testing volatility autocorrelation in the constant elasticity of variance stochastic volatility...

Computational Statistics and Data Analysis 53 (2009) 2201–2218

Contents lists available at ScienceDirect

Computational Statistics and Data Analysis

journal homepage: www.elsevier.com/locate/csda

Testing volatility autocorrelation in the constant elasticity ofvariance stochastic volatility modelGianna Figà-Talamanca ∗Department of Economics, Finance and Statistics, University of Perugia, via Pascoli 20, 06100 Perugia, Italy

a r t i c l e i n f o

Article history:Available online 24 August 2008

a b s t r a c t

The sample autocovariance of the suitably scaled squared returns of a given stock isshown here to be a consistent and asymptotically normal estimator of the theoreticalautocovariance of the mean variance, when the data is generated by the Constant Elasticityof Variance stochastic volatility (CEV SV) process. By computing explicitly the asymptoticvariance of the estimator, confidence bands are obtained for the theoretical autocovariance.For each one of the stock indexes S&P500, CAC40, FTSE, DAX and SMI the estimatedconfidence bands are comparedwith the theoretical autocovariances computed for severalvalues of themodel parameters. The results suggest that the CEV SVmodel is able to capturethe empirical autocovariance detected on the observed data. Analogous results are derivedfor the theoretical autocorrelation function.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

The pioneering paper of Black and Scholes proposing an option pricing model (Black and Scholes, 1973) has beengeneralized in different directions. One of the most widely spread approaches is allowing for stochastic volatility in theprocess describing the underlying stock price changes, see for instance Hull andWhite (1987), Scott (1987), Wiggins (1987),Stein and Stein (1991) and Heston (1993).Stochastic volatility models account for many empirical facts of the stock and the derivative prices, such as the

leptokurtosis of financial log-returns and the so-called smile curve of the implied volatility of the European options whenplotted against the strike price (see Cont (2001)). The estimation of SV models is still a challenging issue: recently, BenHamida and Cont (2005), Ewald and Zhang (2004) and Lindström et al. (2008), among others, use calibration to estimate theparameters in SV models by fitting theoretical to market option prices.This approach has brought a renewed interest in the Heston specification (Heston, 1993) and the GARCH diffusion firstly

introduced in Nelson (1990). In fact, the price of a European option is computed via a quasi-closed formula in the formercase, while in the latter it can be approximated by an analytical formula derived in Barone Adesi et al. (2005). Hence, thecalibration to market option prices is rather straightforward in both settings.From a model risk minimization perspective (see Cont (2006)), before estimating a model, one should perform a

preliminary analysis to test whether the model reflects at least some properties of observed data (e.g. moments, serialdependence etc).This paper focuses on the Constant Elasticity of Variance stochastic volatility (CEV SV) model which nests both the model

by Heston and the GARCH diffusion. By taking advantage of the results in Genon-Catalot et al. (2000), it is proved thatif the data generating process (DGP hereafter) of a stock price is of CEV SV type in continuous time, then the sample

∗ Tel.: +39 0755855297; fax: +39 0755855299.E-mail address: [email protected]: http://www.ec.unipg.it/DEFS/talamanca.html.

0167-9473/$ – see front matter© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2008.08.024

http://www.elsevier.com/locate/csda

http://www.elsevier.com/locate/csda

mailto:[email protected]

http://www.ec.unipg.it/DEFS/talamanca.html








http://dx.doi.org/10.1016/j.csda.2008.08.024

2202 G. Figà-Talamanca / Computational Statistics and Data Analysis 53 (2009) 2201–2218

autocovariance of the suitably scaled squared returns of a given stock is a consistent and asymptotically normal estimator ofthe theoretical autocovariance of the mean variance process. The asymptotic variance of the estimator is derived explicitlyand an approximate confidence band is obtained for the corresponding theoretical value. Similar results are also providedfor the autocorrelation function.Furthermore, the following criterion to validate CEV SV model is suggested: given a finite set of discrete observations

for a stock/index price the CEV SV model is consistent with the data if the theoretical autocovariance (autocorrelation) ofthe mean variance process lies in the confidence band estimated from the sample. In order to illustrate the procedure, thesample autocovariance and the sample autocorrelation functions of the squared returns are calculated and confidence bandsare estimated for 6 major stock indexes (DJIA, S&P500, CAC40, FTSE, DAX and SMI). Since the theoretical autocovariance,computed for several values of the model parameters, mainly lies within the estimated confidence bands it is then arguedthat the CEV model is able to capture the empirical autocovariance detected on the observed data.The paper is organized as follows: sections 2 and 3 describe the model and set up the main definitions, in Section 4 the

results of Genon-Catalot et al. (2000) are briefly reported paying attention to some special cases, in Section 5 it is provedthat the sample autocovariance (autocorrelation) of the squared returns is a consistent and asymptotically normal estimateof the theoretical autocovariance (autocorrelation) function of the mean variance and confidence bands are also derived. InSection 6 the empirical results on the considered time series are described and Section 7 is dedicated to some concludingremarks and possible future developments of this study.

2. The bivariate mean reverting diffusion

The so-called stochastic volatility models for describing the dynamics of the price St of a given stock are usually definedthrough the following bivariate stochastic differential equation (SDE):

dStSt= µdt + σtdBt ,

dσ 2t = b(θ, σ2t )dt + a(θ, σ

2t )dWt ,

(1)

where a and b are suitable functions in order to guarantee the existence of a strong solution to the SDE,µ ∈ R and θ ∈ Rd aremodel parameters and (B,W ) is a possibly correlated bi-dimensional Brownianmotion. In the Constant Elasticity of Variance(CEV) process the second diffusion is chosen as:

dσ 2t = α(β − σ2t )dt + ϕ(σ

2t )dWt , (2)

where ϕ(x) = cxυ with 0 ≤ υ ≤ 1. Here, α and β are positive parameters representing respectively the mean reversionspeed and the long run mean for the variance; parameter c is the so-called ‘‘volatility of volatility’’ and υ is the elasticity ofvariance. Special cases within this framework are the Heston model (Heston, 1993), obtained for υ = 0.5 and the GARCHdiffusion, for υ = 1, introduced in Nelson (1990) as the continuous time limit of the discrete GARCH(1,1)model of Bollerslev(1986). Heston’s specification is a very interesting example in financial literature since it provides a quasi-closed formula forcomputing the price of a European option (some integrals involved in the expression are numerically evaluated). Besides,Barone Adesi et al. (2005) have derived an analytical approximation of the price of a European option also for the GARCHdiffusion.From here on, the following reduced form for the model specification in (2) will be considered, using the centered log-

prices Yt :

dYt = σtdBt , Y0 = 0,

dσ 2t = α(β − σ2t )dt + ϕ(σ

2t )dWt , σ 20 = ν,

(3)

where (B,W ) is a standard bi-dimensional Brownian motion. The stationarity and ergodicity of the entire process isguaranteed if it is also assumed that the initial value ν is a random variable which has the stationary distribution of theprocess and is independent on the Brownian motion (B,W ). It is worth noticing that Heston’s dynamics admits a strongsolution if c > 0 and the origin is unattainable if 2αβ ≥ c2 (see Feller (1966)) and that the process admits a Gamma( 2αβ

c2, 2αc2)

stationary distribution with finite moments of any order given by

E[νp] =(c2

2α

)p 0( 2αβc2+ p)

0(2αβc2).

In the GARCH diffusion the stationary distribution of the process is an Inverse Gamma(1+ 2αc2,2αβc2) and it exists with finite

moment of order p equal to

E[νp] =(2αβc2p

)p 0(1+ 2αc2− p)

0(1+ 2αc2),

G. Figà-Talamanca / Computational Statistics and Data Analysis 53 (2009) 2201–2218 2203

if c > 0 and p < 1+ 2αc2. In both cases E[ν] = β while it is straightforward to derive

Var(σ 20 ) =βc2

2αand

Var(σ 20 ) =β2c2

2α − c2, if 2α > c2,

respectively for the Heston and the GARCH diffusion.The CEVdiffusion process has been used tomodel interest rates in Chan et al. (1992). In their setting theoretical properties

and estimation methods are easier to derive since a univariate process is considered for the short rate which is, at least viaproxies, an observable variable. On the other hand, in SV models, the CEV diffusion is applied to model the dynamics of theinstantaneous variance process of another diffusion which, of course, is not observable and so should be considered as alatent variable for statistical inference purposes. In Kim and Wang (2006) a class of SV models extending Chan et al. (1992)(see also Andersen et al. (2004)) has been evaluated on the daily/weekly released US T-Bill yields data.

3. The mean variance process

Assume that n+1 equally spaced observations S0, S1,S2, . . . , Sn are given for the stock price and denote with∆ the fixedobservation step. For i = 1, 2, . . . , n, define the returns Ri = log Si − log Si−1 and the scaled returns as Xi =

Ri√∆.

Bymakinguse of the theory ofHiddenMarkovModels andofmixing techniques, Genon-Catalot et al. (2000) show that theprocess Xii=1,2,...,n is stationary and ergodic. Moreover, conditionally on the sigma-field F generated by

σ 2s , s ≥ 0

, the

distribution function of the scaled returns (X1, X2, . . . , Xn) is n-dimensional Gaussian with zero mean and with covariancematrixΣ = diag(σ 21 , σ

22 , . . . , σ

2n ), where

σ 2i :=1∆

∫ ∆i

∆(i−1)σ 2s ds, (4)

is themean variance in the interval [(i− 1)∆, i∆[ for i = 1, 2, . . . , n.In Barndorff-Nielsen and Shephard (2001b, 2002) the authors define the integrated volatility process as σ 2

∗

t =∫ t0 σ

2s ds

and the actual volatility as its increment process i.e. the quantity σ 2i = σ2∗i∆ − σ

2∗(i−1)∆, for i ∈ N − 0. In their terminology

the mean variance in (4) is the mean actual volatility.

3.1. Second order properties and autocorrelation function

Denote for clarity Vt = σ 2t . The computation of the first moment Eθ[V1]is straightforward and gives

E[V1] = E[1∆

∫ ∆

0σ 2r dr

]=1∆

∫ ∆

0E[σ 2r ]dr =

1∆

∫ ∆

0βdr = β.

Concerning the variance and the raw autocovariance of order h notice that

E[V12] = E

[1∆2

∫ ∆

0

∫ ∆

0σ 2t σ

2r dt dr

]=1∆2

∫ ∆

0

∫ ∆

0E[σ 2t σ

2r ]dt dr

=2∆2

∫ ∆

0

∫ ∆

tE[σ 2t σ

2r ]drdt

(5)

and

E[V 1V 1+h] = E[1∆2

∫ ∆

0

∫ (h+1)∆

h∆σ 2t σ

2r dt dr

]=1∆2

∫ ∆

0

∫ (h+1)∆

h∆E[σ 2t σ

2r ]dt dr. (6)

In Appendix A it is shown that

E[σ 2t σ2r ] = β

2+ exp(−α(r − t))Var(σ 20 ), for t < r.

Computing integrals in (5) and (6) leads to

Eθ[V12]= β2 + Var(σ 20 )

2(α∆− 1+ exp (−α∆))α2∆2

(7)


and

Eθ[V1V1+h

]= β2 + Var(σ 20 ) exp(−hα∆)

(1− exp (−α∆))2

α2∆2, (8)

which are independent on function ϕ in (3) provided Var(σ 20 ) is finite.By taking ratios, the theoretical autocovariance function γh of the mean variance process is easily obtained as

γh =

Var(σ 20 )∆2α2

e−α∆(h−1)(1− exp (−α∆))2, for h ≥ 1,

2Var(σ 20 )α2∆2

(α∆− 1+ e−α∆), for h = 0,(9)

and the autocorrelation function is

ρh =γh

γ0= e−α∆(h−1)

(1− exp (−α∆))2

2(α∆− 1+ e−α∆). (10)

Notice that the autocovariance function, for h ≥ 1, is linear in λ = Var(σ 20 ). Moreover, its limit when α∆ approaches 0is Var(σ 20 ), hence, for very small values of α∆ the autocovariance function is almost flat at the level Var(σ

20 ).

The autocorrelation functionρh, which depends only upon the product of themean reversion speedα and the observationstep∆, is always positive, is exponentially decaying in h, approaches 0 for α∆ increasing to infinity and its limit is 1 whenα∆ goes to zero.It is remarkable that the independence from Var(σ 20 ) implies that, if all the technical assumptions for ergodicity are

assumed and if Var(σ 20 ) is finite, all models within the CEV framework (3) display the same form of serial dependence forthe mean variance process. In the case of the Heston model the stationary distribution exists with finite moment of anyorder if 2αβ ≥ c2, hence if 0 < λ =

βc2

2α ≤ β2. In the GARCH diffusion process the stationary distribution exists with finite

moment of order p when c2(p − 1) ≤ 2α hence, if it is assumed that the third order moment exists then 2α ≥ 2c2 and0 < λ =

β2c2

2α−c2≤ β2. Furthermore, if it is also assumed that the fourth order moment exists then 0 < λ =

β2c2

2α−c2≤

β2

2and so on. The above results give an upper limit to the model autocovariance function for both the Heston and the GARCHdiffusion setting.Analogous results for the model autocovariance (autocorrelation) structure were obtained in Barndorff-Nielsen and

Shephard (2001a,b) for the actual volatility process in the case of general continuous SV models and assuming that theinstantaneous variance processσ 2t is described by aOrnstein–Uhlenbeck (OU) diffusion: their proof is based on the cumulantgenerating function of the OU process. In a later paper (Barndorff-Nielsen and Shephard, 2002) it is claimed that similarresults hold when the instantaneous variance process σ 2t is described by a CEV process as it is assumed in this paper.Nevertheless, the derivation of the autocovariance (autocorrelation) structure given here is very straightforward and easyto follow since it is simply based on a double integration and higher moments can also be obtained applying the sametechnique.

4. Estimating second order moments

Given a positive integer d and a set of Borel functions g1, g2, . . . , gm with gj : Rd → R, for j = 1, 2, . . . ,m, consider thefunction hgj : (R+)

d→ R defined as:

hgj(u1, u2, . . . , ud) = E[gj(e1√u1, e2

√u2, . . . , ed

√ud)],

where (e1, e2, . . . , ed) are i.i.d. standard Gaussian random variables. If Xii=1,2,...,n is a sequence of scaled returns as definedin the previous section, denote for brevity

Gi,j = gj(Xi+1, Xi+2, . . . , Xi+d).

Under the assumptions (A0)–(A3) reported in Appendix B, provided that E∣∣hgj(V 1, V 2, . . . , V d)∣∣ < +∞ and thatΣ∆ is well

defined and positive definite, the following results are demonstrated in Genon-Catalot et al. (2000):

(a)

1n

n−d∑i=0

Gi,ja.s.−→n→∞

E[hgj(V 1, V 2, . . . , V d)

]form = 1, 2, . . . , k. (11)


(b)

1√n

n−d∑i=0

(Gi,1 − E

[hg1(V 1, V 2, . . . , V d)

])n−d∑i=0

(Gi,2 − E

[hg1(V 1, V 2, . . . , V d)

]). . .

n−d∑i=0

(Gi,k − E

[hgk(V 1, V 2, . . . , V d)

])

law−→n→∞

N (0,Σ∆) , (12)

whereΣ∆(gj, gl) = Cov(Gj,0,Gl,0)+∑∞

i=1 Cov(Gj,0,Gl,i)+∑∞

i=1 Cov(Gj,i,Gl,0).The above limits are analogous, respectively, to the Strong Law of Large Numbers and to the Central Limit Theorem

for standard sums of i.i.d random variables in a multivariate setting. It should be underlined that all the assumptions forthese limits to hold are fulfilled for the model specification in (3) under the parameter restrictions assumed throughout thisanalysis. By defining ψ1(x1) = x21, ψ2(x1) = x

41 and ψ3(x1, x1+h) = x

21x21+h, and applying the limits in (11) and (12), one can

see that

1n

n∑1

X2ia.s.→

n→+∞β,

1n

n∑1

X4ia.s.→

n→+∞3E[V12],

1n

n−h∑1

X2i X2i+h

a.s.→

n→+∞E[V1V1+h

](13)

and that

√n

1n

n∑1

X2i − β

1n

n∑1

X4i − 3E[V12]

1n

n−h∑1

X2i X2i+h − E

[V1V1+h

]

law−→N (0,Σ∆) , (14)

whereΣ∆(ψj, ψl) is to be computed.

4.1. Asymptotic covariance matrix

In order to calculate the entries of Σ∆, Proposition 2.4 in Genon-Catalot et al. (2000) is generalized as follows:

Proposition 1. Assume (A0)–(A4) described in Appendix B; the following properties hold:

(i) if g1(x1, . . . , xd1) = x2p1 with d1 = 1 and E[V

2p(1+ δ2 )0 ] < +∞, then

Σ∆(g1, g1) = C22p(Var(Vp1)+ 2

∞∑i=1

Cov(Vp1, V

p1+i))+ (C4p − C

22p)E[V

2p1 ];

(ii) if g2(x1, . . . , xd2) = x2q1 x2r1+h with d2 = h+ 1, z = maxr, q and E[V

4z(1+ δ2 )0 ] < +∞, then

Σ∆(g2, g2) = C22qC22r(Var[V

q1Vr1+h] + 2

∞∑i=1

Cov(Vq1Vr1+h, V

q1+iV

r1+h+i))

+ E[V2q1 V

2r1+h](C4qC4r − C

22qC

22r)

+ 2E[Vq1V

(q+r)1+h V

r1+2h](C2qC2rC2(r+q) − C

22qC

22r).

Moreover, if g1 and g2 are defined as above, g3(x1, . . . , xd3) = x2u1 with d3 = 1 and g4(x1, . . . , xd4) = x

2t1 x2s1+k with

d4 = k+ 1, then:


(iii) if z = maxp, u and E[V4z(1+ δ2 )0 ] < +∞

Σ∆(g1, g3) = C2(p+u)E[Vp+u1 ] − C2pC2uE[V

p1]E[V

u1] + C2pC2u

∞∑i=1

(E[V

p1Vu1+i] − 2E[V

p1]E[V

u1] + E[V

p1+iV

u1]

);

(iv) if z = maxp, q, r and E[V3z(1+ δ2 )0 ] < +∞,

Σ∆(g1, g2) = E[Vp+q1 V

r1+h]C2(p+q)C2r − E[V

p1]E[V

q1Vr1+h]C2pC2qC2r + C2pC2qC2r

∞∑i=1

(E[Vp1Vq1+iV

r1+h+i]

− 2E[Vp1]E[V

q1Vr1+h] + E[V

q1Vp1+iV

r1+h])+ (C2(p+r)C2q − C2pC2qC2r)E[V

q1Vp+r1+h].

The proof, based on conditional independence, is given in Appendix B.By applying Proposition 1 part (i) for p = 1 and p = 2 one gets

Σ∆(x21, x21) = 3E[V

21] − E[V 1]

2+ 2

∞∑i=1

(E[V 1V 1+i] − E[V 1]2

)and

Σ∆(x41, x41) = 105E[V

41] − 9E[V

21]2+ 18

∞∑i=1

(E[V

21V21+i] − E[V

21]2),

respectively. Besides, part (ii) for p = q = 1 gives

Σ∆(x21x21+h, x

21x21+h) = 9E[V

21V21+h] − E[V 1V 1+h]

2+ 4E[V 1V

21+hV 1+2h]

+ 2∞∑i=1

(E[V 1V 1+hV 1+iV 1+h+i] − E[V 1V 1+h]2

),

part (iii) for p = 1 and u = 2 leads to

Σ∆(x21, x41) = 15E[V

31] − 3E[V 1]E[V

21] + 3

∞∑i=1

(E[V 1V

21+i] − 2E[V 1]E[V

21] + E[V 1+iV

21]

),

while part (iv) for q = r = 1 and p = 1, p = 2 respectively gives

Σ∆(x21, x21x21+h) = 3E[V

21V 1+h] + 2E[V 1V

21+h] − E[V 1]E[V 1V 1+h]

+

∞∑i=1

(E[V 1V 1+iV 1+h+i] − 2E[V 1]E[V 1V 1+h] + E[V 1V 1+iV 1+h])

and

Σ∆(x41, x21x21+h) = 15E[V

31V 1+h] − 3E[V

21]E[V 1V 1+h] + 12E[V 1V

31+h]

+ 3∞∑i=1

(E[V21V 1+iV 1+h+i] + E[V 1V

21+iV 1+h] − 2E[V

21]E[V 1V 1+h]).

By following the same streamline of Section 3.1 the third and fourth order moments appearing in the above formulas canbe expressed as a function of model parameters by multiple integration respectively of E[σ 2t σ

2r σ2s ] and E[σ

2t σ2r σ2s σ2w], for

t < r < s < w.

5. Estimating the autocovariance and the autocorrelation

By using the outcomes of the previous section the following variables

β(n) =1n

n∑1

X2i ,

M(n)0 =

13n

n∑1

X4i

and

M(n)h =

1n− h

n−h∑1

X2i X2i+h, for h ≥ 1,

are consistent and asymptotic normal estimates of β, E[V12] and E

[V1V1+h

]respectively.


The covariance matrix of these estimators is denoted withΛ(h) to stress the dependence in the lag h and its entriesΛi,jare obtained scaling those ofΣ∆.More precisely, Λ11 = Σ∆(x21, x

21),Λ12 =

13Σ∆(x21, x

41),Λ13(h) = Σ∆(x21, x

21x21+h),Λ22 =

19Σ∆(x41, x

41), Λ23(h) =

13Σ∆(x41, x

21x21+h),Λ33(h) = Σ∆(x21x

21+h, x

21x21+h).

Let us define, for h ≥ 0,

γ(n)h = M

(n)h − β

2 (15)

and

ρ(n)h =

γ(n)h

M(n)0 − β

2. (16)

Rearranging terms in (13),

γ(n)h

a.s.→

n→+∞γh

and

ρ(n)h

a.s.→

n→+∞ρh, (17)

for every h ≥ 0, so that γ (n)h and ρ(n)h are consistent estimators of γh and ρh respectively defined in (9) and (10).Notice that γ (n)h is the sample autocovariance of the squared scaled returns process

X2ii while ρ

(n)h is not its sample

autocorrelation sinceM0 is the sample raw moment of order two ofX2iscaled with a factor of 13 . As a consequence |ρ

(n)h |

is not constrained in [0, 1] (|ρ(n)h | ≤ 3 when β = 0 but can be greater in general). On the contrary, in the limit, |ρh| ≤ 1.From now on, however, ρ(n)h will be referred to as autocorrelation of the squared returns.

5.1. Asymptotic distribution of the autocovariance and autocorrelation

The case of β known

If β is known γ (n)h is a simple translation ofM(n)h , thus the asymptotic distribution of

√n(γ(n)h − γh

)is normal withmean

γh and varianceΛ33(∆, h). In order to derive the asymptotic distribution of ρ(n)h let us denote

ρ(n)h = f (M

(n)0 ,M

(n)h ),

where f (u, v) = v−β2

u−β2. By applying Theorem 5.2.3 in Lehmann (1998), given the asymptotic normality of (M0,Mh) and (17),

one can see that√n(ρ(n)h − ρh

)law−→n→+∞

N(0, τ

(E[V1

2], E

[V1V1+h

])), (18)

where

τ(u, v) =(dfdu

)2Λ22 + 2

dfdudfdvΛ23(h)+

(dfdv

)2Λ33(h)

=

(−

v − β2(u− β2

)2)2Λ22 − 2

v − β2(u− β2

)3Λ23(h)+ ( 1u− β2

)2Λ33(h).

The case of β unknownIn this case it is possible to write

γ(n)h = f

∗(β,Mh)

and

ρ(n)h = f

∗∗(β,M0,Mh),

with f ∗(u, v) = v − u2 and f ∗∗(u1, u2, u3) =u3−u21u2−u21

. By using Corollary 5.4.2 in Lehmann (1998) one gets

√n(γ(n)h − γh

)law−→n→+∞

N(0, τ ∗

(β, E

[V1V1+h

]))


and

√n(ρ(n)h − ρh

)law−→n→+∞

N(0, τ ∗∗

(β, Eθ [V1

2], E

[V1V1+h

])), (19)

where

τ ∗(u, v) =(df ∗

du

)2Λ11 + 2

df ∗

dudf ∗

dvΛ13(h)+

(df ∗

dv

)2Λ33(h)

= 4u2Λ11 − 4uΛ13(h)+Λ33,

and

τ ∗∗(u1, u2, u3) =(df ∗∗

du1

)2Λ11 +

(df ∗∗

du2

)2Λ22 +

(df ∗∗

du3

)2Λ33(h)

+ 2df ∗∗

du1

df ∗∗

du2Λ12 + 2

df ∗∗

du1

df ∗∗

du3Λ13(h)+ 2

df ∗∗

du2

df ∗∗

du3Λ23(h)

=2u21(u3 − u2)

2(u2 − u21

)4 Λ11 +

(u3 − u21

)2(u2 − u21

)4Λ22 + 1(u2 − u21

)2Λ33(h)+4u1(u3 − u2)

(u3 − u21

)(u2 − u21

)4 Λ12 +4u1(u3 − u2)(u2 − u21

)3 Λ13(h)+2(u3 − u21

)(u2 − u21

)3 Λ23(h).

5.2. Confidence bands

Approximate confidence bands for the autocovariance and the autocorrelation functions can be easily derived fromthe asymptotic distribution obtained in the previous subsection. For a given confidence level p, define z 1−p

2implicitly as

Φ(z 1−p2) =

1−p2 or, for symmetry, Φ(−z 1−p2

) =1−p2 where Φ and Φ are the cumulative and the survival distribution

function of a standard normal variable, respectively.Given a large sample xii=1,2,...,n of scaled returns, the p-level approximate confidence bands for the autocovariance

function at lag h, respectively for β known and unknown, are[γh − z 1−p

2

√Λ33(h)n

, γh + z 1−p2

√Λ33(h)n

](20)

and [γh − z 1−p

2

√τ ∗

n, γh + z 1−p

2

√τ ∗

n

], (21)

where τ ∗ = τ ∗(β, Eθ

[V1V1+h

]). Similarly, the p-level approximate confidence bands for the autocorrelation function at lag

h, respectively for β known and unknown, are[ρh−z 1−p

2

√τ

n, ρh+z 1−p

2

√τ

n

](22)

and [ρh−z 1−p

2

√τ ∗∗

n, ρh+z 1−p

2

√τ ∗∗

n

], (23)

where τ = τ(Eθ [V1

2], Eθ

[V1V1+h

])and τ ∗∗ = τ ∗∗

(β, Eθ [V1

2], Eθ

[V1V1+h

]).


Table 1Summary statistics for the observed data

DJIA S&P500 CAC40

Mean× 10−3 0.1105 0.0708 0.0389Std. dev. 0.0132 0.0136 0.0170Skew. −0.1866 −0.0376 −0.0402Kurt. 5.8463 4.9475 4.4774Jarque–Bera Stat. 512.2423 235.8006 135.7301Engle’s ARCH Stat. 21.645 36.182 49.437

FTSE DAX SMI

Mean× 10−3 −0.1117 −0.1239 −0.0731Std. dev. 0.0137 0.0191 0.0149Skew. −0.1008 −0.1398 −0.0815Kurt. 4.3699 4.5677 5.6994Jarque–Bera Stat. 118.8338 157.2807 454.5363Engle’s ARCH Stat. 35.539 77.498 39.986

6. Empirical application

6.1. Data description

In this section the results shown above are applied in order to evaluate the choice of CEV SV models as DGP of the timeseries of daily observations of several stock indexes (DJIA, S&P500, CAC40, FTSE, DAX, SMI). For each index six years ofdaily data are considered, from April 21st, 1997 to April 20th, 2003, which are enough to achieve a reasonable accuracy inestimation, avoiding, at the same time, structural breaks in the sample.Table 1 sums up the standard data analysis and also reports the Jarque–Bera and the Engle’s ARCH test statistics values on

the daily log-returns of the time series. Daily log-returns are almost centered with a mean of order 10−4. Mean log-returnsare slightly negative for the FTSE, the DAX and the SMI indexes and all standard deviations are below 2%. Furthermore, all thekurtosis are between 4 and 6 showing a more leptokurtic (fat-tailed) distribution with respect to the Gaussian benchmark.The standard data analysis and the Jarque–Bera normality test reject the null of normal returns. Besides, the Engle’s ARCH

statistics value supports stochastic volatility models.

6.2. Sample autocovariance/autocorrelation function and confidence bands

This empirical illustration is designed as a preliminary analysis to be performed before estimating the model and soestimation issues concerning each parameter are avoided on purpose. Denote θ = α∆ and λ = Var(σ 20 ).After having chosen a family of values θi, λi, i = 1, 2, . . . , K , the model autocovariance and autocorrelation functions

γh(θi, λi), ρh(θi), for i = 1, 2, . . . , K , are computed and plotted against h ≥ 0. For all datasets under analysis θ is chosenin the family θi = 0.1:0.01:2 (200 values) and the model autocovariance is computed for λ = β2,

β2

2 andβ2

4 . This choicesfor the parameter λ is motivated by the fact that λ, as remarked in Section 3.1, should be bounded above by β2 for boththe Heston and the GARCH diffusion specifications. If, further, the existence of the fourth moment is also requested in thelatter case then λ should be less than β2

2 . The parameter β is assumed to be known and equal to its estimated value in eachsample.Since γh is always positive and λ is a scaling factor, a smaller λ leads to a contraction of the graphs towards 0. Moreover,

when θ = α∆ is small themodel autocovariance is almost constant at the valueλ, thus, from the expression of γh, parameterλ gives substantially the maximum value for h ≥ 1 of the model autocovariance.In Figs. 1–6 the autocovariance functions, as described above, are plotted with the relative confidence bands, computed

according to (20) for a 95% confidence level, under the assumption that themodel is the true DGP. Notice that the asymptoticcovariance matrixΛ(h) is estimated by recursively applying (11) (see Appendix D for the details).It is worth noting that the theoretical autocovariance curves for different values of θ are plotted together in each graph

taking up a whole area of the quadrant. However, for the purposes of this analysis, there is no need to distinguish one curvefrom another, while it is important to check if the whole area occupied by the family of curves (or a part of it) lies betweenthe confidence limits. To make the graphs more clear the estimated autocovariance itself is not plotted.From Fig. 1, for the DJIA case, it is evident that thewhole area referred to the family ofmodel autocovariance functions lies

mostly inside the confidence band for high order lags while it is on the lower frontier of the band for h = 0 (the variance).Moreover, part of the area exits the band when λ decreases (middle and bottom graphs). The CEV SV model cannot berejected definitely since there are many possible theoretical functions lying inside the confidence band.The graphs for the SP500, the DAX and the SMI Index are reported in Figs. 2–4 respectively: the autocovariance at lag

h = 0 is out of the confidence band for all the values of λ. However, for h ≥ 1, the model autocovariance functions lie insidethe confidence band.


Fig. 1. Empirical confidence bands (circles) for the autocovariance function in the DIJA case. The model autocovariance is reported for several values of θand for λ = β2 (top), β2/2 (middle), β2/4 (bottom) respectively.

Fig. 2. Empirical confidence bands (circles) for the autocovariance function in the S&P500 case. The model autocovariance is reported for several valuesof θ and for λ = β2 (top), β2/2 (middle), β2/4 (bottom) respectively.

For such three indexes themodel is generally not rejected especially for λ = β2 (top graphs) that is for parameters valueson the borderline of the parameter set if the stationarity of the process is assumed. This result suggests that the observeddata may be non stationary.Concerning the CAC40 and the FTSE (Figs. 5 and 6), all the plotted theoretical functions remain inside the bands

for essentially every positive value of the lag. In this case the CEV SV model gives a good description of the sampleautocovariance.To conclude the analysis, the model autocorrelation functions for several values of the parameter θ are plotted in Figs. 7

and 8 for the DJIA and the S&P500 indexes respectively and such functions are compared with the sample autocorrelation(modified as explained in Section 5).


Fig. 3. Empirical confidence bands (circles) for the autocovariance function in the DAX case. The model autocovariance is reported for several values of θand for λ = β2 (top), β2/2 (middle), β2/4 (bottom) respectively.

Fig. 4. Empirical confidence bands (circles) for the autocovariance function in the SMI case. The model autocovariance is reported for several values of θand for λ = β2 (top), β2/2 (middle), β2/4 (bottom) respectively.

In Fig. 9 the same graphs are reported with the corresponding confidence bands. In agreement with the outcomes forthe autocovariance function the CEV SV model cannot be rejected for the DJIA time series. Notice in particular that allplotted model functions are inside the confidence band meaning that many values of the parameter θ are suitable (or atleast not rejected) for describing the serial dependence of the sample. For the S&P500 index only one among the plottedmodel autocorrelation functions lies inside the confidence band so few values of the parameters could be appropriate fordescribing the autocorrelation of the observed data. Nevertheless, in this prior to estimation analysis, the CEV SV modelcannot be rejected definitely. The results for the autocorrelation of the other time series are analogous.


Fig. 5. Empirical confidence bands (circles) for the autocovariance function in the CAC40 case. The model autocovariance is reported for several values ofθ and for λ = β2 (top), β2/2 (middle), β2/4 (bottom) respectively.

Fig. 6. Empirical confidence bands (circles) for the autocovariance function in the FTSE case. The model autocovariance is reported for several values of θand for λ = β2 (top), β2/2 (middle), β2/4 (bottom) respectively.

7. Concluding remarks and further developments

This study is focused on the estimation of the moments, the autocovariance function and the autocorrelation functionof the mean variance process, when the data are discretely sampled from the Constant Elasticity of Variance stochasticvolatility model described in (3). The main contribution is to derive consistent and asymptotically normal estimates of theautocovariance and of the autocorrelation of any order hwhich are based on the results in Genon-Catalot et al. (2000), brieflydescribed in Section 4.The asymptotic variance for special polynomial functions of the observations is obtained explicitly in Section 4.1 and, as

a special case, the asymptotic variance matrix of the estimators of convenient moments is computed.


Fig. 7. The empirical autocorrelation of the DJIA for h ≥ 1 (stars) is compared to several theoretical autocorrelation functions (solid lines). Theautocorrelation function for θ = 0.01 (triangles) and for θ = 0.05 (squares) are emphasized.

Fig. 8. The empirical autocorrelation of the S&P500 for h ≥ 1 (stars) is compared to several theoretical autocorrelation functions (solid lines). Theautocorrelation function for θ = 0.01 (triangles) and for θ = 0.05 (squares) are emphasized.

By applying the so-called delta-method, the asymptotic distribution of the estimators for the autocovariance and theautocorrelation structure at lag h are also obtained and they are used to compute approximate confidence bands for thecorresponding theoretical values.An application of this study is the evaluation of the CEV SV model on observed time series. Given a finite set of discrete

observations for a stock/index price, a CEV SV model is consistent with the observed data if the theoretical autocovarianceof the mean variance lies in the confidence bands estimated from the sample. Computing the sample autocorrelation andautocovariance of the squared returns for 1500 daily observations of 6 major stock indexes and estimating their confidencebands, it is shown that the CEV SV model generally gives a good description of the sample linear serial dependence.A similar analysis can also be used to validate and to compare the estimationmethods currently available in the literature

for CEV SV type process, which are often under debate: after model parameters are estimated, the autocorrelation andautocovariance process according to formulas (9) and (10) can be plotted against h and compared with the estimatedconfidence band.One can ask, for example, if calibrating the parameters on option prices, as recently proposed, leads to parameter values

which are consistent with the linear serial dependence of the historical observations of the underlying price. Notice that,in the dynamics of the Heston model, the mean reversion speed under the risk neutral probability and the same parameterunder the historical measure differ only by a constant which is the estimate of the volatility risk premium.


Fig. 9. Top/Bottom graph: confidence bands (circles) estimated from the autocorrelation (modified) of the DJIA/S&P500 for h ≥ 1, compared to severaltheoretical autocorrelation functions (solid lines).

Finally, the expression of the model autocovariance and autocorrelation as functions of the model parameters could beexploited in order to obtain an estimate for each parameter, based for instance on a least squares method or on a nonlinearcurve fitting. These possible applications are currently under investigation.

Acknowledgements

I thank the Associate Editor and the anonymous referee for their comments and suggestions which resulted in theimprovement of the paper.

Appendix A

Setting Ut := eαtVt and defining f (t, r) = α∫ rt exp(αs)ds and using Ito’s Lemma:

Ur = Ut + αβ∫ r

texp(αs)ds+

∫ r

texp(αs)ϕ(Vs)dWs

= Ut + βf (t, r)+∫ r

texp(αs)ϕ(Vs)dWs.

Hence,

E[UtUr ] = E[UtE[Ur |Ft ]] = E[Ut(Ut + βf (t, r))]= E[U2t ] + βf (t, r)E[Ut ] = exp(2αt)E[σ

4t ] + β

2f (t, r) exp(αt)

= exp(2αt)(Var(σ 2t )+ β2)+ β2f (t, r) exp(αt)

= β2(exp(2αt)+ exp(αr) exp(αt)− exp(2αt))+ exp(2αt)Var(σ 20 )

= β2 exp(α(r + t))+ exp(2αt)Var(σ 20 ),

from which

E[σ 2t σ2r ] = exp(−α(t + r))E[UtUr ]

= β2 + exp(−α(r − t))Var(σ 20 ).


Appendix B

In this appendix the assumptions are reported under which the results of Genon-Catalot et al. (2000) hold.Let (Yt , Vt)t≥0 be a bi-dimensional diffusion process defined by

dYt = σtdBt , Y0 = 0,dVt = b(Vt)dt + a(Vt)dWt , V0 = ν,

(B.1)

where functions a(x) and b(x) are defined on (l, r) ⊂ (0,+∞).Define, for x0, x ∈ (l, r), the scale and speed densities and the stationary density of Vt respectively as

s(x) = exp(−2

∫ x

x0

b(u)a2(u)

du),

m(x) =1

a2(x)s(x).

Assume the following properties to hold:

• (A0) (B,W ) is a standard Brownian motion in R2 defined on a probability space (Ω,F , P) and ν is a random variabledefined onΩ , independent of (B,W ).• (A1) The functions a(x) and b(x) are defined on (l, r) ⊂ (0,+∞) and satisfy

(i) b ∈ C1(l, r), a2 ∈ C2(l, r), a(x) > 0, ∀x ∈ (l, r)(ii) ∃K > 0 such that, ∀x ∈ (l, r), |b(x)| ≤ K(1+ |x|) and a2(x) ≤ K(1+ x2).

• (A2)∫l s(x)dx = +∞,

∫ r s(x)dx = +∞ and ∫ rl m(x) = M < +∞.• (A3) The initial random variable v has distribution π(dx) = π(x)dx where π(x) = m(x)

M 1x∈(l,r) is the stationary densityof Vt .LetA andB be two sigma-fields included in F and define

α(A,B) = supA∈A,B∈B

|P(A ∩ B)− P(A)P(B)|

as a measure of dependence.For the variance matrix in (12) to be well defined and positive definite the following assumption is in order:

• (A4) ∃δ > 0 such that E∣∣G0,j∣∣ < +∞ and∑k≥1 α

22+δV (k∆) < +∞where αV (t) = sups≥0α(Fs, F s+t),with Fs = σ(Vs, s ≤

t) and F s+t = σ(Vu, u ≥ s+ t).

Appendix C

Proof of Proposition 1. (i) See Genon-Catalot et al. (2000).(ii) By conditional independence,

Var(Φ0) = Var[X2q1 X

2r1+h] = E[X

4q1 X

4r1+h] − E[X

2q1 X

2r1+h]

2

= E[V2q1 V

2r1+h]C4qC4r − E[V

q1Vr1+h]

2C22qC22r

= C22qC22r(E[V

2q1 V

2r1+h] − E[V

q1Vr1+h]

2)+ E[V2q1 V

2r1+h](C4qC4r − C

22qC

22r)

= C22qC22rVar[V

q1Vr1+h] + E[V

2q1 V

2r1+h](C4qC4r − C

22qC

22r).

For i 6= h,

Cov(Φ0,Φi) = Cov(Vq1Vr1+hC2qC2r , V

q1+iV

r1+h+iC2qC2r)

= C22qC22rCov(V

q1Vr1+h, V

q1+iV

r1+h+i),

while, for i = h

Cov(Φ0,Φh) = Cov(X2q1 X

2r1+h, X

2q1+hX

2r1+h+h)

= E[X2q1 X2(q+r)1+h X2r1+2h] − E[X

2q1 X

2r1+h]E[X

2q1+hX

2r1+2h]

= C2qC2rC2(r+q)E[Vq1V

(q+r)1+h V

r1+2h] − C

22qC

22rE[V

q1Vr1+h]

2.


Hence,

Σ∆(g2, g2) = C22qC22rVar[V

q1Vr1+h] + E[V

2q1 V

2r1+h](C4qC4r − C

22qC

22r)+ 2

∞∑i=1

C22qC22rCov(V

q1Vr1+h, V

q1+iV

r1+h+i)

− 2C22qC22rCov(V

q1Vr1+h, V

q1+hV

r1+2h)+ 2C2qC2rC2(r+q)E[V

q1V

(q+r)1+h V

r1+2h] − 2C

22qC

22rE[V

q1Vr1+h]

2

= C22qC22r(Var[V

q1Vr1+h] + 2

∞∑i=1

Cov(Vq1Vr1+h, V

q1+iV

r1+h+i))+ E[V

2q1 V

2r1+h](C4qC4r − C

22qC

22r)

− 2C22qC22r(E[V

2q1 V

2(r+q)1+h V

r1+2h] − E[V

q1Vr1+h]

2)+ C2qC2rC2(r+q)E[Vq1V

(q+r)1+h V

r1+2h] − C

22qC

22rE[V

q1Vr1+h]

2

= C22qC22r(Var[V

q1Vr1+h] + 2

∞∑i=1

Cov(Vq1Vr1+h, V

q1+iV

r1+h+i))

+ E[V2q1 V

2r1+h](C4qC4r − C

22qC

22r)+ 2E[V

q1V

(q+r)1+h V

r1+2h](C2qC2rC2(r+q) − C

22qC

22r).

(iii) By using similar arguments,

Cov(Φ10 ,Φ30 ) = Cov(X

2p1 , X

2u1 ) = E[X

2(p+u)1 ] − E[X2p1 ]E[X

2u1 ]

= E[Vp+u1 E[e2(p+u)1 ]] − E[V

p1E[e

2p1 ]]E[V

u1E[e

2u1 ]]

= C2(p+u)E[Vp+u1 ] − C2pC2uE[V

p1]E[V

u1],

Cov(Φ10 ,Φ3i ) = Cov(X

2p1 , X

2u1+i) = E[X

2p1 X

2u1+i] − E[X

2p1 ]E[X

2u1+i]

= E[Vp1Vu1+iE[e

2p1 e2u1+i]] − E[V

p1E[e

2p1 ]]E[V

u1+iE[e

2u1 ]]

= C2pC2uE[Vp1Vu1+i] − C2pC2uE[V

p1]E[V

u1+i]

= C2pC2u(E[V

p1Vu1+i] − E[V

p1]E[V

u1]

)and

Cov(Φ1i ,Φ30 ) = Cov(X

2p1+i, X

2u1 ) = E[X

2p1+iX

2u1 ] − E[X

2p1+i]E[X

2u1 ]

= E[Vp1+iV

u1E[e

2p1+ie

2u1 ]] − E[V

p1+iE[e

2p1+i]]E[V

u1E[e

2u1 ]]

= C2pC2uE[Vp1+iV

u1] − C2pC2uE[V

p1+i]E[V

u1]

= C2pC2u(E[V

p1+iV

u1] − E[V

p1]E[V

u1]

).

Hence,

Σ∆(g1, g3) = C2(p+u)E[Vp+u1 ] − C2pC2uE[V

p1]E[V

u1]

+ C2pC2u∞∑i=1

(E[V

p1Vu1+i] − E[V

p1]E[V

u1] + E[V

p1+iV

u1] − E[V

p1]E[V

u1]

)= C2(p+u)E[V

p+u1 ] − C2pC2uE[V

p1]E[V

u1] + C2pC2u

∞∑i=1

(E[V

p1Vu1+i] − 2E[V

p1]E[V

u1] + E[V

p1+iV

u1]

).

(iv) Again,

Cov(Φ10 ,Φ20 ) = Cov(X

2p1 , X

2q1 X

2r1+h) = E[X

2(p+q)1 X2r1+h] − E[X

2p1 ]E[X

2q1 X

2r1+h]

= E[Vp+q1 V

r1+hE[e

2(p+q)1 e2r1+h]] − E[V

p1E[e

2p1 ]]E[V

q1Vr1+hE[e

2q1 e2r1+h]]

= E[Vp+q1 V


p1]E[V

q1Vr1+h]C2pC2qC2r ,

Cov(Φ10 ,Φ2i ) = Cov(X

2p1 , X

2q1+iX

2r1+h+i) = E[X

2p1 X

2q1+iX

2r1+h+i] − E[X

2p1 ]E[X

2q1+iX

2r1+h+i]

= E[Vp1Vq1+iV

r1+h+iE[e

2p1 e2q1+ie

2r1+h+i]] − E[V

p1E[e

2p1 ]]E[V

q1+iV

r1+h+iE[e

2q1+ie

2r1+h+i]]

= C2pC2qC2r(E[Vp1Vq1+iV

r1+h+i] − E[V

p1]E[V

q1Vr1+h]), for i ≥ 1,

Cov(Φ1i ,Φ20 ) = Cov(X

2p1+i, X

2q1 X

2r1+h) = E[X

2p1+iX

2q1 X

2r1+h] − E[X

2p1+i]E[X

2q1 X

2r1+h]

= E[Vp1+iV

q1Vr1+hE[e

2p1+ie

2q1 e2r1+h]] − E[V

p1+iE[e

2p1+i]]E[V

q1Vr1+hE[e

2q1 e2r1+h]]

= C2pC2qC2r(E[Vp1+iV

q1Vr1+h] − E[V

p1]E[V

q1Vr1+h]), for i ≥ 1, i 6= h and


Cov(Φ1h ,Φ20 ) = Cov(X

2p1+h, X

2q1 X

2r1+h) = E[X

2q1 X

2(p+r)1+h ] − E[X

2p1+h]E[X

2q1 X

2r1+h]

= E[Vq1Vp+r1+hE[e

2(p+r)1+h e

2q1 ]] − E[V

p1+hE[e

2p1+h]]E[V

q1Vr1+hE[e

2q1 e2r1+h]]

= C2(p+r)C2qE[Vq1Vp+r1+h] − C2pC2qC2rE[V

p1+h]E[V

q1Vr1+h]

= C2(p+r)C2qE[Vq1Vp+r1+h] − C2pC2qC2rE[V

p1]E[V

q1Vr1+h], for i = h.

Hence,

Σ∆(g1, g2) = E[Vp+q1 V


p1]E[V

q1Vr1+h]C2pC2qC2r + C2pC2qC2r

∞∑i=1

(E[Vp1Vq1+iV

r1+h+i]

− 2E[Vp1]E[V

q1Vr1+h] + E[V

q1Vp1+iV

r1+h])+ (C2(p+r)C2q − C2pC2qC2r)E[V

q1Vp+r1+h].

Appendix D

(1) E[V31] is estimated by

115n

∑n1 X6i ; if g(x1) = x

61 with d = 1 then

1n

n−d+1∑i=1

X6i −→ E[g(√V1e1

)]= E[V1

3e61] = C6E[V1

3].

(2) E[V41] is estimated by

1105n

∑n−d+11 X81 ; if g(x1) = x

81 with d = 1 then

1n

∑n−d+1i=1 X8i −→ E[g(

√V1e1)] = E[V1

4e81] =

C8E[V14]

(3) E[V 1V21+j] is estimated by

13n

∑n−i+1i=1 X2i X

4i+j; if g(x1, x1+j) = x

21x41+j with d = j+ 1, then

1n

n−d+1∑i=1

X2i X4i+j −→ E

[g(√V1e1,

√V1+je1+j

)]= E[V1e21V1+j

2e41+j] = C2C4E[V1V1+j

2].

(4) E[V 1+jV21] is estimated by

13n

∑n−ji=1 X

4i X2i+j.

(5) E[V21V21+h] is estimated by

19n

∑n−hi=1 X

4i X4i+h.

(6) E[V 1V31+h] is estimated by

115n

∑n−hi=1 X

2i X6i+h.

(7) E[V 1+hV31] is estimated by

115n

∑n−hi=1 X

6i X2i+h.

(8) E[V 1V 1+jV 1+h+j] is estimated by 1n

∑n−h−ji=1 X2i X

2i+jX

2i+j+h; if g(x1, x1+j, x1+j+h) = x

21x21+j x

21+j+h with d = j+h+1, then

1n

n−j−h∑i=1

X2i X2i+jX

2i+j+h −→ E

[g(√V1e1,

√V1+je1+j,

√V1+j+he1+j+h

)]= E[V1e21V1+je

21+jV1+je

21+j] = C

32 E[V1V1+jV1+j+h].

(9) E[V 1V 1+hV 1+j] is estimated by 1n∑n−max(h,j)i=1 X2i X

2i+min(j,h)X

2i+max(j,h).

(10) E[V 1V21+hV 1+2h] is estimated by

13n

∑n−2hi=1 X

2i X4i+hX

2i+2h.

(11) E[V 1V21+jV 1+h] is estimated by

13n

∑n−max(j,h)i=1 X2i X

4i+jX

2i+h.

(12) E[V21V 1+jV 1+h+j] is estimated by

13n

∑n−j−hi=1 X4i X

2i+jX

2i+h+j.

(13) E[V12V 1+hV 1+k] is estimated by 1

3n

∑n−max(h,k)i=1 X4i X

2i+min(h,k)X

2i+max(h,k).

(14) E[V 1V21+hV 1+h+k] is estimated by

13n

∑n−h−ki=1 X2i X

4i+hX

2i+h+k.

(15) if k < hE[V 1V 1+h−kV21+h] is estimated by

13n

∑n−hi=1 X

2i X2i+h−kX

4i+h.

(16) E[V 1V 1+hV 1+jV 1+h+j] is estimated by 1n

∑n−h−ji=1 X2i X

2i+min(j,h)X

2i+max(j,h)X

2i+j+h; if g(x1, x1+j, x1+h, x1+j+h)

= x21x21+jx

21+hx

21+j+h with d = j+ h+ 1, then

1n

n−h−j∑i=1

X2i X2i+min(j,h)X

2i+max(j,h)X

2i+j+h −→ E[V1e21V1+je

21+jV1+he

21+hV1+je

21+j] = C

42 E[V1V1+jV1+jV 1+j+h].


References

Andersen, T.G., Benzoni, L., Lund, J., 2004, Stochastic volatility, mean drift, and jumps in the short term interest rate (manuscript). Paper n. 432 inEconometric Society 2004 North American Winter Meetings Series.

Barone Adesi, G., Rasmussen, H., Ravanelli, C., 2005. An option pricing formula for the GARCH diffusion model. Computational Statistics and Data Analysis49 (2), 287–310.

Barndorff-Nielsen, O.E., Shephard, N., 2001a. Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.(Eds.), Lévy Processes—Theory and Applications. Birkhäuser, Boston.

Barndorff-Nielsen, O.E., Shephard, N., 2001b. Non-Gaussian Ornstein–Uhlenbech-basedmodels and some of their use in financial economics. Journal of theRoyal Statistical Society B 63 (2), 167–241.

Barndorff-Nielsen, O.E., Shephard, N., 2002. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of theRoyal Statistical Society B 64 (2), 253–280.

Ben Hamida, S., Cont, R., 2005. Recovering volatility from option prices by evolutionary optimization. Journal of Computational Finance 8 (4), 43–76.Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659.Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Chan, K.C., Karolyi, A.J., Longstaff, F.A., Sanders, A.B., 1992. An empirical comparison of alternative models of the short-term interest rate. The Journal ofFinance 47 (3), 1209–1227.

Cont, R., 2001. Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1 (2), 223–236.Cont, R., 2006. Model uncertainty and its impact on derivative instruments. Mathematical Finance 16, 519–542.Ewald, C.O., Zhang, A., 2004, Calibration of stochastic volatility models with gradient methods and Malliaven calculus, Working paper.Feller, W., 1966. An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York.Genon-Catalot, V., Jeantheau, T., Laredo, C., 2000. Stochastic volatilitymodels as hiddenMarkovmodels and statistical applications. Bernoulli 6, 1051–1080.Heston, S.L., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of FinancialStudies 6, 327–343.

Hull, J., White, A., 1987. The pricing of options on assets with stochastic volatility. Journal of Finance 42 (2), 281–300.Kim, M.S., Wang, S., 2006. On the applicability of stochastic volatility models. Computational Statistics and Data Analysis 51 (4), 2210–2217.Lehmann, E.L., 1998. Elements of Large-Sample Theory. Springer.Lindström, E., Ströjby, J., Brodén, M., Wiktorsson, M., Holst, J., 2008. Sequential calibration of options. Computational Statistics and Data Analysis 52 (6),2877–2891.

Nelson, D.B., 1990. ARCH models as diffusion approximations. Journal of Econometrics 45, 7–38.Scott, L., 1987. Option pricing when the variance changes randomly: Theory, estimators and applications. Journal of Financial and Quantitative Analysis 22,419–438.

Stein, E., Stein, J., 1991. Stock price distributions with stochastic volatility. The Review of Financial Studies 4, 727–752.Wiggins, J., 1987. Option values under stochastic volatilities. Journal of Financial Economics 19, 351–372.

testing volatility autocorrelation in the constant elasticity of variance stochastic volatility...

Documents