STABLE MODELING OF MARKET AND CREDIT VALUE AT RISK
3. A finance-oriented description of stable distributions
In this part we describe parameters and some finance-oriented properties of stable distrib- utions. We also examine methods of estimating parameters of stable laws.
3.1. Parameters and properties of stable distributions
A random variableRis said to bestable7if for anya >0 andb >0 there exist constants c >0 andd∈Rsuch that
aR1+bR2
=d cR+d,
whereR1andR2are independent copies ofRand=d denotes the equality in distribution.
In general, stable distributions do not have closed form expressions for the density and distribution functions. Stable random variables (R) are commonly described by their char- acteristic functions:
ΦR(θ )=E
exp(iRθ )
=exp
−σα|θ|α 1−iβsign(θ )tanπ α 2
+iàθ
, ifα =1, ΦR(θ )=E
exp(iRθ )
=exp
−σ|θ| 1+iβ2
πsign(θ )lnθ
+iàθ
, ifα=1, whereαis theindex of stability, 0< α2, βis theskewness parameter,−1β1, σ is the scale parameter,σ 0, andà is the location parameter,à∈R. To indicate the dependence of a stable random variableRon its parameters, we writeR∼Sα(β, σ, à). If
7 OftenRis calledα-stableorPareto stableorPareto–Lévy-stable(forα <2).
256 S.T. Rachev et al.
theindex of stabilityα=2, then the stable distribution reduces to the Gaussian distribution.
In empirical studies, the modeling of financial return data is done typically with stable distributions having 1< α <2.8Stable distributions are unimodal and the smallerαis, the stronger the leptokurtic feature of the distribution (the peak of the density becomes higher and the tails are heavier). Thus, the index of stability can be interpreted as ameasure of kurtosis. Whenα >1, thelocation parameteràmeasures the mean of the distribution. If theskewness parameterβ=0, the distribution ofR is symmetric and the characteristic function is
ΦR(θ )=E
exp(iRθ )
=exp
−σα|θ|α+iàθ .
Ifβ >0, the distribution is skewed to the right. Ifβ <0, the distribution is skewed to the left. Larger magnitudes ofβindicate stronger skewness. Ifβ=0 andà=0, then the stable random variableRis calledsymmetricα-stable(sαs). Thescale parameter(the volatility) σ allows any stable random variable R to be expressed as R=σ R0, where R0 has a unit scale parameter, and the same index of stabilityαand skewness parameterβ as R.
The scale parameter generalizes the definition of standard deviation. The stable analog of variance is thevariation:να=σα.
In VaR estimations we are interested in investigating the behavior of the distributions in the tails. Thetailsof the stable (non-Gaussian) distributions have a power decay and are characterized by the following properties:
λ→+∞lim λαP (R > λ)=kα
1+β 2 σα and
λ→+∞lim λαP (R <−λ)=kα1−β 2 σα, where
kα= 1−α
.(2−α)cos(π α/2), ifα =1, kα= 2
π, ifα=1.9 Thep-th absolute moment,E|R|p=∞
0 P (|R|p> x)dx,is
• finiteifp < αorα=2, and
• infiniteotherwise.
8 The financial returns modeled withα-stable laws exhibit finite means but infinite variances.
9 Note that, in contrast to the normal case, the tails of the non-Gaussian (Pareto) stable distributions are much fatter, which will be an important issue in estimating VaR.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 257
Thus, the second moment of any non-Gaussian stable distribution is infinite.
Stable distributions possess theadditivity property: a linear combination of independent stable random variables with stability indexαis again a stable random variable with the sameα.10
Example. IfR1, R2, . . . , Rnare independent stable random variables with stability index α,Ri∼Sα(βi, σi, ài), thenR=n
i=1wiRi is a stable random variable with the sameα and parameters:
(a) ifα =1, σ=
|w1|σ1α
+ ã ã ã +
|wn|σnα1/α
,
β=sign(wi)β1(|w1|σ1)α+ ã ã ã +sign(wn)βn(|wn|σn)α (|w1|σ1)α+ ã ã ã +(|wn|σn)α , à=w1à1+ ã ã ã +wnàn;
(b) ifα=1,
σ= |w1|σ1+ ã ã ã + |wn|σn,
β=sign(w1)β1|w1|σ1+ ã ã ã +sign(wn)βn|wn|σn
|w1|σ1+ ã ã ã + |wn|σn
, à=w1à1+ ã ã ã +wnàn− 2
π
w1ln|w1|σ1β1+ ã ã ã +wnln|wn|σnβn
.
Since the Pareto-stable distributions have infinite variances, one cannot estimate risk by variance and dependence by correlations. We shall introduce variance- and covariance- similar notions for stable laws. These notions are based on the multivariate assumptions of stable distributions.
A random vectorRof dimensiondisstableif for anya >0 andb >0 there existc >0 and ad-dimensional vectorDsuch that
aR1+bR2 d
=cR+D,
whereR1andR2are independent copies ofR.
If a random vector is stable withα >1, then it means that all components of the vec- tor are stable with the same index of stability and any linear combination (for example, portfolio returns) is again stable.11
10This property is shared only by normal and stable laws, and is the main advantage of the use of stable laws for portfolio returns.
11We shall model the dependence structure of the vector of returns (R1, . . . , Rd) of a portfolio by assuming that (R1, . . . , Rd) is anα-stable vector.
258 S.T. Rachev et al.
The characteristic function of ad-dimensional vector is given by:
(a) ifα =1,
ΦR(θ )=ΦR(θ1, θ2, . . . , θd)
=Eexp iθTR
=exp
−
Sd
θTsα 1−i sign θTs
tanπ α 2
Γ (ds)+iθTà
,
(b) ifα=1, ΦR(θ )=exp
−
Sd
θTs 1+i2 π sign
θTs
lnθTs
Γ (ds)+iθTà
,
whereΓ is a bounded nonnegative measure on the unit sphereSd,sis the integrand unit vector (s∈Sd) andàis the shift vector. The measureΓ is named aspectral measure. Let Hbe the distribution function ofΓ. Then, the characteristic function in polar coordinates is as follows
(a) ifα =1,
ΦR(θ )=exp
−|θ|α π
0
π
0
. . . 2π
0
cos(θ , ψ)α
×
1−sgn
cos(θ , ψ) tanπ α
2 dH (ψ)+iθTà
, (b) ifα=1,
ΦR(θ )=exp
−|θ|α π
0
π
0
. . . 2π
0
cos(θ , ψ)α
×
1−sgn
cos(θ , ψ)2 πln
ρcos(θ , ψ)dH (ψ)+iθTà
, where forθ given by its polar coordinates,θ (ρsinφ1ã ã ãsinφd−1, ρsinφ1ã ã ãsinφd−2ì cosφd−1, ρsinφ1ã ã ãsinφd−3cosφd−2, . . . , ρcosφ1), we denote
cos(θ , ψ)= d−1
i=1
sinφisinψi
+ d−2
i=1
sinφisinψi
cosφd−1cosψd−1
+ ã ã ã +cosφ1cosψ1.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 259
Ifα >1, thenàis the mean vector,à=ER. The scale parameter of a linear combina- tion of the components of a stable vectorRsatisfies the relation:
σα wTR
=σα(w1R1+ ã ã ã +wdRd)=
Sd
wTsαΓ (ds).
ViewingR=(R1, . . . , Rd) as the vector of individual returns in a portfolio with weights w1, . . . , wd,σα(wTR) will be the portfolio risk-measure. As we defined above,να=σα is thevariation, the stable equivalent of variance. Similarly to the traditional interpretation of covariance as an indicator ofdependence, one can use thecovariationto estimate the dependence between twosαsdistributions:
[R1;R2]α= 1 α
∂σα(w1R1+w2R2)
∂w1
w1=0;w2=1
=
Sn
s1s2α−1Γ (ds),
where (R1, R2) is asαsvector (1< α) andxk= |x|ksgn(x)(signed power). The ma- trix of covariations[Ri;Rj]α, 1id, 1j d, determines the dependence structure among the individual returns in the portfolio.
3.2. Estimation of parameters of stable distributions12
We shall examine the methods of estimating the stable parameters and their applicability in VaR computations, where the primary concern is the tail behavior of distributions. It has been proposed that it is more useful to evaluate directly the tail index (the index of stability) instead of fitting the whole distribution. The latter method is claimed to negatively affect the estimation of the tail behavior by its use of “center” observations. We shall describe both approaches: tail estimation and entire-distribution modeling. We suggest a method, which combines the two techniques: it is designed for fitting the overall distribution with greater emphasis on the tails.
3.2.1. Tail estimation
Tail estimators for the index of stabilityαare based on the asymptotic Pareto tail behavior of stable distributions.13We shall consider the following estimators of tail thickness: the Hill, the Pickands, and the modified unconditional Pickands.14
12For additional references on estimation of the four parameters of stable univariate laws, see Chobanov et al. (1996), Gamrowski and Rachev (1994, 1995a, b), Klebanov, Melamed and Rachev (1994), Kozubowski and Rachev (1994), McCulloch (1996), Mittnik and Rachev (1991), Rachev and SenGupta (1993). For the multivariate case estimation of: the spectral measure, the index of stability, the covariation and tests for dependence of stable distributed returns, see Cheng and Rachev (1995), Gamrowski and Rachev (1994, 1995a, b, 1996), Heathcote, Cheng and Rachev (1995), Mittnik and Rachev (b), Rachev and Xin (1993).
13See Section 3.1.
14For details on the Hill, Pickands, and the modified unconditional Pickands estimators, see Mittnik, Paolella and Rachev (1998c) and references therein.
260 S.T. Rachev et al.
The Hill estimator15is described by ˆ
αHill= 1
1 k
k
j=1ln(Xn+1−j:n)−lnXn−k:n,
whereXj:n denotes thej-th order statistic of sampleX1, . . . , Xn;16 the integerkpoints where the tail area “starts”. The selection ofk is complicated by a tradeoff: it must be adequately small so thatXn−k:nis in the tail of the distribution; but if it is too small, the estimator is not accurate. The disadvantage of the estimator is the condition to explicitly determine the order statistic Xn−k:n. It is proved that, for stable Paretian distributions, the Hill estimator is consistent and asymptotically normal. Mittnik, Paolella and Rachev (1998c) found that, the small sample performance ofαˆHilldoes not resemble its asymptotic behavior, even forn >10 000 (see Figure 117). It is necessary to have enormous data series in order to obtain unbiased estimates ofα, for example, withα=1.9, reasonable estimates are produced only forn >100 000 (see Figure 218). Alternatives to the Hill estimator are the Pickands and the modified unconditional Pickands estimators.
The “original” Pickands estimator19takes the form ˆ
αPick= ln 2
ln(Xn−k+1:n−Xn−2k+1:n)−ln(Xn−2k+1:n−Xn−4k+1:n), 4k < n.
The Pickands estimator requires choice of the optimalk, which depends on the true unknownα. Mittnik and Rachev (1996) proposed a new tail estimator named “the modified unconditional Pickands (MUP) estimator”,αˆMUP. An estimate ofαis obtained by applying the nonlinear least squares method to the following system:
k2=X2, X−11k1+ε, where
X1=
X−n−αk+1:n X−n−2αk+1:n X−n−α2k+1:n Xn−−2α2k+1:n
, X2=
X−n−α3k+1:n X−n−2α3k+1:n X−n−α4k+1:n X−n−2α4k+1:n
,
15Hill (1975).
16Given a sample of observationsX1, . . . , Xn, we rearrange the sample in increasing orderX1:nãããXn:n, then thej-th order statistic is equal toXj:n.
17In Figure 1, the true value ofαis 1.9, the sample size isn=10 000; thex-axis shows values ofkfrom 1 ton/2=5000. Notice that the estimator forαˆ= ˆα(k(n), n)is unbiased when limn→∞(k(n)/n)→0. So, unbiasedness of the estimator requires very small values ofk. However, for a small value ofk, the variance of the estimator is large. A close look at the estimatorα(k, n)ˆ suggests value ofαˆaround 2.2, whereasα=1.9.
18In Figure 2, the trueαis again 1.9, the sample size isn=500,000, k=1, . . . , n/2=250,000. One can see that, for very small values ofk,α≈1.9.
19Pickands (1975).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 261
Fig. 1. Hill estimator for 10 000 standard stable observations with indexα=1.9.
Fig. 2. Hill estimator for 500 000 standard stable observations with indexα=1.9.
262 S.T. Rachev et al.
k1= k−1
2k−1
, and k2=
3k−1 4k−1
.
Mittnik, Paolella and Rachev (1998c) found that the optimalkforαˆMUPis far less depen- dent onαthan in the case of either the Hill or Pickands estimators. Studies demonstrated thatαˆMUPis approximately unbiased forα∈ [1.00,1.95)and nearly normally distributed for large sample sizes. The MUP estimator appears to be useful in empirical analysis.
3.2.2. Entire-distribution modeling
We shall describe the following methods of estimating stable parameters with fitting the entire distribution: quantile approaches, characteristic function (CF) techniques, and max- imum likelihood (ML) methods.
Fama and Roll (1971) suggested the first quantile approach based on observed properties of stable quantiles. Their method was designed for evaluating parameters of symmetric sta- ble distributions with index of stabilityα >1. The estimators exhibited a small asymptotic bias. McCulloch (1986) offered a modified quantile technique, which provided consistent and asymptotically normal estimators of all four stable parameters, forα∈ [0.6,2.0]and β∈ [−1,1]. The estimators are derived using functions of five sample quantiles: the 5%, 25%, 50%, 75%, and 95% quantiles. Since the estimators do not consider observations in the tails (below the 5% quantile and above the 95% quantile), the McCulloch method does not appear to be suitable for estimating parameters in VAR modeling.
Characteristic function techniques are built on fitting the sample CF to the theoretical CF.
Press (1972a, b) proposed several CF methods: the minimum distance, the minimumr-th mean distance, and the method of moments. Koutrouvelis (1980, 1981) developed the iter- ative regression procedure. Kogon and Williams (1998) modified the Koutrouvelis method by eliminating iterations and limiting the estimation to a common frequency interval.20CF estimators are consistent and under certain conditions are asymptotically normal.21
Maximum likelihood methods for estimating stable parameters differ in a way of com- puting the stable density. DuMouchel (1971) evaluated the density by grouping data and applying the fast Fourier transform to “center” values and asymptotic expansions – in the tails. Mittnik, Rachev and Paolella (1998) calculated the density at equally spaced grid points via a fast Fourier transform of the characteristic function and at intermedi- ate points – by linear interpolation. Nolan (1998a) computed the density using numeri- cal approximation of integrals in the Zolotarev integral formulas for the stable density.22 DuMouchel (1973) proved that the ML estimator is consistent and asymptotically normal.
In Section 4 we analyze applicability of the ML method in VAR estimations.
20For additional references, see Arad (1980), Feuerverger and McDunnough (1981), Mittnik, Rachev and Paolella (1998), Paulson, Holcomb and Leitch (1975).
21Heathcote, Cheng and Rachev (1995).
22For additional references, see Mittnik et al. (1997).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 263
3.2.3. Tail estimation: Fast Fourier transform method
Tail estimation using the Fourier Transform (FT) method is based on fitting the character- istic function in a neighborhood of the origint=0. Here we use the classical tail estimate:
P X−1 a
P |X|1 a
K
a a
0
1−Re fX(t)
dt, for alla >0, where Re{fX(t)}is the real part of the characteristic functionfX(t)and the constantK= 1/(1−sin 1) <1/7. Precise estimation of the characteristic function guarantees accurate tail estimation, which leads to an adequate evaluation of VaR.
Suppose that the distribution of returnsris symmetric-α-stable,23that is: thecharacter- istic functionofris given by
fr(t)=Eeirt=eit à−|ct|α.
If α >1,24 then, given observations r1, . . . , rn, we estimate à by the sample mean
¯
à= ¯r= n1n
i=1ri. For large values of n, the characteristic function of observations Ri=ri− ¯rapproachesfR(t)=e−|ct|α. Consider theempirical characteristic functionof the centered observations:fˆR,n(t)=1nn
k=1eiRkt. Because the theoretical characteristic function,fR(t), is real and positive, we have that
fˆR,n(t)=Re 1 n
n
k=1
eiRkt
=1 n
n
k=1
cos(Rkt).
Now the problem of estimatingαandcis reduced to determiningαˆ andcˆsuch that M
0
fˆR,n−fR(t,α,ˆ c)ˆ = M
0
1 n
n
k=1
cos(Rkt)−e−(ct )ˆ αˆ dt is minimal, whereMis a sufficiently large value.
The realization of the FT method is performed in the following steps:
Step 1. Given the asset returnsr1, . . . , rn, compute the centered returnsRi=ri− ¯r,i= 1, . . . , n, wherer¯=1nn
i=1ri.
Step 2. Construct the sample characteristic function f (tˆ j)=1
n n
k=1
cos(Rktj),
23Empirical evidence suggests thatβdoes not play a significant role for VAR estimation.
24As we have already observed, in all financial return data, fitting anα-stable model results inα >1, which implies existence of the first moment.
264 S.T. Rachev et al.
wheretj =jκπτ ,j=1, . . . , τ,κπ is the maximal value oft,τ is the number of grid points on(0, κπ].25
Step 3. Do the search for bestαˆ andcˆsuch that τ
j=1
1 n
n
k=1
cos(Rktj)−e−(ctˆj)αˆ is minimal.