Nonlinear GARCH model and 1/f noise

A. Kononovicius and J. Ruseckas

Abstract

Auto-regressive conditionally heteroskedastic (ARCH) family models are still used, bypractitioners in business and economic policy making, as a conditional volatility forecast-ing models. Furthermore ARCH models still are attracting an interest of the researchers.In this contribution we consider the well known GARCH(1,1) process and its nonlinearmodifications, reminiscent of NGARCH model. We investigate the possibility to repro-duce power law statistics, probability density function and power spectral density, usingARCH family models. For this purpose we derive stochastic differential equations fromthe GARCH processes in consideration. We find the obtained equations to be similar toa general class of stochastic differential equations known to reproduce power law statis-tics. We show that linear GARCH(1,1) process has power law distribution, but its powerspectral density is Brownian noise-like. However, the nonlinear modifications exhibit bothpower law distribution and power spectral density of the 1/fβ form, including 1/f noise.

1 IntroductionForecasting volatility opens up a possibility to make better-informed decisions. Thus thisproblem is of high interest to the practitioners in business and economic policy making aswell as to the scientific community. The auto-regressive conditionally heteroskedastic (ARCH)model and its generalization, known as GARCH, were proposed exactly for this purpose [1, 2].Since then numerous modifications of the seminal models proposed by Engle and Bollerslevwere introduced to serve varying purposes: from financial market to macroeconomic modeling(see [3] for a long list of the ARCH family models). Our particular interest, in the context ofthis paper, lies in the continuous-time GARCH model (COGARCH, see [4–7]) and nonlinearvariations of GARCH, such as NGARCH [8,9] or MARCH [10].

In order to evaluate the suitability of a given model one should compare the features of thetime series produced by the model with the actual empirical data. It is known that high fre-quency time series of financial data exhibit some universal statistical properties. Vast amountsof historical stock price data around the world have helped to establish a variety of so-calledstylized facts [11–20] corresponding to the statistical signatures of financial processes. One ofthe stylized facts concerns with autocorrelation function, or, equivalently, power spectral den-sity (PSD) of the time series. There is empirical evidence that trading activity, trading volume,and volatility are stochastic variables with the long-range correlation [21–23] leading to 1/ftype PSD. Successful models should reproduce as many stylized facts as possible. However,the 1/f type PSD is not accounted for in some widely used models, such as ARCH familymodels. Therefore, it would be useful to examine under which conditions power law PSD maybe observed in GARCH(1,1) and in nonlinear modifications of GARCH(1,1) models.

Power law statistics and especially 1/f noise are rather ubiquitous phenomena observed inmany different fields of science ranging from natural phenomena to computer networks andfinancial markets [24–29]. Since the discovery of 1/f noise numerous models and theories havebeen proposed, for a recent review see [30]. A class of nonlinear stochastic differential equations

1

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v2 [

q-fi

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] 1

3 Fe

b 20

15

(SDEs) exhibiting power law probability density function (PDF) and power law PSD in a wideregion of frequencies has been derived in [31–34] starting from the point process model. Suchnonlinear SDEs have been used to describe signals in socio-economical systems [35,36].

Usually ARCH family models are calibrated by retro-fitting historical data and thus mayreplicate patterns observed in the past [37]. Yet other approaches are also possible. For example,one may consider comparing them to successful models from other frameworks. In this paperwe compare GARCH(1,1) and its nonlinear modifications to nonlinear stochastic differentialequations generating signals with power law PSD.

This work is organized as follows: in Section 2 we briefly introduce a class of nonlinear SDEsreproducing 1/f noise; in Section 3 we show that power law distributions with varying exponentsmay be obtained from GARCH(1,1) process; in Section 4 we consider a nonlinear modificationsof GARCH(1,1) process using which we reproduce 1/f noise. Section 5 summarizes our work.

2 Stochastic differential equations generating signals with1/f noise

The nonlinear SDEs generating signals with power law steady state PDF and 1/fβ PSD havebeen previously derived in Refs. [31–33]. The general expression for the proposed class of ItôSDEs is given by

dx = σ2

(η − 1

2λ

)x2η−1dt+ σxηdWt . (1)

In the above x is the signal, η 6= 1 is the exponent of a power law multiplicative noise, λdefines the exponent of the power law steady state PDF of the signal, while Wt is the standardWiener process (one dimensional Brownian motion) and σ is a scaling constant determining theintensity of noise. The nonlinear SDE (1) assumes the simplest form of the multiplicative noiseterm, σxηdWt, although it may take other forms as long as η is the largest power of x for largevalues of x [33,35]. Such nonlinear SDEs have been used to describe signals in socio-economicalsystems [35,36]. In Refs. [38,39] a nonlinear SDE similar to the one given by Eq. (1) was derivedby starting from a simple agent-based herding model, thus providing agent-based reasoning forthis class of SDEs.

The steady state PDF of Eq. (1) has a power law form p(x) ∼ x−λ with the exponent λ. Ifλ > 1 then p(x) diverges as x → 0, therefore the diffusion of the stochastic variable x shouldbe restricted from the side of small values. This can be achieved by modifying Eq. (1). Thesimplest restriction of the diffusion is produced by the reflective boundary conditions at theminimum value x = xmin and the maximum value x = xmax. Alternatively, one can modifyEq. (1) to get rapidly decreasing steady state PDF when the stochastic variable x acquiresvalues outside of the interval [xmin, xmax]. For example, the steady state PDF

p(x) ∼ 1

xλexp

(−x

mmin

xm− xm

xmmax

)(2)

with m > 0 has a power law form inside of the interval xmin � x � xmax and exponentialcut-offs are present outside of this interval. Exponentially restricted diffusion is generated bythe SDE

dx = σ2

[η − 1

2λ+

m

2

(xmmin

xm− xm

xmmax

)]x2η−1dt+ σxηdWt (3)

which differs from Eq. (1) only by a couple of additional terms in the drift part of SDE.One can estimate the PSD of the signals generated by the SDE (1) by using the approximate

scaling properties of the signals [40]. The Wiener process scales as dWat = a1/2dWt, thus by

2

10-10

10-8

10-6

10-4

10-2

100

102

100 101 102 103

p(x)

x

(a)

10-5

10-4

10-3

10-2

10-1

100

101

10-1 100 101 102 103 104 105

S(f)

f

(b)

Figure 1: Statistical properties, PDF (a) and PSD (b), obtained by numerically solving SDE (1)(red squares). In the numerical computation we have used the reflective boundaries placed atxmin and xmax. Black curves show power law approximations (a) x−3 for numerically obtainedPDF and (b) 1/f for the PSD. The following parameter values were used: η = 2, λ = 3,xmin = 1, xmax = 103, σ = 1.

changing the variable x in Eq. (1) to a scaled variable xs = ax or by introducing the scaled timets = a2(η−1)t one obtains exactly the same SDE. This feature indicates that the change of thescale of the stochastic variable x and the change of the time scale are statistically equivalent.Using the transition probability (the conditional probability that at time t the signal has valuex′ with the condition that at time t = 0 the signal had the value x) this equivalence may bemathematically expressed as

aP (ax′, t|ax, 0) = P (x′, aµt|x, 0) , (4)

with the exponent µ being equal to 2(η− 1). The discussed scaling property Eq. (4) as well aspower law form of the steady state PDF p(x) ∼ x−λ lead to the PSD with power law behaviorS(f) ∼ f−β, which is observed in a wide range of frequencies. The power law exponent of thePSD is given by [40]

β = 1 +λ− 3

2(η − 1). (5)

The restrictions imposed on diffusion at x = xmin and x = xmax makes the scaling relation-ship Eq. (4) only approximate. This limits the power law part of the PSD to a certain finiterange of frequencies fmin � f � fmax. Note, that power law behavior 1/fβ of the PSD forall frequencies is physically impossible, because the total power of the signal then would beinfinite. Therefore, it is natural to consider signals with the power law PSD in a limited rangeof frequencies. The frequency range for the PSD of the signal generated by solving SDE (4)was estimated in Ref. [40] as

σ2x2(η−1)min � 2πf � σ2x2(η−1)max , η > 1, (6)

σ2x−2(1−η)max � 2πf � σ2x−2(1−η)min , η < 1.

From Eq. (6) it is evident that the width of the frequency range can be increased by increasingthe ratio between the minimum and the maximum diffusion restriction boundary positionsxmax/xmin. In addition, the width also depends on the multiplicative noise exponent η. Namely,the width is zero if η = 1 and increases with increasing |η − 1| [34].

With λ = 3, from Eq. (5) we obtain β = 1 and SDE (1) should generate a signal exhibiting1/f noise. This case with η = 2 is shown in Fig. 1. In numerical computation we haveused a modified Euler-Maruyama approximation (the original Euler-Maruyama approximationis described in [41]) with a variable time step which decreases with the larger values of x,

3

as is described in [31, 32]. We find a good agreement of the numerical results with analyticalpredictions. Though the range of frequencies with S(f) ∼ 1/f is much narrower than expected,1� f � 103 versus 1� f � 106 predicted by Eq. (6). This discrepancy arises because Eq. (6)is only a qualitative estimation. To obtain a more precise values of limiting frequencies oneneeds to use more precise scaling properties of the nonlinear SDE.

3 GARCH(1,1) process and stochastic differential equa-tions

Almost half a century ago Mandelbrot, Fama and others proposed an idea that volatility fluctu-ations might be responsible for the fluctuating nature of price change (return) dynamics [42–44].Actually intermittency of return time series is usually associated with localized burst in volatil-ity and thus frequently referred to as volatility clustering [27, 45–47]. It was proven that mod-eling temporal dynamics of second-order moment, known as heteroskedasticity [1], may allowbetter performing option-price models [48–50].

In his seminal article [1] R. F. Engle laid foundations to the autoregressive conditionalheteroskedasticity (ARCH) models by proposing to split a heteroskedastic variable z (e.g.,return) into a stochastic part ωt and time dependent volatility (standard deviation) σt,

zt = σtωt . (7)

Stochastic part ωt may follow any distribution, but a common choice is the Gaussian distri-bution, though other distributions such as the q-Gaussian distribution might be also consid-ered [51, 52]. In this paper we will use the Gaussian distribution with zero mean 〈ωt〉 = 0and unit variance 〈ω2

t 〉 = 1. As the process modeling the evolution of the standard deviationtime series σt we choose GARCH(1,1) process, which is defined as an iterative equation of thefollowing form:

σ2t = a+ bz2t−1 + cσ2

t−1 . (8)

Using Eq. (7) the GARCH(1,1) process can be written as

σ2t = a+ bσ2

t−1ω2t−1 + cσ2

t−1 . (9)

GARCH(1,1) process can be approximated as a continuous time SDE by considering thediffusion limit of this process [4–7]. Usually the parameters of the GARCH process are obtainedby retro-fitting empirical data and thus the actual values of the parameters a, b and c are tiedto a particular discretization time step. Taking this into account let us rewrite the above as [5]

σ2kh,h = ah + bhσ

2(k−1)h,hω

2(k−1)h,h + chσ

2(k−1)h,h

= σ2(k−1)h,h + ah − (1− bh − ch)σ2

(k−1)h,h + bhσ2(k−1)h,h

(ω2(k−1)h,h − 1

). (10)

Here h is a time series discretization period (t = kh, where k = 1, 2, 3, . . .), while subscriptsindicate that the parameters depend on h. Assuming that h is infinitesimally small, h→ 0 weintroduce the continuous time parameters A, B and C related to the parameters ah, bh and chby the equations [5]

ah = Ah, 1− bh − ch = Ch, 2b2h = B2h . (11)

Note, that the parameters A, B, C do not depend on h. Using Eq. (11) we can rewrite Eq. (10)as

σ2kh,h = σ2

(k−1)h,h +(A− Cσ2

(k−1)h,h)h+ |B|

√h

2σ2(k−1)h,h

(ω2(k−1)h,h − 1

). (12)

4

10-10

10-8

10-6

10-4

10-2

100

102

10-1 100 101 102 103

p(y)

y

(a)

10-4

10-2

100

102

104

10-6 10-5 10-4 10-3 10-2 10-1 100

S(f)

f

(b)

Figure 2: Statistical properties, PDF (a) and PSD (b), of numerically evaluated linearGARCH(1,1) process Eq. (8). The GARCH(1,1) process parameters were set as follows:a = 0.015, b = 0.1, c = 0.89 (red squares), 0.88 (blue circles) and 0.87 (magenta triangles).

The stochastic variable ω2(k−1)h,h−1 we approximate as a normally distributed stochastic variable

with zero mean and variance equal to 2, since 〈ω2 − 1〉 = 0 and 〈(ω2 − 1)2〉 = kω − 1 = 2. Thisgives

σ2kh,h = σ2

(k−1)h,h +(A− Cσ2

(k−1)h,h)h+ |B|σ2

(k−1)h,h

√hεk , (13)

where εk is a normally distributed stochastic variable with zero mean and unit variance. Note,that the last equation has the form of a difference equation, similar to the Euler-Marujamaapproximation used to numerically solve SDEs. Thus by taking the small time step limith→ 0 one can obtain the SDE for the variable ykh,h = σ2

kh,h from Eq. (13):

dy = (A− Cy)dt+ |B|ydWt . (14)

This equation can be written in the form

dy = B2

(1− 1

2λ+

1

2

ymin

y

)ydt+ |B|ydWt , (15)

where

λ = 2 +2C

B2= 2 +

1− bh − chb2h

, ymin =2A

B2=

a

b2h. (16)

The SDE (15) is a special case of the SDE (1) with η = 1. As is discussed in Section 2, steadystate PDF p(y) thus should have power law tail with the exponent λ, p(y) ∼ y−λ. Analyticalprediction Eq. (5) for the power law exponent in the PSD diverges, but it is worth to notethat the SDE (15) is similar to geometric Brownian motion, PSD of which has the power lawexponent β = 2.

The PDF and the PSD of the time series numerically obtained using GARCH(1,1) process,Eq. (8), are shown in Fig. 2. The analytical predictions of the power law exponents in the PDFand the PSD are in good agreement with the numerical results.

4 Nonlinear GARCH(1,1) process generating signals with1/f noise

The mathematical form of Eq. (5) suggests that it is possible to obtain other values of the powerlaw exponent β as long as η 6= 1. In our previous work we have shown that η > 1 cases work

5

very well for the modeling of high-frequency trading activity as well as high-frequency absolutereturns of the financial markets [35, 53, 54], although theoretically η < 1 is also possible [34].One can obtain the η > 1 case by considering the following modifications of GARCH(1,1)process:

σ2t = a+ bσµt−1ω

µt−1 + cσ2

t−1 , (17)

where µ > 2 is an odd integer, and

σ2t = a+ bσµt−1|ωt−1|µ + σ2

t−1 − cσµt−1 , (18)

where µ may be any positive real number. Nonlinear GARCH model of a similar form wasconsidered by Engle and Bollerslev in [8]. Nonlinear model proposed by Engle and Bollerslevhad a form of Eq. (17), but absolute value of ωt−1 was taken prior to raising it to a generalizedpower µ. Engle and Bollerslev found that µ / 2 for most empirical timeseries they haveconsidered. Another take at nonlinear GARCH model can be found in work by Higgins andBera [9], although they considered dynamics not of the standard deviation (as we do), but ofthe higher order moment, σµt . Note, that the last two terms in Eq. (18) can be seen as thefirst two terms in the power series expansion of a more general function of standard deviationf(σt). In contrast to Eq. (8) for the GARCH(1,1) process, Eqs. (17) and (18) do not ensure thepositivity of σ2

t . In order to avoid negative values of σ2t we consider Eqs. (17) and (18) together

with a reflective boundary at σ2t = 0.

Let us first consider the diffusion limit of Eq. (17). We proceed similarly as in Section 3.Taking into account the relation to physical time via time series sampling, this iterative equationmay be rewritten as follows:

σ2kh,h = σ2

(k−1)h,h + ah − (1− ch)σ2(k−1)h,h + bhσ

µ(k−1)h,hω

µ(k−1)h,h . (19)

Assuming that the time discretization period h is infinitesimally small, h→ 0, we can introducethe coefficients A, B and C via the equations

ah = Ah , 1− ch = Ch , 〈ω2µ〉b2h = B2h (20)

and rewrite Eq. (19) as

σ2kh,h = σ2

(k−1)h,h +(A− Cσ2

(k−1)h,h)h+ |B|

√h

〈ω2µ〉σµ(k−1)h,hω

µ(k−1)h,h . (21)

We approximate the stochastic variable ωµ(k−1)h,h as a normal stochastic variable with zero mean(given that µ is odd) and variance 〈ω2µ〉. This gives

σ2kh,h = σ2

(k−1)h,h +(A− Cσ2

(k−1)h,h)h+ |B|

√hσµ(k−1)h,hεk , (22)

where εk is a normally distributed stochastic variable with zero mean and unit variance. Bytaking the small time step limit h → 0, one can obtain the SDE for the variable ykh,h = σ2

kh,h

from Eq. (22):

dy =

(A

yµ−1− C

yµ−2

)yµ−1dt+ |B|y

µ2 dWt . (23)

By introducing the parameters

y(1) =

(2A

(µ− 1)B2

) 1µ−1

, y(2) =

(2C

(µ− 2)B2

) 1µ−2

(24)

6

10-12

10-9

10-6

10-3

100

103

10-2 100 102 104

p(y)

y

(a)

10-2

100

102

104

106

10-8 10-6 10-4 10-2 100

S(f)

f

(b)

Figure 3: Statistical properties, PDF (a) and PSD (b), of numerically evaluated nonlinearGARCH process Eq. (17) with µ = 3 (red squares). Black curves show power law approxima-tions (a) x−3 for numerically obtained PDF and (b) 1/f for the PSD. Other nonlinear GARCHprocess parameters were set as follows: a = 10−6, b = 10−3, c = 1.

Eq. (23) can be rewritten as

dy = B2

[1

2(µ− 1)

(y(1)

y

)µ−1− 1

2(µ− 2)

(y(2)

y

)µ−2]yµ−1dt+ |B|y

µ2 dWt . (25)

From the Fokker-Planck equation corresponding to Eq. (25) one can obtain the steady statePDF p(y) of the signal generated by Eq. (25). The steady state PDF p(y) has a power law tailwith the exponent λ = µ, while y(1) and y(2) shape the exponential cutoff:

p(y) ∼ 1

yµexp

[−(y(1)

y

)µ−1+

(y(2)

y

)µ−2]. (26)

Eq. (25) has the general form of SDE (1) with the parameters λ = µ and η = µ/2. Thus thePSD of yt time series should have a frequency range with the power law behavior of the PSDS(f) given by

S(f) ∼ 1

fβ, β = 1 +

µ− 3

µ− 2. (27)

Note, that we get 1/f PSD when µ = 3.The PDF and the PSD of the time series numerically obtained using Eq. (17) with µ = 3

are shown in Fig. 3. In the numerical calculations we used reflective boundary at σt = 0. Theanalytical predictions of the power law exponents λ = 3 in the PDF and β = 1 in the PSD arein good agreement with the numerical results. With the chosen parameters in Eq. (17) we areable to reproduce 1/f spectrum over almost 4 decades of frequency f , see Fig. 3(b).

Now let us consider the diffusion limit of Eq. (18). Eq. (18) may be rewritten by takinginto account the relation to physical time:

σ2kh,h = σ2

(k−1)h,h + ah − chσµ(k−1)h,h + bhσµ(k−1)h,h|ω(k−1)h,h|µ . (28)

Approximating |ω(k−1)h,h|µ as a normal stochastic variable with the mean ω̄µ = 〈|ω|µ〉 and thevariance ω̂µ = 〈[|ω|µ − ω̄µ]2〉 yields

σ2kh,h = σ2

(k−1)h,h + ah + (bhω̄µ − ch)σµ(k−1)h,h + bhσµ(k−1)h,h

√ω̂µεk . (29)

Here εk is a normally distributed stochastic variable with zero mean and unit variance. Assum-ing that the time discretization period h is infinitesimally small, h→ 0, we can write Eq. (29)

7

10-12

10-9

10-6

10-3

100

103

10-2 100 102 104

p(y)

y

(a)

10-2

100

102

104

106

10-8 10-6 10-4 10-2 100

S(f)

f

(b)

Figure 4: Statistical properties, PDF (a) and PSD (b), of numerically evaluated nonlinearGARCH process Eq. (18) with µ = 3 (red squares). Black curves show power law approxima-tions (a) x−3 for numerically obtained PDF and (b) 1/f for the PSD. Other nonlinear GARCHprocess parameters were set as follows: a = 10−6, b = 10−3, c = 2

√2π· 10−3 ≈ 1.595769 · 10−3.

The chosen values of the parameters b and c lead to C ≈ 0 and, consequently, y(3) ≈ 0 in theSDE (33) describing the diffusion limit of this nonlinear GARCh process.

asσ2kh,h = σ2

(k−1)h,h +(A+ Cσµ(k−1)h,h

)h+ |B|

√hσµ(k−1)h,hεk , (30)

where the coefficients A, B and C are introduced via the equations

ah = Ah , bhω̄µ − ch = Ch , ω̂µb2h = B2h . (31)

Note, that depending on the values of the parameters bh and ch, the parameter C can benegative as well as positive. By taking the small time step limit h→ 0 we can obtain the SDEfor the variable ykh,h = σ2

kh,h:

dy =

(A

yµ−1+

C

yµ2−1

)yµ−1dt+ |B|y

µ2 dWt . (32)

This equation can be written as

dy = B2

[1

2(µ− 1)

(y(1)

y

)µ−1+

1

2sign(C)

(µ2− 1)(y(3)

y

)µ2−1]

yµ−1dt+ |B|yµ2 dWt , (33)

where

y(1) =

(2A

(µ− 1)B2

) 1µ−1

, y(3) =

(4|C|

(µ− 2)B2

) 2µ−2

. (34)

The steady state PDF p(y) of the signal generated by Eq. (33) has a power law tail with theexponent λ = µ, while y(1) and y(3) shape the exponential cutoff:

p(y) ∼ 1

yµexp

[−(y(1)

y

)µ−1− sign(C)

(y(3)

y

)µ−2]. (35)

Eq. (33) has the general form of SDE (1) with the parameters λ = µ and η = µ/2; theparameters are the same as as in the previous case, Eq. (17). Therefore, the PSD of yt timeseries should have a frequency range where Eq. (27) holds.

The PDF and the PSD of the time series numerically obtained using Eq. (18) with µ = 3are shown in Fig. 4. We have chosen the parameters b and c in Eq. (18) in such a way that the

8

parameter C given by Eq. (31) becomes zero. In the numerical calculations we used reflectiveboundary at σt = 0 by setting σ2

t to zero when σ2t becomes negative. The analytical predictions

of the power law exponents λ = 3 in the PDF and β = 1 in the PSD are in good agreementwith the numerical results. With the chosen parameters in Eq. (18) we are able to reproduce1/f spectrum, but now over only 3 decades of frequency f , see Fig. 4(b).

5 ConclusionsIn summary, we have proposed two possible nonlinear modifications of a GARCH(1,1) process,Eqs. (17) and (18). Comparing the diffusion limit of the proposed nonlinear GARCH processeswith the known nonlinear SDE (1) generating signals with 1/fβ PSD we obtain the conditionswhen the nonlinear GARCH processes yield 1/fβ PSD too. Numerical evaluation of Eqs. (17)and (18) with suitably chosen parameters confirms the presence of a wide power law regionin the PSD of the time series. As should have been expected, the linear GARCH(1,1) process(8) does not reproduce 1/f spectrum. In addition to power law PSD, linear and nonlinearGARCH(1,1) processes resulted in power law distributions.

The results obtained in the paper are especially interesting as 1/f noise is often linked to aconcept of long-range memory, which is considered to be one of the stylized facts of the financialmarkets as well as other socio-economic systems. The obtained results and proposed nonlinearGARCH processes should be useful for creation and application of ARCH family models thatcorrectly reproduce the PSD of the financial time series as well.

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