Bridging the ARCH model for finance and nonextensive entropy

0.62 MB·English

by  Silvio M. Duarte Queiros

Bridging the ARCH model for finance and nonextensive entropy
S´ılvio M. Duarte Queir´os and Constantino Tsallis
∗
Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150,
22290-180, Rio de Janeiro-RJ, Brazil
4 (February 2, 2008)
0
Engle’sARCHalgorithmisageneratorofstochastictimeseriesforfinancialreturns(andsimilar
0
quantities) characterized by a time-dependent variance. It involves a memory parameter b (b = 0
2
corresponds to no memory), and the noise is currently chosen to be Gaussian. We assume here
n a generalized noise, namely qn-Gaussian, characterized by an index qn ∈ R (qn = 1 recovers the
a
J Gaussian case, and qn > 1 corresponds to tailed distributions). We then match the second and
fourth momentaof theARCHreturndistribution with those associated with theq-Gaussian distri-
14 butionobtainedthroughoptimizationoftheentropySq = 1−q−i1piq,basisofnonextensivestatistical
mechanics. The outcome is an analytic distribution for the rPeturns, where an unique q ≥qn corre-
] sponds to each pair (b,qn) (q =qn if b =0). This distribution is compared with numerical results
h andappearstoberemarkablyprecise. Thissystemconstitutesasimple,low-dimensional,dynamical
c mechanism which accommodates well within the current nonextensiveframework.
e
m
PACS numbers: 05.40.-a, 05.90.+m, 89.65.Gh
-
t
a
t Time series are ubiquitous in nature. They appear in zt =σt ωt, (1)
t.s geoseismic phenomena, El Nin˜o, finance, electrocardio- (cid:26)σt2 =a+ si=1bi zt2−i ,
a and electroencephalographic profiles, among many oth-
P
m ers. Someoftheseseriescanbeconstructedbyusing,for or, equivalently,
- successive values ot time t, random variables associated
d with the same distribution for all times: they are called
n homoskedastic. SuchisthecaseoftheordinaryBrownian zt = a+ si=1bi zt2 i ωt (2)
co motion. But many phenomena exist in nature which do ( σt2 =aq+ siP=1bi σt2 i−ωt2 i ,
− −
[ not accomodate with such a property, i.e., the distribu-
P
tion associated with each value of t depends on t. Such wherea,b ,σ 0,andω representsanindependentand
2 i t ≥ t
v randomvariables are then called heteroskedastic. A sim- identicallydistributedstochasticprocesswithmeanvalue
1 ple illustration would be to use say a centered Gaussian nullandunitvariance,(i.e.,hωti=0and ωt2 =1),cur-
8 at all steps, but with a randomly varying width. rentlychosentofollowaGaussiandistribution,butother
1 (cid:10) (cid:11)
Nowadays, time series that are intensively studied are choices are possible. In this manuscript we will discuss
1
the financial ones, where nonstationary volatility (tech- in detail the usual case s = 1, ARCH(1), which will be
0
4 nical name for second-order moment of say returns) is a from now on simply designated by ARCH. In general,
0 common feature [1–3]. In order to mimic explicitly and the distribution, Pn(ω), together with parametersa and
t/ analyze this type of time series, R.F. Engle introduced {bi}, specify the particular ARCH process (n stands for
a in 1982 the autoregressive conditional heteroskedasticity noise).
m
(ARCH) process[4]. Its prominencecanbe measuredby As canbe seenfromEqs. (2), parameters b charac-
i
- { }
its wide use, by the amount of its extensions introduced terize the memory mechanism. For b = 0 ( i), there is
d i
∀
n later [5,6], and — last but not least — by the fact that no memory effect, and consequently the ARCH process
o the 2003 Nobel Prize for Economics was awardedto En- forreturnsreducestogeneratingthenoiseω (multiplied
t
c gle “for methods of analyzing economic time series with by√a). Wecanverifythat,ingeneral, z = σ ω =
: h ti h tih ti
v time-varying volatility (ARCH)”. This and similar pro- 0, and ztzt′ = σtσt′ ωtωt′ = σtσt′ δtt′. Therefore,
h i h ih i h i
i cessesarecommonlyusedinthe implementationofasset even for nonvanishing b ’s, the returns z remain uncor-
X i t
pricingtheories,marketmicrostructuremodels,andpric- related, whereas correlations do exist in the variance σ2
r t
a ing derivative assets [6,7] (see also [8]). Specifically, the [5]. We observe that this stochastic process captures the
ARCH(s) [4] process generates, for the returns z , a dis- tendency for the so called volatility clustering, i.e., large
t
crete time series whose variance σ2, at each time step, (small) values of z are followed by other large (small)
t t
depends linearly on the s previous values of [z ]2. It is values of z , but of arbitrary sign. In other words,
t t+1
defined as follows: zt zt′ by no means is proportional to δtt′.
h| || |i
∗[email protected], [email protected]
1
Hereon we focus on ARCH(1), i.e., b b and b = FIG.2. Time dependence of the volatility σ for the same
1 i
≡ two ARCH processes shown in Fig. 1. The largest value for
0 (i = 2, 3,...). From Eqs. (1) it is simple to obtain
σ occurs at t=238.
the n-th moment for the P(z) stationary distributions,
particularly the second moment
Weshallnowestablishapossibleanalyticalconnection
a
σ2 ≡ zt2 =hσt2i= 1 b (b<1), (3) betweenthememoryparameterb,P (z)andPn(ω)under
the framework of the nonextensive statistical mechanics
−
(cid:10) (cid:11)
and the fourth moment, introduced by one of us [13]. Consider a probability dis-
tributionP (z)suchthatitmaximizestheentropicform
1+b ′
z4 =a2 ω4 . (4)
t (1 b)(1 b2 ω4 )
Withou(cid:10)t (cid:11)loss of(cid:10)gen(cid:11)eral−ity, we−canhatssiume that the Sq = 1− −+∞∞[P′(z)]qdz (q ), (6)
q 1 ∈ ℜ
ARCH procedure generates a time series with unit vari- R −
ance,i.e.,σ2 =1,hencea=1 b. Now,forz ,thefourth
t +
m(aopmoesnsitbilse mnuemaseurriceaollfyneoqnu-gaalu−tsositahneitkyuorrtopsiesakkexdn≡eshshxx2f4oii2r wsthtiaetnhcdolsinmfsotqrr→aBi1noStlsqtzm=+a∞n−nPR-G−′(∞i∞zb)bdPsz)′.(=zM)1lanxaPnim′d(izz)idn+zg∞E≡zq2P.SB′((6Gz))d(wzBitG=h
probability distributions), and we can easily get, 1 we have −∞ −∞
R R
k 1
z4 =k =k 1+b2 ω − (k b2 <1), (5)
t z ωh 1−kωb2i ω P′(z)= [1+ (qA1)z2]q−11 (q <5/3), (7)
Itis(cid:10)str(cid:11)aightforwardlyverifiedthat,whatevertheformof B −
Pn(ω),theARCHprocessgeneratesprobabilitydistribu- with
tions P (z) with a slower decay and consequently with a
kurtosis kz > kw [9–12]. See in Figs. 1 and 2, typical Γ 25q−32q
runs for a Gaussian noise. − , (8)
B ≡ 2(q h1)Γ i3−q
− 2q 2
−
h i
and
Γ 1
q 1
− (2q 1) . (9)
A≡ √2πΓh 3−iq − B
2q−2 p
h i
In the q 1 limit, the normal distribution is recovered.
→
The fourth moment (which for this case coincides with
the kurtosis) is given by,
FIG.1. Two examplesofARCHtimeseries obtained fora z4 =3Γ 72−q−52q Γ 23q−−q2 (1<q < 7). (10)
tGhaeupssrioabnabnioliitsye (foi.re.l,aqrnge=va1l)u.esMoefm|zotr|y, i(.ie.e.,.,fabt>ta0il)sinincrPea(sze)s. (cid:10) (cid:11) h Γ 52i−q32qh 2 i 5
−
The large value at t = 237 is virtually never observed for n h io
b=0. Letus now makethe ansatz P (z) P (z). Consistently
′
≃
we impose the matching of Eqs. (5) and (10) . This
yields q as a function of b and k . Assuming that noise
ω
ω follows a generalizeddistribution, Eq. (7), defined by
t
3.0 b = 0.0 an entropic index qn (such that kw(qn)b2 <1) we are able
b = 0.5 to establish a relation between the memory parameter
t
2.5 b and the entropic indexes q and qn, which character-
izerespectivelythe distributionsP (z)andP (w). This
′ n
2.0 connection is straightforwardly given by
1.5 2 2
1.0 b= rG(q)nΓh52−qn3−qn2io −G(qn)nΓh25q−−32qio , (11)
2
0 150 t 300 sG(qn)(cid:18)3G(q)−nΓh52−q−32qio (cid:19)
2
where G(x) Γ 7 5x Γ 3 x . This connection, de- comparing the sixth-order moment. It is simple to ver-
≡ 2−x 2 2x−2
− − ify that these momenta for P(z) and P (z) exhibit quite
pictedinFig. 3,chonstituitesh,tothiebestofourknowledge, ′
smalldiscrepancies. Forexample,forthecasesillustrated
the first ever found which analytically expresses the re-
in Figs. 4 and 5, we have obtained discrepancies never
turn distribution in terms of the noise distribution and
largerthan 2.9%(occuring in factfor b 0.5)for q =1
the memory parameterb. In what followswe shallverify n
≃
andthan2.7%(occuringinfactforb 0.3)forq =1.3.
thattheaboveansatzisindeedsatisfiedwithinaremark- n
≃
Such minor discrepancies are, in financial practice, com-
able precision. To do this for typical values of b and q ,
n
pletely inocuous (for example, we may check in [14,15]
we first generate, through a standard algorithm based
that realistic return distributions are larger than 10 6,
on Eq.(1) (with s = 1), a set of ARCH time series and −
whereas in our present Figs. 4 and 5 we have simulated
theircorrespondingprobability densityfunctions(PDF’s)
down to 10 8).
P(z). The results are indicated in Figs. 4 and 5. Then −
we compare with a histogram (with a conveniently cho-
sen unit interval δ) associated with the distribution (7),
with q satisfying Eq. (11). In other words, we compare
1
with 1 (1 q)x2 1−q, where is given by Eq.
δ
A −B − B
(8), and
(cid:2) (cid:3)
= +δ/2p(z)dz = δ Γ[q−11] Γ[25q−−32q]
Aδ −δ/2 √2πΓ[23q−−q2]rΓ[23q−−q2] (12)
R F 1, 1 ;3, δ2Γ[25q−−32q]
×2 1(cid:18)2 q−1 2 − 8Γ[23q−−q2] (cid:19)
FIG. 4. PDFs for a qn = 1 noise and typical values of
b. a) b = 0, q = 1 (χ2 = 1.14 × 10−11); b) b = 0.1,
q = 1.01976 (χ2 = 2.34×10−10); c) b = 0.4, q = 1.242424
(χ2=1.82×10−10);d) b=0.5, q= 4 (χ2 =2.98×10−10).
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FIG. 3. Diagram (q,qn,b) for the ARCH process with
z2 =1 (for b=0 we havethe straight line q=qn). FIG. 5. PDFs for a qn = 1.3 noise and typical val-
(cid:10) (cid:11) ues of b. a) b = 0, q = 1.3 (χ2 = 3.93 × 10−10); b)
We verify that the agreement is quite satisfactory. b = 0.1, q = 1.30762 (χ2 = 8.86 × 10−10); c) b = 0.2,
In order to quantify the small discrepancy between the q = 1.32950 (χ2 = 9.72×10−10); d) b = 0.3, q = 1.36306
ARCH PDF’s P(z) and the analytical ones P′(z), we (χ2=2.68×10−10).
have indicated in the captions of Figs. 4 and 5 the val-
N Let us conclude by saying that the fact that a close
ues of χ2 1 [P(z ) P (z )]2, N being the number
≡ N i − ′ i connectionhas beenshownto existbetweenthe possibly
i=1
of points. An aPlternative evaluation of the discrepancy ubiquitous ARCH stochastic processes and the nonex-
of the ARCH and analyticaldistribution can be done by tensive entropy Eq.(6) strongly suggests further possible
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