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EstimatingtheWishartAffineStochasticCorrelationModelUsingtheEmpiricalCharacteristicFunction

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Electronic copy available at: http://ssrn.com/abstract=1054721

Estimating the Wishart Affine Stochastic Correlation Model

using the Empirical Characteristic Function∗

Jose Da Fonseca† Martino Grasselli‡ Florian Ielpo§

First draft: November 27, 2007

This draft: November 10, 2008

Abstract

This paper provides the first estimation strategy for the Wishart Affine StochasticCorrelation (WASC) model. We provide elements to show that the utilization of em-pirical characteristic function-based estimates is advisable: this function is exponentialaffine in the WASC case. We use a GMM estimation strategy with a continuum ofmoment conditions based on the characteristic function. We present the estimationresults obtained using a dataset of equity indexes. The WASC model captures mostof the known stylized facts associated with financial markets, including the leverageand asymmetric correlation effects.

Keywords: Wishart Process, Empirical Characteristic Function, Stochastic Correlation,

Generalized Method of Moments.

∗ Acknowledgements: We are particularly indebted to Marine Carrasco for remarkable insights andhelpful comments. We are also grateful to Christian Gourieroux, Fulvio Pegoraro, Francois-Xavier Vialardand the CREST seminar participants for useful remarks. We are thankful to the seminar participants ofthe 14th International Conference on Computing in Economics and Finance, Paris, France (2008), the 11thconference of the Swiss Society for Financial Market Research, Zurich (2008), Mathematical and StatisticalMethods for Insurance and Finance, Venice, Italy (2008), the 2nd International Workshop on Computa-tional and Financial Econometrics, Neuchatel, Switzerland (2008), the First PhD Quantitative FinanceDay, Swiss Banking Institute, Zurich (2008), Inference and tests in Econometrics, in the honor of RusselDavidson, Marseille, France (2008), the Inaugural conference of the Society for Financial Econometrics(SoFie), New York, USA (2008), the 28th International Symposium on Forecasting, Nice, France (2008),the ESEM annual meeting, Milano, Italy (2008), the Oxford-Man Institute of Quantitative Finance VastData Conference, Oxford, UK (2008), the Courant Institute Mathematical Finance seminar, New York,USA (2008) and the Bloomberg Seminar, New York, USA (2008) for their comments and remarks. Anyerrors remain ours.†Ecole Superieure d’Ingenieurs Leonard de Vinci, Departement Mathematiques et Ingenierie Financiere,

92916 Paris La Defense, France. Email: jose.da fonseca@devinci.fr and Zeliade Systems, 56, Rue Jean-Jacques Rousseau, 75001 Paris.‡Universita degli Studi di Padova , Dipartimento di Matematica Pura ed Applicata, Via Trieste 63,

Padova, Italy. E-mail: grassell@math.unipd.it and ESILV.§Pictet & Cie, Route des Acacias 60, CH-1211 Geneve 73. E-mail: florian.ielpo@ensae.org.

1

Electronic copy available at: http://ssrn.com/abstract=1054721

1 Introduction

The estimation of continuous time processes under the physical measure attracted a lot

of attention over the few past years, and several estimation strategies have been proposed

in the literature. When the transition density is known in closed form, it is possible to

perform a maximum likelihood estimation of the diffusion parameters, as presented e.g.

in Lo (1988). However, the number of models for which the transition density is known

in a closed form expression is somewhat limited. Moreover, the existence of unobserved

factors such as the volatility process in the Heston (1993) model makes it difficult – if not

impossible – to estimate such models using a conditional maximum likelihood approach.

A possible solution consists in discretizing and simulating the unobservable process: for

example, Duffie and Singleton (1993) used the Simulated Methods of Moments to esti-

mate financial Markov processes (methods of this kind are reviewed in Gourieroux and

Monfort, 1996). However, as pointed out in Chacko and Viceira (2003), even though these

methods are straightforward to apply, it is difficult to measure the numerical errors due

to the discretization. What is more, the computational burden of this class of methods

precludes its use for multivariate processes.

For the special class of affine models, another estimation strategy can be used. The

affine models present tractable exponentially affine characteristic functions that can in

turn be used to estimate the parameters under the historical measure. Singleton (2001)

and Singleton (2006) present a list of possible estimation strategies that can be applied

to recover these parameters from financial time series, using the characteristic function of

the process. Methodologies of this kind have been applied to one-dimensional processes,

like the Cox-Ingersoll-Ross process (e.g. Zhou, 2000), the Heston process and a mixture

of stochastic volatility and jump processes (e.g. Jiang and Knight, 2002, Rockinger and

Semenova, 2005 and Chacko and Viceira, 2003) and affine jump diffusion models (e.g.

Yu, 2004), yielding interesting results. Still, it involves additional difficulties. First, as

remarked in Jiang and Knight (2002) and Rockinger and Semenova (2005), numerically

integrating the characteristic function of a vector of the state variable is computationally

intensive. In our multivariate case the state variable is already a vector: for this type of

methodology, the integral discretization is likely to lead to numerical errors. Second, with

the Spectral GMM method presented in Chacko and Viceira (2003), the use of a more

limited number of points of the characteristic function settles the numerical problem, but

leads to a decrease in estimates efficiency. Carrasco and Florens (2000) and Carrasco

2

et al. (2007) present a method that uses a continuum of moment conditions built from the

characteristic function. With this method, the estimates obtained reach the efficiency of

the maximum likelihood method, thanks to the instrument used in this strategy. These

features make this methodology particularly well-suited for the estimation of affine multi-

variate continuous time processes.

Here, we propose to use a Spectral Generalized Method of Moments estimation strategy to

estimate the Wishart Affine Stochastic Correlation model, an affine multivariate stochas-

tic volatility and correlation model introduced in Da Fonseca et al. (2007). Based on the

previous models of Gourieroux and Sufana (2004), this affine model can be regarded as a

multivariate version of the Heston (1993) model: in fact, the volatility matrix is assumed

to evolve according to the Wishart dynamics (mathematically developed by Bru, 1991),

the matrix analogue of the mean reverting square root process. In addition to the Heston

model, it allows for a stochastic conditional correlation, which makes it very promising

process for financial applications. Buraschi et al. (2006) independently proposed a related

model corresponding to a constrained correlation version of the WASC model.

Multivariate stochastic volatility models have recently attracted a great deal of attention.

Bauwens et al. (2006) and Asai et al. (2006) present a survey of the existing models, along

with estimation methodologies. When compared to the previously mentioned processes,

several important differences must be underlined. (1) The volatility being a latent factor,

the observable state variable (the asset log returns) is not Markov anymore and the ML

efficiency cannot be reach. (2) What is more, as we discuss it in the paper, since the

process involves latent volatilities and correlations, the instrument must be set to be equal

to one for the usual GMM methodology to be used. This precludes the use of the Double

Index instruments procedure presented in Carrasco et al. (2007). (3) Since correlations

are also stochastic, there are more latent variables than in the stochastic volatility models,

making simulation-based methods useless. (4) Finally, the dimensionality of the problem

makes the characteristic function difficult to invert.

In view of these difficulties, we propose to estimate the WASC model using its charac-

teristic function, following an approach that is closed to the ones presented in Chacko

and Viceira (2003) and Carrasco et al. (2007). We present a Monte Carlo investigation of

the estimates’ behavior in a small sample and we discuss the empirical results obtained

using a real dataset composed of the prices of the SP500, FTSE, DAX and CAC 40. Our

3

results unfold as follow. (1) The estimated WASC parameters are comparable to what

is obtained in the univariate empirical literature. (2) Thanks to its ability to describe

dynamic correlation, the WASC model can encompass most of the desired features of fi-

nancial markets: it reveals asymmetric correlation and leverage effects. (3) Our estimates

reject systematically the particular correlation structure chosen by Gourieroux and Sufana

(2004) and Buraschi et al. (2006), favoring the flexibility of the specification presented in

Da Fonseca et al. (2007).

The paper is organized as follows. First we present the WASC process, along with the

computation of its conditional characteristic function. In Section 3, we present the esti-

mation methodology used in this paper and briefly review the main theoretical results.

Finally, in Section 4, we present the estimation results obtained with both simulated and

real datasets and discuss their interpretation.

2 The model

In this section, we present the Wishart Affine Stochastic Correlation model introduced in

Da Fonseca et al. (2007): we detail the diffusion that drives this multidimensional process

and present the conditional characteristic function together with its derivatives.

2.1 The dynamics

The Wishart Affine Stochastic Correlation (WASC) model is a new continuous time process

that can be considered as a multivariate extension of the Heston (1993) model, with a more

accurate correlation structure. The framework of this model was introduced in Gourieroux

and Sufana (2004). It relies on the following assumption.

Assumption 2.1. The evolution of asset returns is conditionally Gaussian while the

stochastic variance-covariance matrix follows a Wishart process.

In formulas, we consider a n-dimensional risky asset St whose risk-neutral dynamics are

given by

dSt = diag[St](µdt+

√ΣtdZt

), (1)

where µ is the vector of returns and Zt ∈ Rn is a vector Brownian motion. Following

Gourieroux and Sufana (2004), we assume that the quadratic variation of the risky assets

4

is a matrix Σt which is assumed to satisfy the following dynamics:

dΣt =(

ΩΩ> +MΣt + ΣtM>)dt+

√ΣtdWtQ+Q> (dWt)

>√Σt, (2)

with Ω,M,Q ∈ Mn, Ω invertible, and Wt ∈ Mn a matrix Brownian motion (> denotes

transposition). In the present framework we assume that the above dynamics are inferred

from observed asset price time series, hence the stochastic differential equation is written

under the historical measure.

Equation (2) characterizes the Wishart process introduced by Bru (1991), and represents

the matrix analogue of the square root mean-reverting process. In order to ensure the

strict positivity and the typical mean-reverting feature of the volatility, the matrix M is

assumed to be negative semi-definite, while Ω satisfies

ΩΩ> = βQ>Q, β > n− 1, (3)

with the real parameter β > n− 1 (see Bru, 1991 p. 747).

In full analogy with the square-root process, the term ΩΩ> is related to the expected long-

term variance-covariance matrix Σ∞ through the solution to the following linear equation:

−ΩΩ> = MΣ∞ + Σ∞M>. (4)

Moreover, Q is the volatility of the volatility matrix, and its parameters will be crucial in

order to explain some stylized observed effects in equity markets.

Last but not least, Da Fonseca et al. (2007) proposed a very special yet tractable corre-

lation structure that is able to accommodate the leverage effects found in financial time

series and option prices. Since it is well known that it is possible to approximately repro-

duce observed negative skewness within the Heston (1993) model by allowing for negative

correlation between the noise driving returns and the noise driving variance, they proposed

the following assumption:

Assumption 2.2. The Brownian motions of the asset returns and those driving the co-

variance matrix are linearly correlated.

Da Fonseca et al. (2007) proved that Assumption 2.2. leads to the following relation:

dZt = dWtρ+√

1− ρ>ρdBt, (5)

5

with Zt = (dZ1, dZ2, . . . , dZn)>, B a vector of independent Brownian motions orthogonal

to W , as defined in equation (2), and ρ = (ρ1, ρ2, . . . , ρn)>.

With this specification, the model is able to generate negative skewness, given the possibly

negative correlation between the noise driving the log returns of the assets and the matrix-

sized noise perturbating the covariance matrix. This is easy to show in the special case of

two assets (n = 2), fow which the variance-covariance matrix is given by

Σt =

Σ11t Σ12

t

Σ12t Σ22

t

. (6)

The correlations between assets’ returns and their volatilities admit a closed form expres-

sion, highlighting the impact of the ρ parameters on its value and positivity:

corr(d logS1, dΣ11

)=ρ1Q11 + ρ2Q21√

Q211 +Q2

21

(7)

corr(d logS2, dΣ22

)=ρ1Q12 + ρ2Q22√

Q212 +Q2

22

, (8)

where we recall that√

Σ11 (resp.√

Σ22) represents the volatility of the first (resp. second)

asset. Therefore, the sign and magnitude of the skew effects are determined by both the

matrix Q and the vector ρ. When Q is diagonal, we obtain the following skews for asset

1 and 2:

corr(d logS1, dΣ11

)= ρ1 corr

(d logS2, dΣ22

)= ρ2, (9)

thus allowing a negative skewness for each asset whenever ρi < 0,∀i (see Gourieroux and

Jasiak (2001) on this point). This correlation structure is similar the one obtained in an

Heston model.

Other less general specifications close to the WASC model, actually nested within the

WASC correlation structure, have been proposed in the literature. First, Gourieroux and

Sufana (2004) imposed ρ = 0n ∈ Rn, a choice that leads to a zero correlation case (see

equations (5), (7) and (8)) by analogy with the well known properties of the Heston model.

With this specification, the log returns’ univariate distribution is symmetric.

Second, Buraschi et al. (2006) proposed to impose ρ = (1, 0)>. Their model is thus able

to display negative skewness for asset 1 (resp. asset 2), depending on the positivity of

Q11 (resp. Q12). This model is actually close to the WASC and is able to display similar

6

features. Their choice of ρ is less restrictive than it seems, in so far as this parameter

is only defined up to a rotation. Thus, their hypothesis is reduced to ||ρ|| = 1. With

these settings, the vector-sized noise in the returns is fully generated by the Brownian

motions of the covariances W . This hypothesis having no a priori justification, the WASC

model of Da Fonseca et al. (2007) eliminates it, assuming the more general correlation

structure compatible with an exponential affine characteristic function (see Proposition 1

in Da Fonseca et al. (2007) on this point).

2.2 The Characteristic functions

In the WASC model, the characteristic function of Σt and Yt = logSt is an exponential

affine function of the state variables. For the log returns, the characteristic function of

Yt+τ conditional on Yt and Σt is denoted:

ΦYt,Σt(τ, ω) = E[ei〈ω,Yt+τ 〉|Σt, Yt

], (10)

where E[.|Σt, Yt] denotes the conditional expected value with respect to the historical

measure, ω ∈ Rn, i2 = −1 and 〈., .〉 is the scalar product in Rn.

Proposition 2.1. (Da Fonseca et al., 2007) The characteristic function of the asset re-

turns in the WASC model is given by

ΦYt,Σt(τ, ω) = E[ei〈ω,Yt+τ 〉|Σt, Yt

](11)

= exp Tr [A(τ)Σt] + 〈iω, Yt〉+ c(τ) , (12)

where ω = (ω1, . . . , ωn)> ∈ Rn and the deterministic function A(t) ∈Mn is as follows:

A (τ) = A22 (τ)−1A21 (τ) , (13)

with

A11 (τ) A12 (τ)

A21 (τ) A22 (τ)

= exp τ

M +Q>ρiω> −2Q>Q

12

((iω)(iω)> −

∑nj=1 iωjejj

)−(M> + iωρ>Q

) .

(14)

7

The function c(τ) can be obtained by direct integration, thus giving:

c(τ) = −β2

Tr[log (A22(τ)) + τM> + τiω(ρ>Q)

]+ τTr

[µiω>

]. (15)

The characteristic function of the Wishart process is defined as:

ΦΣt(τ,∆) = E[eiTr[∆Σt+τ ]|Σt

], (16)

where ∆ ∈Mn.

Proposition 2.2. (Da Fonseca et al., 2007) Given a real symmetric matrix D, the con-

ditional characteristic function of the Wishart process Σt is given by:

ΦΣt(τ,∆) = E[eiTr[∆Σt+τ ]|Σt

]= exp Tr [B(τ)Σt] + C(τ) , (17)

where the deterministic complex-valued functions B(τ) ∈Mn(Cn), C(τ) ∈ C are given by

B (τ) = (i∆B12 (τ) +B22 (τ))−1 (i∆B11 (τ) +B21 (τ)) (18)

C(τ) = −β2

Tr[log(i∆B12(τ) +B22(τ)) + τM>

],

with B11 (τ) B12 (τ)

B21 (τ) B22 (τ)

= exp τ

M −2Q>Q

0 −M>

.

What is more, these characteristic functions can be derived with respect to β and the

elements of M , Q and ρ, using the results of Daleckii (1974), on the derivative of a matrix

function. We provide detailed calculations of these derivatives in the Appendix.

3 Spectral GMM in the WASC setting

In this section, we present the detailed estimation methodology used for the WASC model.

In this paper, we are in the special setting where the correlation and variance processes are

unobserved. In this case, the feasible estimation strategies are: (1) to filtrate covariances

8

out of return time series either (a) using DCC estimates, (b) a GARCH-like discretization

of the continuous time process or (c) a linearized Kalman filter; (2) to estimate the process

using the conditional characteristic function. We favor the second type of methodologies

since (1) DCC-based estimates of the ρ parameter are biased1 and (2) any type of dis-

cretization or linearization will lead to additional estimation errors. In the WASC case,

the characteristic function is known in a closed form expression, thus being a very suitable

tool for the estimation of vector-processes, especially when compared to simulation-based

estimators. For further discussion on the estimation strategies of Wishart-based models,

see Gourieroux (2006).

Recent articles presented estimation methodologies using the empirical characteristic func-

tion as an estimation tool, since this function has a tractable expression for many con-

tinuous time processes. In this section, we present how to estimate the WASC in this

framework, building on the approaches developed in Chacko and Viceira (2003) and Car-

rasco et al. (2007).

The usual way to present the generalized method of moments based on spectral moment

conditions unfold as follows. Let ht be the conditional moment condition such that

ht = ei〈w,Yt+τ−Yt〉 −Xt, (19)

with the notations developed earlier and Xt a stochastic process such that E[ht|Yt] = 0.

Therefore Xt = E[ei〈w,Yt+τ−Yt〉|Yt]. Then the estimation can be based on unconditional

moment conditions of the form E[htg(Yt)] = 0. However, this approach can not be im-

plemented here because E[ei〈w,Yt+τ−Yt〉|Yt] does not have a known expression, principally

because the distribution of Σt given Yt is unknown. The solution we adopt is to use uncon-

ditional moment condition, which is equivalent to set the instrument g(Yt) equal to one,

as in Chacko and Viceira (2003) (see page 272)2. This setting stems from the fact that we

integrate the volatility out when computing X. Were Σt observable, a more general form

of instruments would be readily used.

‘

Now ht is simply

ht = ei〈w,Yt+τ−Yt〉 − E[ei〈w,Yt+τ−Yt〉|Σ0] (20)1We ran Monte Carlo test to prove this point empirically. The tables are available upon request.2We thank Marine Carrasco for pointing out this fact.

9

where the initial value of Σ0 is treated as an unknown parameter to be estimated. We

have

E[ei〈w,Yt+τ−Yt〉|Σ0] = ec(τ)E[e〈A(τ),Σt〉|Σ0] = ec(τ)ΦΣ0(t,−iA(τ)), (21)

with c(τ) defined in equation (15), A(τ) defined in equation (13) and ΦΣt(.) defined in

equation (17) (the conditional expectation (21) can also be computed using twice the

function (10)).

In order to increase the efficiency of our estimates, we use a continuum of moment condi-

tions, as presented in Carrasco et al. (2007). Note that the fact that we set the instruments

to be equal to 1 naturally prevents us from reaching the ML efficiency of CGMM estimates

of Carrasco et al. (2007). Anyway, Yt conditionally upon its past is no longer a Markov

process, since the covariance matrix is unobservable. ML efficiency cannot be reach for

non-Markov process: the special instruments chosen here does not necessarily jeopardize

the estimation results.

Let now h(.) be the sample mean of the moment condition, that is a function from R2n to

C. In an infinite conditions framework, Carrasco et al. (2007) showed that the objective

function to minimize is:

θ = arg minθ‖K−1/2h(θ)‖, (22)

where K is the covariance operator, that is the counterpart of the covariance matrix in

finite dimension – as in standard GMM approach and ‖.‖ is the weighted norm

‖f‖2 =∫

Rn

∫Rnf(ω)f(ω)π(ω)dω

where π denotes any probability measure. As in Carrasco et al. (2007), we chose it to be

the normal distribution. Carrasco et al. (2007) showed that the covariance operator K

can be written as follows:

Kf(ω) =∫k(ω, τ)f(τ)π(τ)dτ

where the function k is the so called kernel of the integral operator K and is defined by:

k(ω, τ) =+∞∑j=−∞

Eθ0(ht(ω; θ0)ht−j(τ ; θ0)

).

Since our approach is nested within the Carrasco et al. (2007)’s, we now thoroughly follow

their settings. Our approach can also be related to the methodology presented in Rockinger

10

and Semenova (2005). In order to construct an estimator of the covariance operator,

Carrasco et al. (2007) proposed a two-step procedure. The first step consists in finding:

θ1 = arg minθ‖h(θ)‖. (23)

Since ht is autocorrelated the second step consists in estimating the kernel k as follows:

k(ωs, ωr, ωv, ωw) =T

T− q

T−1∑j=−T+1

ω

(j

ST

)ΓT(j), (24)

with

ΓT(j) =

1T

∑Tt=j+1 ht(ωs, ωr, θ1)ht−j(ωv, ωw, θ1), j ≥ 0

1T

∑Tt=−j+1 ht+j(ωs, ωr, θ1)ht(ωv, ωw, θ1), j < 0,

, (25)

where w(.) is any kernel satisfying some regularity conditions (see Carrasco et al. (2007)

Appendix A.6) and ST is a bandwidth parameter.

Once the covariance operator is estimated, the minimization in equation (22) requires

the computation of the inverse of K. Unfortunately, K has typically a countable infinity

of eigenvalues decreasing to zero, so that its inverse is not bounded. We need then to

regularize the inverse of K, which can be done by replacing K by a nearby operator that

has a bounded inverse, due to the presence of a penalizing term. Carrasco et al. (2007)

used the Tikhonov approximation of the generalized inverse of K. Let α be a strictly

positive parameter, then K−1 is replaced by:

(Kα)−1 = (K2 + αI)−1K. (26)

As outlined in Carrasco et al. (2007), the choice of α is important but does not jeopardize

the consistency of the estimates. Carrasco and Florens (2000) investigated an empirical

method to select its value, and the optimal value for it should represent a trade-off be-

tween the instability of the generalized inverse (for small values of α) and the distance

from the true inverse as α increases. Furthermore we found much more convenient to

compute (Kα)−1 using the Cholesky’s decomposition than the spectral decomposition: it

is sufficient for the evaluation of equation (22) and avoids the numerically difficult problem

of eigenvectors computation and requires the discretization of the integrals.

Under mild regularity conditions (conditions A.1. to A.5. in Carrasco et al., 2007), it can

be proved that the optimal C-GMM estimator of θ is obtained by:

θ = arg minθ‖(Kα

T )−1/2hT (θ)‖ (27)

11

and is asymptotically Normal with

√T (θT − θ0) L→ N

(0, (〈Eθ0(∇θh), Eθ0(∇θh)〉K)−1

), (28)

as T and T aα5/4T go to infinity and α goes to zero. (∇θh denotes the Jacobian matrix of

h(.)).

Finally, it is important to mention that Carrasco et al. (2007) present a matrix-based

version of their estimation method that may be more appealing than the one presented

here for a WASC model based on more than two assets or for other models.

4 Empirical Results

We now review the empirical results obtained with the aforementioned estimation method-

ology. First, we provide insight into the model and the parameters interpretation. Then

we review the results of a Monte Carlo experiment investigating the empirical behavior

of the estimation methodology. Finally, we present the estimates obtained using equity

indexes and discuss the results obtained.

Before moving to the detailed presentation of the results, it is noteworthy to mention that

with this type of model, no forecasting exercise can be performed for two main reasons.

First, with this kind of continuous time stochastic covariances process and the chosen

estimation strategy, we are restricted to the estimation of the parameters driving the

process: we cannot filtrate correlation or volatility time series out of returns and hence

forecast these quantities. Second, since the volatility and correlation are unobserved on

financial markets, it would naturally be impossible to compare – when existing – any

forecast to ”true” values. For these reasons, we cannot perform any test of the model

based on forecasts.

4.1 Preliminary considerations

Unlike the Heston (1993) model, the Wishart Affine Stochastic Correlation model is a new

model for which the parameters interpretation is not immediate. Such an interpretation is

however essential to the understanding of the model and for its estimation. For the sake of

simplicity, we focus on the case where n = 2, i.e. the case for which we observe two assets.

Yt is the vector containing the log of the asset prices, and Σt is its covariance matrix given

12

by equation (6). Y 1t being the log return of the first asset, its volatility is given by

√Σ11t .

In the WASC framework, individual parameters can hardly be interpreted on their own:

on the contrary, combinations of these parameters have standard financial interpretations,

such as the mean-reverting parameter or the volatility of volatility. Now, we review the

computation of these quantities.

For the first asset, the quadratic variation of the volatility can be computed as follow:

d〈Σ11,Σ11〉t = 4Σ11t (Q2

11 +Q221)dt. (29)

Therefore the first column of Q parametrizes the volatility of volatility of the first asset.

Similar results can be obtained for the second asset.

Then, as presented in Section 2,

corr(dY1, dΣ11

)=Q11ρ1 +Q21ρ2√

Q211 +Q2

21

, (30)

where corr(.) is the correlation coefficient. As already mentioned, the short term behavior

of the smile and the skewness effect heavily depend on the correlation structure given by

the vector ρ. If Q and ρ are such that this quantity is negative, then the volatility of S1 will

rise in response to negative shocks in returns of this asset. We expect this correlation to

be large and negative, in order to account for the large skewness found in financial datasets.

The Gindikin coefficient β insures the positiveness of the Wishart process. What is more,

an increase of it will shift the distribution of the smallest eigenvalue to positive values.

Thus, this parameter can be interpreted as a global variance shift factor. From equations

(3) and (4), if β is multiplied by a factor α, the long term covariance matrix Σ∞ will

be multiplied by the same factor. β also impacts the mean reverting and variance of the

correlation process. The higher the β parameter and the lower the persistence and the

variance of the correlation process. Thus, there is a trade-off in the WASC model between

volatility of the returns and volatility of the correlation process.

The M matrix can be compared to the mean reverting parameter in the Cox-Ingersoll-

Ross model. Like for the parameters previously investigated, the elements of this matrix

can hardly be interpreted directly. However, we can compute in a closed form expression

13

the drift part of the dynamics of Σij . In the case of the first asset:

dΣ11t = . . .+ Σ11

t

[2M11 + 2M12

√Σ22t√

Σ11t

ρ12t

]+ . . . , (31)

where ρ12t is the instantaneous correlation between the log-returns of the two assets. Thus,

the mean reverting parameter for Σ11 is a combination of the elements of M . What is

more, this drift term is made of two parts: a deterministic part (2M11) and a stochastic

correction (2M12

√Σ22t√

Σ11t

ρ12t ), linked to the joint dynamics of both assets. Thus, the drift

term of Σ11t is influenced by one of the off-diagonal elements of M . This feature cannot

be replicated by most of the multivariate GARCH-like models. We can perform similar

calculations for Σ12t and Σ22

t . These quantities can then be used to compare the half life

of the variances and covariance processes and thus evaluate their relative persistence in

financial markets.

The instantaneous correlation between assets has also a closed form expression:

dρ12t =

(At(ρ12t

)2 +Btρ12t + Ct

)dt+

√1−

(ρ12t

)2(...)d(Noiset) (32)

with At, Bt, Ct recursive functions of Σ11t , Σ22

t and the model parameters. We present the

drift coefficients and the diffusion term in the Appendix. The drift associated to the cor-

relation is quadratic, and the linear term has a negative coefficient Bt < 0, thus presenting

the typical mean reverting behavior of ρ12t (at least around zero where the quadratic part

is negligible). The linear part can thus be used to analyze the persistence of the correlation

and its mean-reverting characteristics, during low correlation periods. When the absolute

value of the correlation is higher, the quadratic part of the drift get the upper hand and

the correlation process looses most of its persistence. By comparing the values of Bt and

At, when can thus compare the correlation behavior during low and high correlation cy-

cles. This information has not been documented until now, whereas it is important to

understand the joint behavior of financial assets.

The WASC model can also be used to investigate potential contagion effects in financial

markets. By computing the correlation between the correlation process and the returns, we

can discuss under which condition the model is able to display an asymmetric correlation

effect3. Asymmetric correlation effect leads correlation to go up whilst returns are getting3On asymmetric correlation effects, see Roll (1988) and Ang and Chen (2002).

14

down. As already noticed in Da Fonseca et al. (2007), we have:

d〈Y 1, ρ12〉t =

√Σ11t

Σ22t

(1−(ρ12t

)2)× (Q12ρ1 +Q22ρ2)︸ ︷︷ ︸Sign of asset 2 skew

. (33)

Thus, the sign of the skews determines the one of the covariance between correlation and

returns. Were the skew to be negative and the model would also display increases in

the correlation following negative returns. Thus, the WASC model is also able to display

an asymmetric correlation effect, whose sign is driven by the skewness associated to the

returns. Since the asset returns are negatively correlated to their own volatility (leverage

effect), we thus expect volatilities to be positively correlated to correlation: negative

returns periods correspond to both higher correlation and higher volatility periods. In

fact, simple computations given in Appendix lead to

d⟨ρ12,Σ11

⟩t

=

√Σ11t

Σ22t

(1−

(ρ12t

)2)Q12

(Q11 + Q22

)dt.

where Q is the symmetric positive definite matrix associated to the polar decomposition

of Q4. A positive value for Q12 would mean that the WASC model is able to accomodate

stylized effects of the type mentioned earlier. Due to the increase in the drift term of the

correlation dynamics, situation of this kind are expected not to last for long.

We now turn our attention toward a series of Monte Carlo experiments, so as to investigate

the empirical performance of the chosen estimation strategy.

4.2 Monte Carlo study

Following Carrasco et al. (2007), we present the results of a Monte Carlo study of the

CGMM estimation methodology applied to the WASC. We first present the technical de-

tails of the simulation and then we review the results obtained.

For the ease of the presentation, we restrict to the two-assets case. The parameters used4Any invertible matrix Q can be uniquely written as the product of a rotation matrix and a symmetric

positive definite matrix Q, see the Appendix.

15

in the simulation are the following:

Σ0 =

0.0225 −0.0054

−0.0054 0.0144

(34)

M =

−5 −3

−3 −5

(35)

ρ =[

0.3 0.4]

(36)

Q =

0.1133137 0.03335871

0.0000000 0.07954368

(37)

β = 15. (38)

The Q matrix is obtained by inverting the relation that links Q to M , Σ∞ and β:

Q>Q = − 1β

MΣ∞ + Σ∞M>

. (39)

This ensure the stationarity of the correlation process. When Q is selected arbitrary and

given the mean reverting property of Σt, the first part of the simulated sample will be

tainted by the collapse of the process toward its long term average. Situations of this

kind should be discarded. The Figure 1 presents a simulated path for both volatilities

and correlation, using the previous parameter values. The figure displays mean-reverting

dynamics for each of these moments.

The Figure 2 shows the characteristic function used in the spectral GMM method used in

the paper, as presented in equation (??). The grid used for the numerical integration of

the objective function ranges on the real line from -300 to 300. We used Gaussian kernels

with appropriate variance parameter to maintain as much information as possible. The

integral is computed numerically using the Trapezoidal Rule that seemed to performed

well over the simulated dataset. The objective function is minimized using a simulated

annealing method, as described in Belisle (1992).

We present the results of different Monte Carlo experiments. Each of them comes out after

1000 iterations, but they differ by the length of the simulated sample and the sampling

frequency: daily, weekly and monthly. For each sampling frequency, we used two different

samples, one of which contains 500 observations and the other one 1500 observations. The

Table 6.2 presents the Mean Bias and the Root Mean Square Error (RMSE) obtained. We

did not reported the median bias insofar it was close to the median bias, thus indicating

16

that the estimators have a symmetric empirical distribution.

The results can be analyzed as follows. The Monte Carlo results obtained for Q, β and ρ

show that an increase in the sample depth globally results in a reduction of the variance of

the estimates. The bias obtained are small and not significative. For the weekly frequency,

ρ1 displays a noticeable difference as the variance of the estimate grows with the sample

size. This feature will have to be considered when analyzing the real dataset-based esti-

mation results. The M parameter also presents this variance increase feature. However,

this behavior is not very surprising: a large number of articles emphasize the difficulties

involved by the estimation of the mean reverting parameter in continuous time diffusions

(see e.g. Gourieroux and Monfort, 2007). The Monte Carlo results indicate that this mean

reverting parameter is estimated with less volatility with daily series. What is more, di-

agonal elements (resp. off-diagonal elements) of M are estimated with a small positive

(resp. negative) bias and thus may be underestimated (resp. overestimated) when working

with a real-time dataset. Finally, the correlation vector displays a remarkably small bias

and small RMSE for the daily datasets, even in the small sample version. This fact is

somewhat constant for each sampling frequency. This point is important for the WASC

model, given that we are interested in the analysis of the fine correlation structure implicit

in asset dynamics.

We now detail the empirical results obtained with stock indexes.

4.3 Estimation on stock indexes

In this last subsection, we present the empirical results obtained when estimating the

WASC using the C-GMM method on a real dataset. We used the following stock indexes:

SP500, FTSE, DAX and CAC 40. For each stock, the time series starts on January 2nd

1990 and ends on June 30th 2007. This period excludes the 1987 crash and the subprime

crisis. It nonetheless includes a lot of financial turmoils, as pointed in Rockinger and

Semenova (2005). The table 2 presents the descriptive statistics for the sample used in the

estimation, at a weekly sampling frequency. We used daily and weekly time series. We dis-

carded the use of monthly ones since the sample would be far too small. In many articles

devoted to the estimation of continuous time models, the change in the sampling frequency

usually leads to an interesting analysis of the subtle dynamics of financial markets (see

e.g. Chacko and Viceira, 2003). Since the characteristic-function based estimators do

17

not suffer from discretization errors, we can actually use any sampling frequency. Like in

Carrasco et al. (2007), α = 0.02 were found to perform well. The integration grid is the

same as the one used for the previous simulation exercises. We chose to use the Bartlett

kernel for the GMM covariance matrix estimation, following the procedure presented in

Newey and West (1994).

For numerical sake, we focus again on a two-assets case (n = 2). We estimated the pa-

rameters driving the WASC process for the following couples of indexes: (SP500,FTSE),

(SP500,DAX), (SP500,CAC), (DAX/CAC), (DAX,FTSE) and (FTSE/DAX). This way,

we will be able to compare the characteristics specific to individual stock while estimated

with each of the others. For example, we will be able to compare the volatility of volatility

of the SP500, when it is estimated with the DAX, CAC and FTSE as a second asset. It

will highlight the impact of joint dynamics on idiosyncratic behaviors, which has hardly

be documented until now.

The estimation results are presented in table 3 for the daily results and in table 4 for the

weekly results. Most of the estimates are significative up to a 5 or 10% risk level. What

is more, in the weekly observation case, the size of the sample is strongly reduced and so

is the efficiency of the estimation method. Nevertheless, the estimation results yield in-

teresting information both about the WASC process and the dynamics of the stock indexes.

As presented in the previous subsection, it is difficult to compare the individual parameters

and we will thus focus on combinations of these parameters most of which are comparable

with the ones of the Heston (1993) model.

For the estimated parameters presented in Table 3, the associated volatility of volatility

are presented in Table 5. The estimation of this quantity is essential to test the ability

of the model to capture financial market features: as pointed out in Chacko and Viceira

(2003), this parameter controls the kurtosis of the underlying process. Several remarks can

be made. First, the global results match what is generally expected from stock indexes.

Such markets are known to lead to a volatility of volatility ranging from 5% to 25%.

Second, the results obtained for the SP500 are remarkably stable when the second asset

changes at least for the daily sample: it ranges from 14.6% to 24.4%, thus matching the

results obtained in Eraker et al. (2003). Still, it is below the estimates obtained in Chacko

and Viceira (2003) and Rockinger and Semenova (2005): this may be explained by the

18

fact that the model that is estimated here is multivariate, whereas existing attempts to

capture stochastic volatility has been made in a univariate framework. Da Fonseca et al.

(2007) showed that the correlation between the volatility of each stock is non 0 insofar as

d〈Σ11,Σ22〉 = Σ12dt. (40)

Hence, whenever Σ12 is positive, the WASC model is able to model volatility transmis-

sion phenomena among assets. It is noteworthy to remark that these results are globally

stable across the datasets and close to the existing results. Third, when the sampling

frequency reduces, the volatility of volatility parameter globally increases. The few lacks

of consistency for this fact may be due to the fact that the number of observations in the

weekly dataset is far below the one used in the daily dataset. These results are different

from those obtained in Chacko and Viceira (2003). However, this is in line with what is

observed for the volatility of the log-returns when reducing the sampling frequency. This

divergence may also be explained by the effect of the correlation between variances that

cannot be mimicked in a Heston-like framework.

Now, let us discuss an important parameter for the specification of the WASC model, that

is the correlation between the returns and the volatility. We already mentioned that this

parameter is essential to have a model that is consistent with many stylized facts, such as

negative skewness and thus skewed implied volatility surfaces. It can be computed using

its expression given in equation (30). The Table 6 displays the results obtained.

This time, we have results that are comparable to the one obtained in the existing liter-

ature, and especially for the SP500. The correlation for this index is reported in the first

line of the previous Table. In the literature, it actually ranges from -0.27 (Rockinger and

Semenova, 2005) to -0.62 (Chacko and Viceira, 2003), which is close to what is obtained

here. The parameter values obtained for the CAC, DAX and FTSE indexes are not sur-

prising either, since their sign is negative. The main problem here lies in the instability

of the ρ parameter for the different estimation involved, when comparing both the sample

frequency and the couple of indexes that are estimated. The change in sampling frequency

does not lead to a similar behavior across the datasets: depending upon the couples and

the sampling frequency, the skewness in the dataset can considerably differ, underlining

the fact that correlation processes implicit in financial markets are complex. Beyond the

remarks made in the previous paragraph on the importance of the dataset depth, we also

emphasize that the computation of this parameter dwells on the inverse of the square root

19

of a quantity that is small. In this situation, the inverse of something small can be found

to be very variable: any error in the estimation of Q11 and Q21 will have a strong impact

on 1√Q2

11+Q221

. Thus, this skewness quantity must be cautiously interpreted. Last but not

least, since the skews are negative, the fitted WASC models also display asymmetric cor-

relation effects: negative returns are likely to be followed by a higher correlation between

the two assets.

We now turn our attention toward the mean reverting matrix M . For SP500, CAC, DAX

and FTSE, we find the same structure for the matrix M . They are definite negative thus

ensuring a mean reverting behavior for the Wishart process and have positives off diagonal

terms. As presented in the previous subsection, the drift can be decomposed into two dif-

ferent part: an idiosyncratic part (denoted κ1 in the tables) and a joint part (denoted κ2 in

the tables). For univariate stochastic volatility models, the estimation results usually lead

to an estimate of κ = κ1 + κ2, that is the sum of the two preceding elements. Thanks to

the complexity of the WASC process, we are now able to disentangle and analyse these two

different elements. The estimation results are reported in table 7. The κ values should

be compared with the mean reverting value of the Heston. We are close to Rockinger

and Semenova (2005) results who found 6.3352 (see their Table 1) for the S&P500, even

though their results are obtained on a different sample and using daily data. However,

when analysing κ1 and κ2, we find that the idiosyncratic mean reverting component is

always higher than the usual Heston parameter. This idiosyncratic element is dampened

by the negative joint mean reverting component: its negativity is to be related to the

negative non-diagonal elements of the estimated M matrices. Again, when the sampling

frequency changes, each of these values vary, suggesting that the mean reverting parame-

ter associated to the volatility strongly depends on the sampling frequency, as pointed out

in Chacko and Viceira (2003). Globally, the associated half lives are around one month,

which is a realistic value.

As presented in the previous section, it is possible to perform similar computations for

the drift term of the stochastic correlation. This drift is a non linear function of ρ12t and

the usual comments have to be adapted. We present in figure 3 and 4 the instantaneous

variation of ρ12t as a function of ρ12

t , highlighting the contribution of the quadratic term

when the correlation gets very high – that is during crisis period. Our results suggest

that the correlation process is much more persistent than volatility when the correlation is

below its long term level, since in such a case its mean reverting parameter can be reduced

20

to (−Bt): in this situation, the half life is around 2 months, which is again realistic. When

correlation is high, the quadratic term gets higher and the persistence goes down, since

At is added to Bt as both these elements are negative. This is consistent with what is

empirically observed during financial market crises: the correlation gets very high on a

very short period, to finally go back to its long term behavior rapidly. The aforementioned

figures display reaction functions of this kind, underlining the ability of the WASC model

to encompass this standard feature of financial markets.

Another quantity of importance is ||ρ||, since the WASC can be seen as a generalization

of the processes proposed in Gourieroux and Sufana (2004) and Buraschi et al. (2006),

with a more complex correlation structure. Since the WASC model is only defined up to

a rotation matrix, the model presented in Buraschi et al. (2006) encompasses any correla-

tion structure that satisfies ||ρ|| = 1. Testing such an assumption is thus of a tremendous

importance to judge whether the complexity of the WASC is empirically justified. The

table 10 presents the norm of this vector parameter. Each of the estimated value strongly

differs from 1, suggesting the general correlation structure imposed in Da Fonseca et al.

(2007) is empirically grounded.

As presented earlier, a contagion effect can be handled by the WASC through a positive

value for Q12. In table 11 we report the estimated values for this parameter. We found

positive values for all couples of indexes as expected. Therefore, the estimated WASC

model is able to detect the existence of potential contagion effects in the dataset. As

mentioned earlier, these findings may be due to the fact that the dataset includes several

financial crises, periods during which dramatic contagion effects are expected.

5 Conclusion

In this paper we investigated the estimation of a new continuous time model: the Wishart

Affine Stochastic Correlation model, presented in Da Fonseca et al. (2007). After hav-

ing presented the problem that arise when trying to estimate a discrete version of this

model, this paper proposes to estimate the process using its exponential affine character-

istic function. The estimation method uses a continuum of spectral moment conditions

in a GMM framework. After a preliminary Monte Carlo investigation of the estimation

methodology, we show that real-dataset estimation results bring support to the WASC

21

process. First, the empirical results are comparable to those obtained in the literature

(when comparable). Second, the general correlation structure of the WASC casts light on

not-so-well documented features of international equities, allowing us to discuss for exam-

ple the persistence of the correlation process, contagion effects or asymmetric correlation

effects. Third, the generality of the correlation structure is not rejected by the dataset,

bringing empirical support to the model presented in Da Fonseca et al. (2007).

References

Ang, A. and Chen, J. (2002). Asymmetric Correlations of Equity Portfolios. Journal of

Financial Economic, (63):443–494.

Asai, M., McAleer, M., and Yu, J. (2006). Multivariate Stochastic Volatility: a Review.

Econometric Review, 25(2-3):145–175.

Bathia, R. (2005). Matrix Analysis. Graduate Texts in Mathematics, Springer-Verlag.

Bauwens, L., Laurent, S., and Rombouts, J. (2006). Multivariate garch models: a survey.

Journal of Applied Econometrics, 21(1):79–109.

Belisle, C. J. P. (1992). Convergence Theorems for a Class of Simulated Annealing Algo-

rithms. Rd J Applied Probability, 29:885–895.

Bru, M. F. (1991). Wishart Processes. Journal of Theoretical Probability, 4:725–743.

Buraschi, A., Porchia, P., and Trojani, F. (2006). Correlation risk and optimal portfolio

choice. Working paper, SSRN-908664.

Carrasco, M., Chernov, M., Florens, J.-P., and Ghysels, E. (2007). Efficient Estimation

of Jump Diffusions and General Dynamic Models with a Continuum of Moment Condi-

tions. Journal of Econometrics, (140):529–573.

Carrasco, M. and Florens, J. (2000). Generalization of GMM to a Continuum of Moment

Conditions. Econometric Theory, (16):797–834.

Chacko, G. and Viceira, L. M. (2003). Spectral GMM Estimation of Continuous-Time

Processes. Journal of Econometrics, 116(1-2):259–292.

Da Fonseca, J., Grasselli, M., and Tebaldi, C. (2007). Option Pricing when Correlations

are Stochastic: an Analytical Framework. Review of Derivatives Research, 10(2):151–

180.

22

Da Fonseca, J., Grasselli, M., and Tebaldi, C. (2008). A Multifactor Volatility Heston

Model,. Quantitative Finance, 8(6):591–604. An earlier version of this paper circulated

in 2005 as ”Wishart multi–dimensional stochastic volatility”, RR31, ESILV.

Daleckii, J. (1974). Differentiation of Non-Hermitian Matrix Functions Depending on a

Parameter. AMS Translations, 47(2):73–87.

Daleckii, J. and Krein, S. (1974). Integration and Differentiation of Functions of Hermitian

Operators and Applications to the Theory of Perturbations. AMS Translations, 47(2):1–

30.

Donoghue, W. J. (1974). Monotone matrix functions and analytic continuation. Springer-

Verlag.

Duffie, D. and Singleton, K. (1993). Simulated Moments Estimation of Markov Models of

Asset Prices. Econometrica, 61:929–952.

Eraker, B., Johannes, M., and Polson, N. (2003). The Impact of Jumps in Volatility and

Returns. The Journal of Finance, 58(3):1269–1300.

Faraut, J. (2006). Analyse sur les groupes de Lie. Calvage & Mounet.

Gourieroux, C. (2006). Continuous Time Wishart Process for Stochastic Risk. Econometric

Review, 25(2-3):177–217.

Gourieroux, C. and Jasiak, J. (2001). Financial Econometrics. Princeton University Press.

Gourieroux, C. and Sufana, R. (2004). Derivative Pricing with Multivariate Stochastic

Volatility: Application to Credit Risk. Les Cahiers du CREF 04-09.

Hall, B. C. (2003). Lie Groups, Lie Algebras, and Representations: An Elementary intro-

duction. Graduate Texts in Mathematics 222, Springer-Verlag.

Heston, S. (1993). A Closed-Form Solution for Options with Stochastic Volatility with

Applications to Bond and Currency Options. The Review of Financial Studies, 6(2).

Jiang, G. J. and Knight, J. L. (2002). Estimation of Continuous-Time Processes via

the Empirical Characteristic Function. Journal of Business & Economic Statistics,

20(2):198–212.

Lo, A. W. (1988). Maximum Likelihood Estimation of Generalized Ito Processes with

Discretely Sampled Data. Econometric Theory, 4:231–247.

23

Newey, W. K. and West, K. D. (1994). Automatic lag selection in covariance matrix

estimation. Review of Economic Studies, 61(4):631–53.

Rockinger, M. and Semenova, M. (2005). Estimation of Jump-Diffusion Process via Empir-

ical Characteristic Function. FAME Research Paper Series rp150, International Center

for Financial Asset Management and Engineering.

Roll, R. (1988). The International Crash of October, 1987. Financial Analysts Journal,

(September-October):19–35.

Singleton, K. (2001). Estimation of Affine Pricing Models Using the Empirical Character-

istic Function. Journal of Econometrics, (102):111–141.

Singleton, K. J. (2006). Empirical Dynamic Asset Pricing: Model Specification and Econo-

metric Assessment. Princeton University Press.

24

6 Appendix

6.1 Computing the gradient

The gradient of the characteristic function is needed to study the asymptotic distribution

of the estimates but also in the optimization process underlying the estimation procedure.

Therefore we turn our attention to the differentiation of matrix function depending on a

parameter. We illustrate the theoretical framework with the characteristic function of the

assets’ log returns and we give without technical details the results for the forward char-

acteristic function needed in our empirical study. We mainly rely on the work of Daleckii

(1974) for the general case (i.e. in the non-Hermitian matrix case) and to Daleckii and

Krein (1974), Donoghue (1974) and Bathia (2005) for the Hermitian matrix case.

Let us first state some basic results on linear algebra. Denote by λi; i = 1..n the set of

eigenvalues of a matrix X ∈Mn and mi the multiplicity of λi as a root of the characteristic

polynomial of X. Define Li = Ker(X − λiI) and Pi the projection operator from Cn onto

Li, then we have∑n

i=1 Pi = I. Define also Ji such that (X − λiI)Pi = Ji. The Jordan

normal form of X if given by the well known decomposition X =∑n

i=1(λiPi + Ji).

Let f be a function from Mn into Mn: the derivative of f at X in direction H, denoted

Df,X(H), is by definition ‖f(X+tH)−f(X)−Df,X(H)‖ = to(‖H‖) and can be computed

using the following formula Daleckii (1974):

Df,X(H) =n∑

j1,j2

mj1−1∑r1=0

mj2−1∑r2=0

1r1r2

∂r1+r2

∂λr1j1∂λr2j2

[f(λj1)− f(λj2)

λj1 − λj2

]Pj1J

r1j1HPj2J

r2j2. (41)

Remark 6.1. Whenever λj1 = λj2 the term within the bracket should be replaced by

f ′(λj1).

Remark 6.2. When X can be diagonalized then mj = 1 for each j and we are lead to the

very simple form

Df,X(H) =n∑

j1,j2

[f(λj1)− f(λj2)

λj1 − λj2

]Pj1HPj2 . (42)

If X is Hermitian then P−1 = P ∗ and we recover the result presented for example in

Daleckii and Krein (1974).

25

Simple algebra leads to Df,X(H) = PMf (P−1HP )P−1 where P is the matrix of the

eigenvectors of X5, is the Schur product6 and Mf = (Mf (λk, λl))k=1...n,l=1...n is the

Pick matrix associated to the function f , which is defined by

Mf (λ, µ) =

f(λ)−f(µ)

λ−µ if λ 6= µ

f ′(λ) if λ = µ(43)

This formulation is well known and can be found for example in Donoghue (1974) p. 79

and Bathia (2005) p123-124.

The gradient of the characteristic function involves ∂αA where A is given by (13). In fact

for any parameter value α which may be equal to Mkl, Qkl, Rkl or β the gradient is given

by

∂αΦYt,Σt(τ, z) = (Tr(∂αAΣt) + ∂αc(τ)) ΦYt,Σt(τ, z).

From (13) and ∂α(L−1) = −L−1(∂αL)L−1 implied by the derivation of L−1L = I we

conclude that

∂αA (τ) = −A22 (τ)−1 ∂α(A22 (τ))A(τ) +A22(τ)−1∂αA21 (τ) ,

Therefore we are lead to the computation of ∂αA11 (τ) ∂αA12 (τ)

∂αA21 (τ) ∂αA22 (τ)

,

that is the derivative with respect to a parameter of a function of a matrix (in this case

the exponential function). In order to use formula (41) we specify in the following table

for each parameter of the WASC model the choice of the matrice X and H. As usual

el; l = 1 . . . n resp. ekl; k, l = 1 . . . n stands for the canonical basis of Rn resp. Mn(R)

(the function f being f(x) = ex).

Parameter X H

Mkl τG τ

ekl 0

0 −ekl>

Qij τG τ

(ekl)>ρiω> −2(Q>ekl + (ekl)>Q)

0 −iωρ>ekl

ρl τG τ

Q>eliω> 0

0 −iωe>l Q

5If pi is the ith eigenvector of X and qi is the ith row of P−1 then the projection operator on Li is given

by Pi = piq>i

6Given two matrix of same size Xkl and Y kl then X Y = XklY kl

26

To fulfill the analytical computation of the gradient we need the derivative of c(τ) with

respect to a model parameter and particularly the term logA22(τ).

∂αc(τ) = −β2

[Tr (∂α(logA22(τ))) + τTr

(∂αM

> + iω∂α(ρ>Q))]

(44)

In order to apply (41) we just need to define P22 from M2n into Mn such that P22L = L22

with

L =

L11 L12

L21 L22

. (45)

Then it is easy to see that

∂α logA22(τ) = Dlog,A22(τ)(P22Dexp,τG(H)) (46)

Once again using (41) with the log function gives the result.

Remark 6.3. If f is the exponential function then we can also compute the derivative of

the exponential of a matrix using the Baker-Hausdorff’s formula (see e.g. Hall (2003) p.

71 formula (3.10) for the details)

∂αeτG = D expτG ∂α(τG), (47)

where D expX = eX I−e−adX

adXand adXY = [X,Y ] = XY − Y X is the Lie bracket.

The empirical study is based on the forward characteristic function of assets’ log returns

defined by

ΦΣ0(t,−iA(τ))ec(τ) = exp Tr [B(t)Σ0] + C(t) + c(τ) (48)

with

B (t) = (A(τ)B12(t) +B22(t))−1(A(τ)B11(t) +B21(t))

C(t) = −β2

Tr[log(A(τ)B12(t) +B22(t)) + tM>

]c(τ) = −β

2Tr[log(A22(τ)) + τM> + τiγ(ρ>Q)

]As for the characteristic function of assets’ log returns it is straightforward to show that

for any given model parameter α we have:

∂αΦΣ0(t,−iA(τ))ec(τ) = ΦΣ0(t,−iA(τ))ec(τ) (Tr(∂αB(t)Σ0) + ∂αC(t) + ∂αc(τ)))

27

where the matrix derivatives are given by

∂αB(t) = −(A(τ)B12(t) +B22(t))(∂αA(τ)B12(t) +A(τ)∂αB12(t) + ∂αB22(t))−1B(t)

+ (A(τ)B12(t) +B22(t))−1(∂αA(τ)B11(t) +B21(t)),

∂αC(t) = −β2

Tr(Dlog,A(τ)B12(t)+B22(t)(∂αA(τ)B12(t) +A(τ)∂αB12(t) + ∂αB22(t))).

This completes the analytical computation of the gradient.

6.2 Dynamics of the correlation process

In this Appendix we compute in the 2-dimensional case the drift and the diffusion coeffi-

cients of the correlation process ρ12t defined by

ρ12t =

Σ12t√

Σ11t Σ22

t

. (49)

We differentiate both sides of the equality(ρ12t

)2 = (Σ12t )2

Σ11t Σ22

t. We refer to Da Fonseca et al.

(2008) for the explicit computation of all covariations involved in the below formulas. We

obtain:

2ρ12t dρ

12t =

2Σ12t

Σ11t Σ22

t

dΣ12t +

(Σ12t

)2( 1Σ22t

d

(1

Σ11t

)+

1Σ11t

d

(1

Σ22t

))+ (.)dt,

so that

dρ12t =

1√Σ11t Σ22

t

(dΣ12

t −Σ12t

2Σ11t

dΣ11t −

Σ12t

2Σ22t

dΣ22t

)+ (.)dt.

By using the covariations among the Wishart elements we have

d⟨ρ12⟩t

=1

Σ11t Σ22

t

[Σ11t

(Q2

12 +Q222

)+ 2Σ12

t (Q11Q12 +Q21Q22) + Σ22t

(Q2

11 +Q221

)+(Σ12t

)2(Q211 +Q2

21

Σ11t

+Q2

12 +Q222

Σ22t

+ 2Σ12t

Σ11t Σ22

t

(Q11Q12 +Q21Q22))

− 2Σ12t

Σ11t

(Σ11t (Q11Q12 +Q21Q22) + Σ12

t

(Q2

11 +Q221

))−2

Σ12t

Σ22t

(Σ12t

(Q2

12 +Q222

)+ Σ22

t (Q11Q12 +Q21Q22))]dt,

which leads to:

d⟨ρ12⟩t

=(

1−(ρ12t

)2)(Q212 +Q2

22

Σ22t

+Q2

11 +Q221

Σ11t

− 2ρ12t (Q11Q12 +Q21Q22)√

Σ11t Σ22

t

)dt.

Now let us compute the drift of the process ρ12t .

28

We differentiate both sides of the equality ρ12t = Σ12

t√Σ11t Σ22

t

and we consider the finite

variation terms:

dρ12t =

1√Σ11t Σ22

t

dΣ12t + Σ12

t d

(1√

Σ11t Σ22

t

)+ d

⟨Σ12,

1√Σ11Σ22

⟩t

=1√

Σ11t Σ22

t

(Ω11Ω21 + Ω12Ω22 +M21Σ11

t +M12Σ22t + (M11 +M22) Σ12

t

)dt

+ Σ12t

1√Σ22t

− 1

2√(

Σ11t

)3(Ω2

11 + Ω212 + 2M11Σ11

t + 2M12Σ12t

)

+1√Σ11t

− 1

2√(

Σ22t

)3(Ω2

21 + Ω222 + 2M21Σ12

t + 2M22Σ22t

)+

3

8√

Σ11t Σ22

t

(Σ11t

)2d ⟨Σ11⟩t+

3

8√

Σ11t Σ22

t

(Σ22t

)2d ⟨Σ22⟩t

+1

4√(

Σ11t Σ22

t

)3d ⟨Σ11,Σ22⟩t

dt+1√Σ22t

− 1

2√(

Σ11t

)3 d

⟨Σ11,Σ12

⟩t

+1√Σ11t

− 1

2√(

Σ22t

)3 d

⟨Σ12,Σ22

⟩t+ Diffusions.

Now we use the formulas of the covariations of the Wishart elements and we arrive to an

expression which can be written as follows:

dρ12t =

(At(ρ12t

)2 +Btρ12t + Ct

)dt+ Diffusions,

where7:

At =1√

Σ11t Σ22

t

(Q11Q12 +Q21Q22)−

√Σ22t

Σ11t

M12 −

√Σ11t

Σ22t

M21

Bt = −Ω211 + Ω2

12

2Σ11t

− Ω221 + Ω2

22

2Σ22t

+Q2

11 +Q221

2Σ11t

+Q2

12 +Q222

2Σ22t

〈0

Ct =1√

Σ11t Σ22

t

(Ω11Ω21 + Ω12Ω22 − 2 (Q11Q12 +Q21Q22))

+

√Σ22t

Σ11t

M12 +

√Σ11t

Σ22t

M21.

From the definition of Ω =√βQ> and the Gindikin condition we deduce that Bt is

negative. As a by-product, we easily deduce the instantaneous covariation between the7Notice that the diffusion term and both the expressions for Bt and Ct are different from the ones

obtained by Buraschi et al. (2006).

29

0 200 400 600 800 1000

0.5

1.0

1.5

2.0

2.5

Index

Vol

atili

tyTime Varying Volatilities

0 200 400 600 800 1000

0.0

0.4

0.8

Index

Cor

rela

tion

Time Varying Correlation

Figure 1: Time varying (simulated) volatilities (top) and correlations (bottom).

This figure displays simulated volatilities and correlation in the two dimensional case(n = 2). The simulation has been produced using the parameters used in the Monte Carloexperiments. Given Σt the dynamic covariance matrix, the volatilities are

√Σ11t ,√

Σ22t .

The correlation is obtained by computing Σ12t /(√

Σ11t +

√Σ22t

).

Wishart element Σ11t and the correlation process:

d⟨ρ12,Σ11

⟩t

=1√

Σ11t Σ22

t

(d〈Σ11,Σ12〉t −

Σ12t

2Σ11t

d〈Σ11,Σ11〉t −Σ12t

2Σ22t

d〈Σ22,Σ22〉t)

= 2

√Σ11t

Σ22t

(1−

(ρ12t

)2) (Q11Q12 +Q21Q22) dt.

Using the fact that Q ∈ GL(n,R)8 there exists a unique couple (K, Q) ∈ O(n)×Pn9 such

that Q = KQ. We refer to Faraut (2006) for basic results on matrix analysis. The law of

the Σt being invariant by rotation of Q we rewrite this covariation as

d⟨ρ12,Σ11

⟩t

= 2

√Σ11t

Σ22t

(1−

(ρ12t

)2)Q12

(Q11 + Q22

)dt.

8GL(n,R) is the linear group, the set of invertible matrices.9O(n) stands for the orthogonal group ie O(n) = g ∈ GL(n,R)|g>g = In and Pn is the set of

symmetric definite positive matrices.

30

w

−300−200

−100

0

100

200

300

w

−300

−200

−100

0

100200

300O

bjective

0.02

0.04

0.06

0.08

0.10

Figure 2: Integrand of the C-GMM estimation criterion.

The figure displays the characteristic function with the integrated volatility presented inequation (??). The parameters used for to compute this characteristic function are thoseused in the Monte Carlo experiments.

31

Daily Weekly

Number of obs. 500 1500 500 1500

M11 Bias 0.0133 0.0382 0.227 0.0706RMSE 1.5126 1.5213 1.529 1.6011

M12 Bias -0.04 -0.014 -0.048 -0.036RMSE 1.0136 1.0341 1.098 1.0896

M21 Bias -0.056 -0.04 -0.085 -0.036RMSE 0.981 1.0205 1.104 1.1101

M22 Bias 0.1667 0.1784 0.070 0.0657RMSE 1.5378 1.5502 1.511 1.5778

Q11 Bias 0.1077 0.1069 0.132 0.1166RMSE 0.088 0.0872 0.476 0.3568

Q12 Bias 0.0309 0.0287 0.022 0.0208RMSE 0.0664 0.0675 0.554 0.4467

Q21 Bias -3E-04 -6E-04 -0.059 -0.008RMSE 0.0876 0.0862 0.553 0.371

Q22 Bias 0.0792 0.0772 0.057 0.0574RMSE 0.0678 0.0661 0.455 0.4424

β Bias 0.0632 -0.066 0.044 -0.107RMSE 4.335 4.3472 4.546 4.4364

ρ1 Bias 0.0055 -0.007 0.035 0.0268RMSE 0.4617 0.4913 0.641 0.8031

ρ2 Bias -0.004 0.0265 0.063 0.0445RMSE 0.5231 0.4734 0.653 0.6405

Table 1: Results of the Monte Carlo experiments

This table displays the results for the Monte Carlo simulations performed using the following parameters:

Σ0 =

[0.0225 −0.0054−0.0054 0.0144

].M =

[−5 −3−3 −5

], ρ =

[0.3 0.4

], (50)

Q =

[0.1133137 0.033358710.0000000 0.07954368

], β = 15. (51)

Two different types of simulations are presented: one of the sample includes 500 daily observations and asecond one uses 1500 daily observations, as in Carrasco et al. (2007).

SP500 FTSE DAX CAC 40

Min. :-0.128129 Min. :-0.141420 Min. :-0.197775 Min. :-0.1491491st Qu.:-0.009940 1st Qu.:-0.010858 1st Qu.:-0.014283 1st Qu.:-0.015173Median : 0.002491 Median : 0.002101 Median : 0.003727 Median : 0.002117Mean : 0.001535 Mean : 0.001014 Mean : 0.001523 Mean : 0.001095

3rd Qu.: 0.013506 3rd Qu.: 0.013424 3rd Qu.: 0.019416 3rd Qu.: 0.018025Max. : 0.123746 Max. : 0.135879 Max. : 0.171546 Max. : 0.166252

Table 2: Descriptive statistics for the real dataset.

The table summarizes the descriptive statistics for the available dataset. This dataset is made of theSP500, FTSE, DAX and CAC time series, on a weekly sampling frequency. The dataset starts on January2nd 1990 and ends on June 30th 2007.

32

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

M11

-3.4

4-2

.94

-2.6

5-3

.22

-2.6

2-2

.61

Std

.D

ev.

(0.2

39)

(0.2

17)

(0.3

09)

(0.1

5)(0

.158

)(0

.181

)M

12

1.17

0.33

0.74

1.51

0.04

0.5

Std

.D

ev.

(0.2

11)

(0.2

67)

(0.5

19)

(0.2

66)

(0.3

58)

(0.2

36)

M21

0.22

0.96

1.79

1.51

0.74

0.76

Std

.D

ev.

(0.4

69)

(0.3

72)

(0.4

91)

(0.1

68)

(0.1

53)

(0.1

62)

M22

-2.8

7-3

.41

-3.0

4-3

.49

-3.5

3-2

.48

Std

.D

ev.

(0.2

1)(0

.261

)(0

.384

)(0

.193

)(0

.215

)(0

.165

)Q

11

-0.0

5-0

.07

-0.0

10.

060.

060.

12S

td.

Dev

.(0

.01)

(0.0

05)

(0.0

08)

(0.0

04)

(0.0

05)

(0.0

03)

Q12

-0.1

0.01

-0.0

8-0

.04

-0.0

40.

04S

td.

Dev

.(0

.012

)(0

.007

)(0

.007

)(0

.004

)(0

.004

)(0

.003

)Q

21

0.07

0.02

-0.0

8-0

.08

0.09

0.01

Std

.D

ev.

(0.0

07)

(0.0

05)

(0.0

07)

(0.0

07)

(0.0

09)

(0.0

04)

Q22

0.07

0.12

-0.0

3-0

.10.

080.

06S

td.

Dev

.(0

.014

)(0

.005

)(0

.01)

(0.0

04)

(0.0

05)

(0.0

03)

β10

.65

12.2

512

.711

.34

10.7

410

.87

Std

.D

ev.

(0.7

33)

(0.7

76)

(1.2

36)

(0.6

64)

(0.7

38)

(0.6

38)

ρ1

0.3

0.6

0.4

0.16

-0.2

-0.3

Std

.D

ev.

(0.0

67)

(0.1

03)

(0.0

98)

(0.0

76)

(0.0

81)

(0.0

69)

ρ2

-0.2

-0.4

0.28

0.4

-0.2

6-0

.81

Std

.D

ev.

(0.0

71)

(0.0

63)

(0.0

81)

(0.0

6)(0

.076

)(0

.11)

Tab

le3:

His

tori

cal

par

amet

eres

tim

ates

for

the

WA

SC

mod

elu

sin

gth

ed

aily

dat

aset

.T

he

table

pre

sents

the

para

met

ers

esti

mate

dusi

ng

the

C-G

MM

met

hod

dev

elop

edin

Carr

asc

oet

al.

(2007),

usi

ng

asi

mple

index

pro

cedure

.T

he

inte

gra

tion

gri

disω

=[−

300;3

00].

The

data

sets

are

made

of

daily

obse

rvati

ons,

from

01/01/1990

unti

l09/12/2007,

usi

ng

act

ive

day

scl

osi

ng

pri

ces.

Sta

ndard

dev

iati

ons

are

giv

enb

etw

een

bra

cket

s.

33

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

M11

-2.8

3-2

.67

-3.4

3-3

.14

-2.8

9-3

.42

Std

.D

ev.

(0.6

71)

(1.0

77)

(0.6

71)

(0.3

26)

(0.2

79)

(0.2

67)

M12

1.87

1.28

1.13

0.44

0.83

0.01

Std

.D

ev.

(0.3

76)

(0.5

66)

(0.5

38)

(0.3

43)

(0.5

64)

(0.5

27)

M21

0.42

1.54

0.82

0.63

0.89

1.09

Std

.D

ev.

(1.0

02)

(1.6

61)

(0.7

2)(0

.38)

(0.2

64)

(0.2

6)M

22

-3.2

9-3

.79

-3.2

1-3

.9-1

.27

-3.7

6S

td.

Dev

.(0

.305

)(0

.393

)(0

.291

)(0

.264

)(0

.326

)(0

.312

)Q

11

0.04

-0.0

50.

070.

010.

04-0

.12

Std

.D

ev.

(0.0

03)

(0.0

12)

(0.0

06)

(0.0

03)

(0.0

11)

(0.0

06)

Q12

0.05

0.02

0.08

-0.1

50.

06-0

.05

Std

.D

ev.

(0.0

05)

(0.0

14)

(0.0

05)

(0.0

02)

(0.0

07)

(0.0

05)

Q21

00.

03-0

.02

-0.1

10.

09-0

.02

Std

.D

ev.

(0.0

05)

(0.0

04)

(0.0

04)

(0.0

07)

(0.0

06)

(0.0

06)

Q22

-0.1

20.

13-0

.05

-0.1

0.01

-0.0

7S

td.

Dev

.(0

.007

)(0

.006

)(0

.004

)(0

.008

)(0

.005

)(0

.004

)β

12.4

413

.58

13.1

714

.85

11.5

614

.37

Std

.D

ev.

(1.2

05)

(1.3

76)

(1.9

29)

(0.3

92)

(1.8

47)

(1.2

9)ρ

1-0

.75

0.01

-0.4

0.39

-0.2

30.

36S

td.

Dev

.(0

.067

)(0

.041

)(0

.031

)(0

.038

)(0

.078

)(0

.062

)ρ

20.

35-0

.62

0.64

0.26

-0.4

90.

11S

td.

Dev

.(0

.035

)(0

.039

)(0

.058

)(0

.076

)(0

.065

)(0

.054

)

Tab

le4:

His

tori

cal

par

amet

eres

tim

ates

for

the

WA

SC

mod

elu

sin

gth

ew

eekly

dat

aset

.T

he

table

pre

sents

the

para

met

ers

esti

mate

dusi

ng

the

C-G

MM

met

hod

dev

elop

edin

Carr

asc

oet

al.

(2007),

usi

ng

asi

mple

index

pro

cedure

.T

he

inte

gra

tion

gri

dis

ω=

[−300;3

00].

The

data

sets

are

made

of

wee

kly

obse

rvati

ons,

from

01/01/1990

unti

l09/12/2007,

usi

ng

act

ive

day

scl

osi

ng

pri

ces.

Sta

ndard

dev

iati

ons

are

giv

enb

etw

een

bra

cket

s.

34

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

Dai

lyV

olvo

lA

sset

10.

172

0.14

60.

161

0.2

0.21

60.

241

Vol

vol

Ass

et2

0.24

40.

241

0.17

10.

215

0.17

90.

144

Wee

kly

Vol

vol

Ass

et1

0.08

0.11

70.

146

0.22

10.

197

0.24

3V

olvo

lA

sset

20.

260.

263

0.18

90.

361

0.12

20.

172

Tab

le5:

Vol

atilit

yof

vola

tility

esti

mat

edfr

omth

ed

iffer

ent

dat

aset

s,w

ith

diff

eren

tsa

mp

lefr

equ

ency

.

Inth

eta

ble

,th

efirs

tline

corr

esp

onds

toth

evola

tility

of

vola

tility

of

the

firs

tass

et.

For

exam

ple

,fo

rth

eco

lum

nSP

500/C

AC

,th

efirs

tline

corr

esp

onds

toth

evola

tility

of

vola

tility

of

the

SP

500.

The

seco

nd

line

dis

pla

ys

the

one

obta

ined

for

the

CA

C.

The

vola

tility

of

vola

tility

isobta

ined

when

com

puti

ng:

Vol/

vol

Ass

et1

=2√ Q

2 11

+Q

2 21

Vol/

vol

Ass

et2

=2√ Q

2 12

+Q

2 22.

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

Dai

lyS

kew

Ass

et1

-0.3

37-0

.687

-0.3

27-0

.224

-0.3

27-0

.366

Ske

wA

sset

2-0

.36

-0.3

49-0

.468

-0.4

31-0

.143

-0.8

4W

eekly

Ske

wA

sset

1-0

.75

-0.3

28-0

.56

-0.2

19-0

.541

-0.3

73S

kew

Ass

et2

-0.6

12-0

.611

-0.6

78-0

.466

-0.3

07-0

.299

Tab

le6:

Ske

wes

tim

ated

from

the

diff

eren

td

atas

ets,

wit

hd

iffer

ent

sam

ple

freq

uen

cy.

Inth

eta

ble

,th

efirs

tline

corr

esp

onds

toth

esk

ewass

oci

ate

dto

the

firs

tass

et.

For

exam

ple

,fo

rth

eco

lum

nSP

500/C

AC

,th

efirs

tline

corr

esp

onds

toth

esk

ewof

the

SP

500.

The

seco

nd

line

dis

pla

ys

the

one

obta

ined

for

the

CA

C.

The

skew

isobta

ined

when

com

puti

ng:

Corr( dY

1 t,d

Σ11t

) =Q

11ρ1

+Q

21ρ2

√ Q2 11

+Q

2 21

Corr( dY

2 t,d

Σ22t

) =Q

12ρ1

+Q

22ρ2

√ Q2 12

+Q

2 22

.

35

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

Ass

et1

Ass

et2

Ass

et1

Ass

et2

Ass

et1

Ass

et2

Ass

et1

Ass

et2

Ass

et1

Ass

et2

Ass

et1

Ass

et2

Dai

lyκ

16.

885.

745.

886.

825.

36.

086.

446.

985.

247.

065.

224.

96κ

2-1

.493

-0.7

4-0

.363

-0.2

08-0

.612

-0.5

86-2

.149

-2.4

89-0

.038

-0.0

74-0

.589

-0.9

83κ

5.38

75

5.51

76.

612

4.68

85.

494

4.29

14.

491

5.20

26.

986

4.63

13.

977

Wee

kly

κ1

5.66

6.58

5.34

7.58

20.8

66.

426.

287.

85.

782.

546.

847.

52κ

2-3

.739

-1.8

04-2

.288

-1.3

08-1

.509

-1.5

36-0

.672

-0.7

96-0

.844

-1.7

81-0

.012

-0.0

21κ

1.92

14.

776

3.05

26.

272

19.3

514.

884

5.60

87.

004

4.93

60.

759

6.82

87.

499

Tab

le7:

Mea

nre

vert

ing

par

amet

eras

soci

ated

toΣt.

Thi

sta

ble

disp

lays

the

mea

nre

vert

ing

para

met

eras

soci

ated

toth

evo

lati

litie

s.Fo

rth

efir

stas

set,

we

have

:

dΣ

11t

=...+

Σ11t

2M11 ︸︷︷︸ −κ 1

+2M

12

√ Σ22t

Σ11t

(ρ12t

)

︸︷︷

︸−κ2

︸︷︷

︸−κ

dt

+...

κ1

isth

eid

iosy

ncra

tic

part

ofth

em

ean

reve

rtin

gpa

ram

eter

.κ

2is

the

cros

s-co

mpo

nent

inth

epa

ram

eter

andκ

isth

est

anda

rdH

esto

n-lik

em

ean

reve

rtin

gpa

ram

eter

.(ρ

12)

=Corr(dlogS

1,dlogS

2)

isth

elo

ngte

rmco

rrel

atio

nbe

twee

nth

eas

set

log

retu

rns.

The

drift

isev

alua

ted

atit

sst

atio

nary

valu

e,us

ing

the

valu

espr

esen

ted

inta

ble

7.

36

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

Dai

ly√

Σ11

0.15

80.

158

0.15

80.

221

0.22

10.

206

√Σ

22

0.22

10.

206

0.15

80.

206

0.15

80.

158

ρ12

0.44

90.

417

0.40

50.

766

0.66

90.

761

Wee

kly

√Σ

11

0.18

0.18

0.18

0.26

0.26

0.23

8√

Σ22

0.26

0.23

80.

180.

238

0.18

0.18

ρ12

0.69

40.

676

0.67

40.

831

0.73

90.

781

Tab

le8:

Lon

gte

rmco

vari

ance

mat

rix

imp

lied

by

the

esti

mat

es.

The

long

term

cova

riance

matr

ixis

esti

mate

dth

rough

the

usu

al

mom

ent

esti

mato

rs.

Its

dia

gonal

elem

ents

are

Σ11t

and

Σ22t

.(ρ

12t

)is

the

long

term

corr

elati

on

com

pute

dfr

om

the

esti

mate

dlo

ng

term

cova

riance

matr

ix.

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

Dai

lyA

(t)

-1.5

22-1

.108

-2.3

76-2

.902

-0.9

18-1

.199

B(t

)-3

.022

-3.2

46-3

.345

-2.5

44-2

.779

-2.7

82C

(t)

4.37

11.

722

3.92

94.

212.

283

2.87

4W

eekly

A(t

)-2

.922

-2.7

6-1

.056

-0.8

68-1

.761

-1.2

1B

(t)

-2.4

85-3

.723

-3.8

9-7

.545

-1.9

86-4

.761

C(t

)3.

629

3.98

84.

664

3.94

52.

849

4.57

1

Tab

le9:

Mea

nre

vert

ing

par

amet

ers

asso

ciat

edto

ρ12t

.T

his

tabl

edi

spla

ysth

edr

iftpa

rtof

the

corr

elat

ion

coeffi

cien

t(ρ

12t

)in

the

two-

asse

tca

se,

for

the

valu

ees

tim

ated

inth

epr

evio

usta

bles

.T

hedr

iftis

give

nby

:

dρ

12t

=(A

t(ρ

12t

)2+Bt(ρ

12t

)+Ct)dt

+di

ffusi

onpa

rt.

The

com

puta

tion

ofth

isdr

iftis

pres

ente

din

the

appe

ndic

es.

The

drift

isco

mpu

ted

atit

sst

atio

nary

valu

e,us

ing

the

valu

espr

esen

ted

inta

ble

7.

37

0.8

0.9

1.0

1.1

−1.00.00.51.01.52.0

SP

500

DA

X

rho

drho

0.40

0.45

0.50

0.55

−0.4−0.20.00.20.4

SP

500

CA

C

rho

drho

0.60

0.65

0.70

0.75

0.80

0.85

0.90

−1.00.00.51.01.52.0

SP

500

FTS

E

rho

drho

0.70

0.80

0.90

1.00

−1012

DA

XC

AC

rho

drho

0.55

0.60

0.65

0.70

0.75

0.80

−0.6−0.20.20.6

DA

XFT

SE

rho

drho

0.65

0.70

0.75

0.80

0.85

0.90

−0.50.00.51.0

CA

CFT

SE

rho

drho

Fig

ure

3:d(ρ

12t

)as

afu

nct

ion

of(ρ

12t

)u

sin

gth

ed

aily

dat

aset

.T

his

figur

edi

spla

ysth

edr

iftte

rmas

soci

ated

tod(ρ

12)

asa

func

tion

of(ρ

12),

usin

gda

ilyes

tim

atio

nre

sult

s.T

his

rela

tion

issp

ecifi

edin

the

follo

win

geq

uati

on:

dρ

12t

=(A

t(ρ

12t

)2+Bt(ρ

12t

)+Ct)dt

+di

ffusi

onpa

rt.

The

blue

line

isth

elin

ear

term

ofth

ela

tter

rela

tion

.T

here

don

eis

the

tota

lexp

ecte

dva

riat

ion

ofd(ρ

12t

)de

pend

ing

upon

the

(ρ12t

)va

lues

.T

hegr

een

line

sign

als

the

long

term

corr

elat

ion

impl

ied

byth

ees

tim

ated

WA

SCm

odel

.T

his

long

term

corr

elat

ion

isth

eon

lypo

siti

vero

otof

the

poly

nom

ial

defin

edby

the

stat

iona

rydr

iftof

the

corr

elat

ion,

usin

gth

eva

lues

pres

ente

din

tabl

e7.

38

0.65

0.70

0.75

0.80

0.85

0.90

−1.00.00.51.01.52.0

SP

500

DA

X

rho

drho

0.60

0.65

0.70

0.75

0.80

0.85

−1.00.00.51.01.52.0

SP

500

CA

C

rho

drho

0.8

0.9

1.0

1.1

−1.00.00.51.01.5

SP

500

FTS

E

rho

drho

0.40

0.45

0.50

0.55

0.60

−0.50.00.51.0

DA

XC

AC

rho

drho

0.65

0.75

0.85

0.95

−0.50.00.51.01.5

DA

XFT

SE

rho

drho

0.65

0.70

0.75

0.80

0.85

0.90

0.95

−1.0−0.50.00.51.01.5

CA

CFT

SE

rho

drho

Fig

ure

4:d(ρ

12t

)as

afu

nct

ion

of(ρ

12t

)u

sin

gth

ew

eekly

dat

aset

.T

his

figur

edi

spla

ysth

edr

iftte

rmas

soci

ated

tod(ρ

12t

)as

afu

ncti

onof

(ρ12t

),us

ing

wee

kly

esti

mat

ion

resu

lts.

Thi

sre

lati

onis

spec

ified

inth

efo

llow

ing

equa

tion

:

dρ

12

=(A

t(ρ

12t

)2+Bt(ρ

12t

)+Ct)dt

+di

ffusi

onpa

rt.

The

blue

line

isth

elin

ear

term

ofth

ela

tter

rela

tion

.T

here

don

eis

the

tota

lexp

ecte

dva

riat

ion

ofd(ρ

12t

)de

pend

ing

upon

the

(ρ12t

)va

lues

.T

hegr

een

line

sign

als

the

long

term

corr

elat

ion

impl

ied

byth

ees

tim

ated

WA

SCm

odel

.T

his

long

term

corr

elat

ion

isth

eon

lypo

siti

vero

otof

the

poly

nom

ial

defin

edby

the

stat

iona

rydr

iftof

the

corr

elat

ion,

usin

gth

eva

lues

pres

ente

din

tabl

e7.

39

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

Dai

ly0.

361

0.72

10.

484

0.43

10.

328

0.86

4W

eekly

0.82

80.

620

0.75

50.

466

0.54

10.

376

Tab

le10

:N

orm

ofth

eρ

vect

or.

The

norm

ofρ

isco

mpute

das||ρ||

=√ (ρ

>ρ),

whic

his

the

Eucl

idia

nnorm

.

SP

500/

DA

XS

P50

0/C

AC

SP

500/

FT

SE

DA

X/C

AC

FT

SE

/DA

XF

TS

E/C

AC

Dai

ly0.

058

0.00

90.

020.

028

0.02

50.

03W

eekly

0.01

20.

016

0.04

90.

034

0.02

20.

038

Tab

le11

:O

ffd

iago

nal

term

ofth

ep

olar

dec

omp

osit

ion

ofQ

esti

mat

edfr

omth

ed

iffer

ent

dat

aset

s,w

ith

diff

eren

tsa

mp

lefr

equ

ency

.

Usi

ng

the

fact

thatQ∈

GL

(n,R

),w

her

eG

L(n,R

)is

the

linea

rgro

up,

i.e.

the

set

of

inver

tible

matr

ices

,th

ere

exis

tsa

uniq

ue

couple

(K,Q

)∈O

(n)×Pn,

wher

e

O(n

)st

ands

for

the

ort

hogonal

gro

up

ieO

(n)

=g∈

GL

(n,R

)|g>g

=I n

andPn

isth

ese

tof

sym

met

ric

defi

nit

ep

osi

tive

matr

ices

,su

chth

atQ

=KQ

.T

his

dec

om

posi

tion

ofQ

isuniq

ue

and

isre

fere

dto

as

pola

rdec

om

posi

tion.

We

refe

rto

Fara

ut

(2006)

for

furt

her

resu

lts

on

matr

ixanaly

sis.

40