings, is evaluated by Z −VaR P(w < −VaR) = P(w)dw = 1 − c.

(74)

−∞

5.2. Generalized framework with the Beck model Afterwards we generalize the above portfolio risk assessment framework by use of the Beck model. Consider the multivariable Beck model Eq. (19) where we identify u i = yi − µi . For the simplicity of theoretical treatment, we consider the case β1 ≡ · · · ≡ β N ≡ β, namely, all assets share the single fluctuating volatility (indicated as the single β model in the following). In this case, the joint distribution of asset returns under a fixed β becomes   1 β p(u|β) = exp − u> J u , (75) Z (β) 2  N 1 β 2 Z (β) = |J | 2 , (76) 2π which is identical with that of the variance–covariance method, Eq. (70). The covariance of u is obtained as Cij (β) =

(J −1 )ij , β

(77)

where we explicitly describe the dependency of Cij (β) on β. If we take the fluctuation of β into consideration, the distribution of u is evaluated as   Z ∞ β f (β) exp − u> J u dβ, (78) p(u) = Z (β) 2 0 which is calculated in a similar way as the one-variable Beck model and turns into  −α− N 2 1 β0 > p(u) = 1+ u Ju . Z (β0 ) 2α

(79)

The covariance of u is obtained by integrating Eq. (77) with β-distribution as Cij =

α (J −1 )ij . α − 1 β0

(80)

In the same way as the variance–covariance method, one can evaluate the distribution of portfolio return w. The variance of w under a fixed β is given by D

w2

E β

− hwi2β =

x> J −1 x , β

the distribution of w is then evaluated by

(81)

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Fig. 9. The return distribution of the portfolio P1. The line indicates the distribution evaluated by the single β model. ∞

Z p(w) = 0

  f (β) β (w − µw )2 dβ, exp − Z (β) 2 x> J −1 x

(82)

which is calculated in the same way as Eq. (79):  −α− 12 1 β0 (w − µw )2 p(w) = 1+ . Z (β0 ) 2α x> J −1 x

(83)

5.3. Empirical results In order to confirm the effectiveness of the single β model, we experimented with imaginary portfolios composed of stock market indices, whether their distributions and VaR can be well-expressed by the single β model. We examined Dow Jones Industrial, Transportation and Utility Averages for the periods from Apr. 1997 to Dec. 2006. We regarded them as certain assets and considered portfolios with various component weightings. At first, on the assumption of all indices sharing the single volatility fluctuation, we estimate the parameter α corresponding to the intensity of fluctuations. To this end, one can perform MLE about Eq. (79) with return time series of the indices, but the degree of freedom for parameters is so large that the estimation turns out to be inefficient. Therefore, we utilize the covariance matrix C which can be directly calculated from the data. Substituting C into J −1 /β0 in Eq. (83) yields 1 p(u) = Z



−α− N 2 1 > −1 u C u 1+ . 2(α − 1)

(84)

Then, we calculate the empirical covariance matrix Cˆ and apply MLE in order to estimate the parameter α. If one obtains Cˆ and α, ˆ by substituting them into Eq. (83), the distribution of a portfolio with arbitrary weightings can be evaluated. For example, a comparison between the empirical distribution of a portfolio with component weightings x = (1/3, 1/3, 1/3)> (named P1) and the distribution evaluated by Eq. (83) is shown in Fig. 9. VaR of a portfolio is also evaluated by Eq. (56) with qˆ = 1 + 1/(αˆ + 1/2). In Fig. 10, we present the result of testing VaR estimation with the period of 1 day by the variance–covariance method and the single β model. The performance of prediction is as follows: VC suffered violations 24 times out of 2045 predictions (violation rate is 1.174%) while Beck suffered 19 times (0.929%). The result indicates that the single β model appropriately estimates VaR which the variance–covariance method underestimates by the amount of tail risk. 6. Discussion 6.1. Role of the Beck model in VaR evaluation methods In Section 4, we considered an application of the Beck model and q-Gaussian distributions to VaR evaluation. Here we discuss where the method of the Beck model is placed among established VaR evaluating methods.

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Fig. 10. The experiment of VaR prediction for variance–covariance (VC) method and the single β model (Beck) with the period of 1 day. For each method, VaR with the confidence level 99% for the following day has been estimated based on recent 250 samples.

The derivation of q-Gaussian from the Beck model revealed that fluctuations of volatility produces the fat-tailness of return distributions. In Section 4, we exhibited the insufficiency of the variance–covariance method, that is an underestimation of downside risk due to the existence of tail risk, and showed the measurement of the effect of tail risk quantified by q on VaR. If one vanishes a volatility fluctuation in the Beck model, namely taking the limit of q → 1, the q-Gaussian distribution converges into Gaussian. This obviously agree with the scheme of the variance–covariance method. Accordingly, the Beck model can be seen as a generalization of the variance–covariance method with tail risk. Moreover, the Beck model approach can be taken as an approximation theory for Stochastic Volatility (SV) models employed in the Monte Carlo method. In order to explain this point, we consider the discretization of the Beck model. Let u t be the daily return on t = 1, 2, . . . . In the Beck model, as γ −1 is a much smaller time scale than 1 day, u t obeys p u t = γ /βt εt , (85) βt = a1 (βt−1 ) + a2 (βt−1 )ηt , (86) where εt and ηt are independently and identically distributed standard Gaussian noises. This is an analogy of a typical SV model [30] u t = σ t εt ,

(87)

σt = b1 (σt−1 ) + b2 (σt−1 )ηt .

(88)

Therefore, as long as the time scale of volatility fluctuations is much longer than that of returns, one can apply the Beck model approach to SV models alike. Consequently, the distribution of an SV model in long terms and the correlation structure in time such as T -autocorrelation discussed in Section 2 can be evaluated. Thus the Beck model approach is useful to design or select an SV model for market returns. 6.2. Applicability of the single β model According to the empirical results in Section 5, the single β model, which suppose assets to be governed by a single volatility fluctuation, turned out to be effective in portfolio risk assessment. In order to investigate the applicability of the single β model, we examined the volatility correlations between Dow Jones indices adopted in Section 5. When the single β model is appropriate for portfolios composed of the indices, they are expected to be highly correlated in their volatilities. On the contrary, volatility correlations actually calculated with index returns, shown in Table 2 exhibit that the degree of correlations is not excessive. Nevertheless, as shown in Section 5, the single β model has good consistency with a portfolio of Dow Jones indices. This result implies the robustness of the single β model against disagreement between the volatility fluctuations of assets. Namely, the model can be applied to such assets that do not possess highly correlated volatilities. Moreover, the single β model is regarded as a generalization of the variance–covariance method, which enables the same risk factor analysis with covariance matrix, while the fat-tailness, omitted in variance–covariance method, is considered.

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Table 2 The volatility correlations of Dow Jones indices

INDU TRANS UTIL

INDU

TRANS

UTIL

1.000

0.571 1.000

0.330 0.285 1.000

7. Conclusion In this paper, we applied the Beck model, which has been developed for turbulent flows, to stock markets. Since the Beck model is a model with fluctuating temperature, or volatility in finance, and thus consistent with heteroskedasticity observed in financial markets, the application of the Beck model to markets seems worthwhile. We investigated the adequacy of representing stock markets as the Beck model with data of S&P500 index, from the viewpoint of a relaxation time, distributions of volatility, and distributions of returns. As the result of empirical analysis, we confirmed that the time constant of volatility is of the order of months and that the inverse temperature β approximately obeys 0-distribution, which supports the assumption of the Beck model. Also, returns obey q-Gaussian distribution in broad time scales, as expected by the Beck model. We then explained the specific applications of the Beck model, namely, the evaluation method of VaR and the portfolio risk assessment. The VaR evaluation scheme of the variance–covariance method was generalized by considering the tail risk represented by Tsallis’ q-index, where we quantified the contribution of tail risk to VaR by its linear approximation. The framework of portfolio risk assessment was also generalized with the multivariable Beck model. We introduced the single β model which assumes that all assets under consideration share a common volatility fluctuation and examined the effectiveness of the model by an imaginary portfolio composed of Dow Jones indices. The Beck model approach to financial markets has several advantages: (1) it considers the fat-tailness of return distributions, which enables improvement in the accuracy of predictions compared to the Gaussian-based method, (2) the Beck model also allows theoretical treatments similar to the variance–covariance method and thus coincides with no computational difficulties in contrast to the Monte Carlo methods, and (3) the analytical approach used in the Beck model can similarly be applied to the SV models widely adopted in the Monte Carlo methods, which might provide a step to evaluate stock market models from the viewpoint of the stationary distribution and the correlation function. The Beck model approach which we employed in this paper is a first approximation for the non-Gaussian property of market returns. For further development of the Beck model approach, it is required to consider the characteristics of time evolution of market returns, which are not sufficiently exploited except autocorrelations in the present paper. For example, we have to construct a bridge between the Beck model and several studies on time evolution of returns from the view point of Markov properties [31,32]. Especially, treating the fact that market returns exhibit parabolic form for the second-order Kramers–Moyal moment [33,19] is an interesting problem, leading to a progress in risk estimation methods. Acknowledgements The authors express their sincere gratitude to Professor T. Munakata for giving helpful information and advice. The authors appreciate the valuable comments and suggestions by the Associate Professor A. Igarashi. This work was partially supported by “Initiatives for Attractive Education in Graduate Schools” from the MEXT (Ministry of Education, Culture, Sports, Science and Technology) of Japan. Appendix A. Autocorrelation functions of the Beck model In the Beck model, the time scale of β fluctuation is supposed to be far longer than the typical time scale of order γ −1 . Hence we can obtain the transition probability before arrival at local equilibrium, s   (u − u 0 e−γ τ ) 1 exp − , (A.1) p(u, t + τ ; u 0 , t) = 2πat (τ ) 2at (τ )

M. Kozaki, A.-H. Sato / Physica A 387 (2008) 1225–1246

at (τ ) =

σ (t + τ )2 (1 − e−2γ τ ), γ

1243

(A.2)

where we incorporate time dependency of σ at longer time scale of fluctuating temperature. The autocorrelation functions of u(t) and u 2 (t) can be obtained as follows. First, we calculate t-dependent autocorrelation functions: e−γ τ , β(t) D E e−2γ τ 1 − e−2γ τ (2) Rt (τ ) = u 2 (t + τ )u 2 (t) = 3 . + β(t)β(t + τ ) β(t)2 Rt (τ ) = hu(t + τ )u(t)i =

(A.3) (A.4)

Next, assuming ergodicity of β fluctuation, we replace time average of autocorrelation functions by ensemble average with respect to β. This procedure yields Z 1 T R(τ ) = lim Rt (τ )dt = hRt (τ )i T →∞ T 0 −γ τ = hT (0)i e , (A.5) Z T E D 1 (2) (2) Rt (τ )dt = Rt (τ ) R (2) (τ ) = lim T →∞ D T E0 = 3 T (0)2 e−2γ τ − hT (0)T (τ )i (1 − e−2γ τ ). (A.6) Here we obtain t-independent autocorrelation functions. Normalizing them gives corresponding autocorrelation coefficients ρ(τ ) =

R(τ ) = e−γ τ , R(0)

ρ (2) (τ ) =

1 hT (0)T (τ )i R (2) (τ )  (1 − e−2γ τ ).

= e−2γ τ + 3 T (0)2 R (2) (0)

(A.7) (A.8)

Appendix B. Linear approximation of VaR at q = 1 Consider a random variable U which obeys q-Gaussian distribution − 1 1  q−1 1 + β(q − 1)u 2 , Z or, in the convenient expression with parameters α and λ, −α 1  p(u; α, λ) = 1 + λu 2 . Z p(u; q, β) =

Instead of directly evaluating the VaR of U , we deal with normalized variable Y = U/

 evaluated by u 2 = (1/λ)(1/2α − 3), the distribution of scaled variable y becomes −α y2 , 1+ 2α − 3   1 1 1 Z = (2α − 3) 2 B α − , , 2 2 1 p(y; α) = Z

(B.1)

(B.2) q  u 2 . As the variance of U is



where B(a, b) is a beta function defined by Z 1 0(a)0(b) B(a, b) = = t a−1 (1 − t)b−1 dt. 0(a + b) 0

(B.3) (B.4)

(B.5)

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The VaR for scaled variable y is given by  −α Z y2 1 −VaR 1+ dy = 1 − c. P(y < −VaR) = Z −∞ 2α − 3 We regard the rhs of Eq. (B.6) as a function of q and v = VaR, namely,  −α Z 1 −VaR y2 f (q, v) = 1+ dy, Z −∞ 2α − 3 with a constraint f (q, v) = ε for a fixed confidence level c = 1 − ε. For an infinitesimal increment of q, the variation of v in order to keep f (q, v) constant is evaluated by   ∂f ∂q ∂f ∂f dq + dv = 0 ⇔ dv = −   dq, df = ∂f ∂q ∂v

(B.6)

(B.7)

(B.8)

∂v

where ∂f ∂ = ∂v ∂v

−v

Z

p(y; α)dy = − p(v; α),

(B.9)

−∞

∂f ∂f α ∂f = = −α 2 . ∂q ∂α dq ∂α

(B.10)

The partial derivative of f with respect to α is calculated as follows. Z −v ∂ ln p(y; α) ∂ ln Z ∂f = p(y; α)dy = −ε − (I1 − I2 ), ∂α ∂α ∂α −∞

(B.11)

where 1 I1 = Z

Z

1 Z

Z

I2 =

−α   y2 y2 1+ ln 1 + dy, 2α − 3 2α − 3  −α−1 2αy 2 y2 1 + dy. 2α − 3 (2α − 3)2

∞ v ∞ v

(B.12) (B.13)

We evaluate Eq. (B.11) for each term. For the first term, we partially differentiate ln Z by α as   ∂ ∂ 1 1 1 1  ln Z = +  B α− , ∂α 2α − 3 2 2 B α − 1 , 1 ∂α 2 2

=

3 3 − + o(α −2 ). 2α(2α − 3) 8α(α + 1)

The first integral of the second term in the brackets I1 is evaluated as   1 1   ∞ 1X 1 k − 2 ··· 2 1 1 I1 = Bc α − , k + , 2 k=1 k (α + k − 1) · · · α 2 2 where Bx is an incomplete beta function Z x 1 Bx (a, b) = t a−1 (1 − t)b−1 dt. B(a, b) 0 Similarly, the second integral of the second term I2 is evaluated as   1 3 1 1 Bc α − , . I2 = 2 2α − 3 2 2

(B.14)

(B.15)

(B.16)

(B.17)

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Hence the second term becomes       1 3 3 1 1 3 1 5 I1 − I2 = − Bc α − , + Bc α − , + ··· . 2 2α(2α − 3) 2 2 8 α(α + 1) 2 2

1245

(B.18)

Now we should evaluate incomplete beta functions. To this end, we make a backward transformation of variable, t to y, and obtain   1 3 1 = vp(v; α) + ε, (B.19) Bc α − , 2 2 2   1 2α 1 5 v3 = Bc α − , p(v; α) + vp(v; α) + ε. (B.20) 2 2 2 2 2α − 3 + v 3 Therefore, Eq. (B.18) turns into I1 − I2 =

v3 3 1 2α 3 2α + 7 {vp(v; α) + ε} + · · · . p(v; α) − 2 8 α(α + 1) 2α − 3 + v 3 8 α(α + 1)(2α − 3)

(B.21)

As a consequence, we can obtain the evaluation of Eq. (B.10) at q = 1, ∂f ∂ ∂f = −α 2 = εα 2 ln Z + α 2 (I1 − I2 ), ∂q ∂α ∂α

α → ∞.

(B.22)

The rhs of Eq. (B.22) can be calculated as 3 3 v3 p(v) − {vp(v) − ε} , 8 3 8 ∂ α→∞ 3 ln Z −−−→ ε, εα 2 ∂α 8 α→∞

α 2 (I1 − I2 ) −−−→

(B.23) (B.24)

where  2 1 v p(v) = p(v; α → ∞) = √ exp − . 2 2π Therefore, taking the limit of q → 1, or equally α → ∞ of Eq. (B.22) leads to   ∂f 3 v3 = p(v) − vp(v) . ∂q 8 3 Finally, the infinitesimal increment of v(q) at q = 1 is given by     ∂f 3 v3 ∂q 8 3 − v p(v) −   =− , ∂f − p(v) ∂v

(B.25)

(B.26)

(B.27)

q=1

namely, Eq. (B.8) turns out to be   3 v3 dv = − v dq. 8 3

(B.28)

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