Seediscussions,stats,andauthorprofilesforthispublicationat:https://www.researchgate.net/publication/228266805
EstimatingtheWishartAffineStochastic
CorrelationModelUsingtheEmpirical
CharacteristicFunction
ArticleinStudiesinNonlinearDynamics&Econometrics·August2012
DOI:10.2139/ssrn.1054721
CITATIONS
READS
16
217
3authors:
JoseDaFonseca
MartinoGrasselli
AucklandUniversityofTechnology
UniversityofPadova
23PUBLICATIONS488CITATIONS
25PUBLICATIONS586CITATIONS
SEEPROFILE
SEEPROFILE
FlorianIelpo
UniversitédeParis1Panthéon-Sorbonne
44PUBLICATIONS195CITATIONS
SEEPROFILE
Allin-textreferencesunderlinedinbluearelinkedtopublicationsonResearchGate,
lettingyouaccessandreadthemimmediately.
Availablefrom:FlorianIelpo
Retrievedon:11August2016
Estimating the Wishart Affine Stochastic Correlation Model
using the Empirical Characteristic Function∗
José 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 Stochastic
Correlation (WASC) model. We provide elements to show that the utilization of empirical characteristic function-based estimates is advisable: this function is exponential
affine in the WASC case. We use a GMM estimation strategy with a continuum of
moment conditions based on the characteristic function. We present the estimation
results obtained using a dataset of equity indexes. The WASC model captures most
of the known stylized facts associated with financial markets, including the leverage
and 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 and
helpful comments. We are also grateful to Christian Gourieroux, Fulvio Pegoraro, François-Xavier Vialard
and the CREST seminar participants for useful remarks. We are thankful to the seminar participants of
the 14th International Conference on Computing in Economics and Finance, Paris, France (2008), the 11th
conference of the Swiss Society for Financial Market Research, Zürich (2008), Mathematical and Statistical
Methods for Insurance and Finance, Venice, Italy (2008), the 2nd International Workshop on Computational and Financial Econometrics, Neuchâtel, Switzerland (2008), the First PhD Quantitative Finance
Day, Swiss Banking Institute, Zürich (2008), Inference and tests in Econometrics, in the honor of Russel
Davidson, 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 Vast
Data 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. Any
errors remain ours.
†
Ecole Supérieure d’Ingénieurs Léonard de Vinci, Département Mathématiques et Ingénierie Financière,
92916 Paris La Défense, France. Email: jose.da [email protected] and Zeliade Systems, 56, Rue JeanJacques Rousseau, 75001 Paris.
‡
Università degli Studi di Padova , Dipartimento di Matematica Pura ed Applicata, Via Trieste 63,
Padova, Italy. E-mail: [email protected] and ESILV.
§
Pictet & Cie, Route des Acacias 60, CH-1211 Genève 73. E-mail: [email protected].
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 estimate financial Markov processes (methods of this kind are reviewed in Gouriéroux 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
Electronic copy available at: http://ssrn.com/abstract=1054721
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 multivariate 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 stochastic 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 characteristic 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 financial 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 estimation 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
p
dSt = diag[St ] µdt + Σt dZt ,
(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:
p
p
dΣt = ΩΩ> + M Σt + Σt M > dt + Σt dWt Q + 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 longterm variance-covariance matrix Σ∞ through the solution to the following linear equation:
−ΩΩ> = M Σ∞ + 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 correlation 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 reproduce 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 covariance matrix are linearly correlated.
Da Fonseca et al. (2007) proved that Assumption 2.2. leads to the following relation:
dZt = dWt ρ +
p
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 matrixsized 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


12
Σ11
Σ
t
t 
Σt = 
.
12
Σt Σ22
t
(6)
The correlations between assets’ returns and their volatilities admit a closed form expression, highlighting the impact of the ρ parameters on its value and positivity:
ρ1 Q11 + ρ2 Q21
(7)
corr d log S1 , dΣ11 = p 2
Q11 + Q221
ρ1 Q12 + ρ2 Q22
corr d log S2 , dΣ22 = p 2
,
(8)
Q12 + Q222
√
√
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 log S1 , dΣ11 = ρ1
corr d log S2 , 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 = log St 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:
h
i
ΦYt ,Σt (τ, ω) = E eihω,Yt+τ i |Σt , Yt ,
(10)
where E[.|Σt , Yt ] denotes the conditional expected value with respect to the historical
measure, ω ∈ Rn , i2 = −1 and h., .i is the scalar product in Rn .
Proposition 2.1. (Da Fonseca et al., 2007) The characteristic function of the asset returns in the WASC model is given by
h
i
ΦYt ,Σt (τ, ω) = E eihω,Yt+τ i |Σt , Yt
(11)
= exp {Tr [A(τ )Σt ] + hiω, Yt i + c(τ )} ,
(12)
where ω = (ω1 , . . . , ωn )> ∈ Rn and the deterministic function A(t) ∈ Mn is as follows:
A (τ ) = A22 (τ )−1 A21 (τ ) ,
(13)
with




Q> ρiω >
−2Q> Q

 A11 (τ ) A12 (τ ) 
 M+
.

 = exp τ 



 P
n
1
>−
> + iωρ> Q
(iω)(iω)
iω
e
−
M
A21 (τ ) A22 (τ )
j jj
j=1
2
(14)
7
The function c(τ ) can be obtained by direct integration, thus giving:
i
h
i
β h
c(τ ) = − Tr log (A22 (τ )) + τ M > + τ iω(ρ> Q) + τ Tr µiω > .
2
(15)
The characteristic function of the Wishart process is defined as:
h
i
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 conditional characteristic function of the Wishart process Σt is given by:
h
i
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)
i
β h
C(τ ) = − Tr log(i∆B12 (τ ) + B22 (τ )) + τ M > ,
2
with



 B11 (τ ) B12 (τ ) 
 M

 = exp τ 



0
B21 (τ ) B22 (τ )

−2Q> Q
−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 discretization 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 function as an estimation tool, since this function has a tractable expression for many continuous 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 Carrasco 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 = eihw,Yt+τ −Yt i − Xt ,
(19)
with the notations developed earlier and Xt a stochastic process such that E[ht |Yt ] = 0.
Therefore Xt = E[eihw,Yt+τ −Yt i |Yt ]. Then the estimation can be based on unconditional
moment conditions of the form E[ht g(Yt )] = 0. However, this approach can not be implemented here because E[eihw,Yt+τ −Yt i |Yt ] does not have a known expression, principally
because the distribution of Σt given Yt is unknown. The solution we adopt is to use unconditional 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 = eihw,Yt+τ −Yt i − E[eihw,Yt+τ −Yt i |Σ0 ]
1
We ran Monte Carlo test to prove this point empirically. The tables are available upon request.
2
We thank Marine Carrasco for pointing out this fact.
9
(20)
where the initial value of Σ0 is treated as an unknown parameter to be estimated. We
have
E[eihw,Yt+τ −Yt i |Σ0 ] = ec(τ ) E[ehA(τ ),Σt i |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 conditions, 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 ĥ(.) 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:
θ̂ = arg min kK −1/2 ĥ(θ)k,
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 k.k is the weighted norm
Z Z
2
kf k =
f (ω)f (ω)π(ω)dω
Rn
Rn
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:
Z
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 ) =
+∞
X
Eθ0 ht (ω; θ0 )ht−j (τ ; θ0 ) .
j=−∞
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:
θ̂1 = arg min kĥ(θ)k.
(23)
θ
Since ht is autocorrelated the second step consists in estimating the kernel k as follows:
T−1
X
T
j
k̂(ωs , ωr , ωv , ωw ) =
ω
Γ̂T (j),
(24)
T−q
ST
j=−T+1
with
Γ̂T (j) =



1
T
1
T
PT
t=j+1 ht (ωs , ωr , θ̂1 )ht−j (ωv , ωw , θ̂1 ), j
PT
≥0
,
(25)
t=−j+1 ht+j (ωs , ωr , θ̂1 )ht (ωv , ωw , θ̂1 ), j < 0,
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:
(K α )−1 = (K 2 + αI)−1 K.
(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 between 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 (K α )−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:
θ̂ = arg min k(KTα )−1/2 ĥT (θ)k
θ
11
(27)
and is asymptotically Normal with
√
5/4
as T and T a αT
L
T (θ̂T − θ0 ) → N 0, (hE θ0 (∇θ h), E θ0 (∇θ h)iK )−1 ,
(28)
go to infinity and α goes to zero. (∇θ h 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 methodology. 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). Yt1 being the log return of the first asset, its volatility is given by
p
Σ11
t .
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:
2
2
dhΣ11 , Σ11 it = 4Σ11
t (Q11 + Q21 )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,
Q11 ρ1 + Q21 ρ2
,
corr dY1 , dΣ11 = p 2
Q11 + Q221
(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 S 1 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-IngersollRoss 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:
"
#
p
22
Σ
11
dΣ11
2M11 + 2M12 p t ρ12
+ ...,
t = . . . + Σt
t
Σ11
t
(31)
where ρ12
t 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
√ 22is made of two parts: a deterministic part (2M11 ) and a stochastic
Σ
correction (2M12 √ t11 ρ12
t ), linked to the joint dynamics of both assets. Thus, the drift
term of
Σ11
t
Σt
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
22
calculations for Σ12
t and Σt . 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:
q
2
12 2
12
dρ12
=
A
ρ
+
B
ρ
+
C
1 − ρ12
dt
+
(...)d(N oiset )
t
t t
t
t
t
t
(32)
22
with At , Bt , Ct recursive functions of Σ11
t , Σt and the model parameters. We present the
drift coefficients and the diffusion term in the Appendix. The drift associated to the correlation is quadratic, and the linear term has a negative coefficient Bt < 0, thus presenting
the typical mean reverting behavior of ρ12
t (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 cycles. 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 getting
3
On asymmetric correlation effects, see Roll (1988) and Ang and Chen (2002).
14
down. As already noticed in Da Fonseca et al. (2007), we have:
s
2
Σ11
1 12
t
dhY , ρ it =
(1 − ρ12
) × (Q12 ρ1 + Q22 ρ2 ) .
t
22
{z
}
|
Σt
(33)
Sign of asset 2 skew
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
s
12 11 Σ11
12 2
t
1
−
ρ
Q̃
Q̃
+
Q̃
d ρ ,Σ t =
12
11
22 dt.
t
Σ22
t
where Q̃ is the symmetric positive definite matrix associated to the polar decomposition
of Q4 . A positive value for Q̃12 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 details 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 used
4
Any invertible matrix Q can be uniquely written as the product of a rotation matrix and a symmetric
positive definite matrix Q̃, see the Appendix.
15
in the simulation are the following:

Σ0 = 

M =
0.0225
−0.0054
−0.0054
0.0144
−5 −3


(34)


−3 −5
h
i
ρ = 0.3 0.4


0.1133137 0.03335871

Q=
0.0000000 0.07954368
(35)
β = 15.
(38)
(36)
(37)
The Q matrix is obtained by inverting the relation that links Q to M , Σ∞ and β:
Q> Q = −
o
1n
M Σ∞ + Σ∞ M > .
β
(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 estimation 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. Gouriéroux and Monfort, 2007). The Monte Carlo results indicate that this mean
reverting parameter is estimated with less volatility with daily series. What is more, diagonal 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 discarded 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 parameters 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 interesting 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
dhΣ11 , Σ22 i = Σ12 dt.
(40)
Hence, whenever Σ12 is positive, the WASC model is able to model volatility transmission 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 literature, 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 surprising 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
.
Q211 +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 correlation 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 different 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 parameter 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 ρ12
t and
the usual comments have to be adapted. We present in figure 3 and 4 the instantaneous
12
variation of ρ12
t as a function of ρt , 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 correlation 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 Q̃12 . 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 having 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 characteristic 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 example 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 Algorithms. 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 Conditions. 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. SpringerVerlag.
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 introduction. 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 Empirical 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 Characteristic Function. Journal of Econometrics, (102):111–141.
Singleton, K. J. (2006). Empirical Dynamic Asset Pricing: Model Specification and Econometric Assessment. Princeton University Press.
24
6
6.1
Appendix
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 characteristic 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 − λi I) and Pi the projection operator from Cn onto
P
Li , then we have ni=1 Pi = I. Define also Ji such that (X − λi I)Pi = Ji . The Jordan
P
normal form of X if given by the well known decomposition X = ni=1 (λi Pi + 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 kf (X +tH)−f (X)−Df,X (H)k = to(kHk) and can be computed
using the following formula Daleckii (1974):
j1 −1 mj2 −1
n mX
X
X 1 ∂ r1 +r2 f (λj ) − f (λj ) 1
2
Df,X (H) =
Pj1 Jjr11 HPj2 Jjr22 .
r1 r2 ∂λrj11 ∂λrj22
λj1 − λj2
j1 ,j2 r1 =0
(41)
r2 =0
Remark 6.1. Whenever λj1 = λj2 the term within the bracket should be replaced by
f 0 (λj1 ).
Remark 6.2. When X can be diagonalized then mj = 1 for each j and we are lead to the
very simple form
n X
f (λj1 ) − f (λj2 )
Pj1 HPj2 .
Df,X (H) =
λj1 − λj2
(42)
j1 ,j2
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) = P Mf ◦ (P −1 HP )P −1 where P is the matrix of the
eigenvectors of X 5 , ◦ 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

 f (λ)−f (µ) if λ =
6 µ
λ−µ
Mf (λ, µ) =
 f 0 (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 ∂α A 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
∂α ΦYt ,Σt (τ, z) = (Tr(∂α AΣt ) + ∂α c(τ )) ΦYt ,Σt (τ, z).
From (13) and ∂α (L−1 ) = −L−1 (∂α L)L−1 implied by the derivation of L−1 L = I we
conclude that
∂α A (τ ) = −A22 (τ )−1 ∂α (A22 (τ ))A(τ ) + A22 (τ )−1 ∂α A21 (τ ) ,
Therefore we are lead to the computation of


∂ A (τ ) ∂α A12 (τ )
 α 11
,
∂α A21 (τ ) ∂α A22 (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

Qij
ρl
5
τG τ 
τG
ekl
0
(ekl )> ρiω >
0

τ

0
−ekl
>
−2(Q> ekl
Q> el iω >
0
+
(ekl )> Q)
−iωρ> ekl

0

−iωe>
Q
l


If 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 = pi qi>
6

Given two matrix of same size X kl and Y kl then X ◦ Y = X kl Y 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 log A22 (τ ).
∂α c(τ ) = −
i
βh
Tr (∂α (log A22 (τ ))) + τ Tr ∂α M > + iω∂α (ρ> Q)
2
(44)
In order to apply (41) we just need to define P22 from M2n into Mn such that P22 L = L22
with

L=
L11 L12
L21 L22

.
(45)
Then it is easy to see that
∂α log A22 (τ ) = Dlog,A22 (τ ) (P22 Dexp,τ G (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)
∂α eτ G = D expτ G ∂α (τ G),
−adX
where D expX = eX I−eadX
(47)
and adX Y = [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 {T r [B(t)Σ0 ] + C(t) + c(τ )}
(48)
with
B (t) = (A(τ )B12 (t) + B22 (t))−1 (A(τ )B11 (t) + B21 (t))
i
β h
C(t) = − Tr log(A(τ )B12 (t) + B22 (t)) + tM >
2
i
β h
c(τ ) = − Tr log(A22 (τ )) + τ M > + τ iγ(ρ> Q)
2
As for the characteristic function of assets’ log returns it is straightforward to show that
for any given model parameter α we have:
∂α ΦΣ0 (t, −iA(τ ))ec(τ ) = ΦΣ0 (t, −iA(τ ))ec(τ ) (Tr(∂α B(t)Σ0 ) + ∂α C(t) + ∂α c(τ )))
27
where the matrix derivatives are given by
∂α B(t) = −(A(τ )B12 (t) + B22 (t))(∂α A(τ )B12 (t) + A(τ )∂α B12 (t) + ∂α B22 (t))−1 B(t)
+ (A(τ )B12 (t) + B22 (t))−1 (∂α A(τ )B11 (t) + B21 (t)),
β
∂α C(t) = − Tr(Dlog,A(τ )B12 (t)+B22 (t) (∂α A(τ )B12 (t) + A(τ )∂α B12 (t) + ∂α B22 (t))).
2
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 coefficients of the correlation process ρ12
t defined by
Σ12
t
p
ρ12
=
.
t
11
Σt Σ22
t
We differentiate both sides of the equality ρ12
t
2
(49)
2
=
(Σ12
t )
22
Σ11
t Σt
. We refer to Da Fonseca et al.
(2008) for the explicit computation of all covariations involved in the below formulas. We
obtain:
12
2ρ12
t dρt
2Σ12
12 2
= 11 t 22 dΣ12
t + Σt
Σt Σt
so that
1
dρ12
t = p 11 22
Σt Σt
1
d
Σ22
t
1
Σ11
t
1
+ 11 d
Σt
1
Σ22
t
+ (.)dt,
Σ12
Σ12
t
t
11
22
dΣ12
−
dΣ
−
dΣ
+ (.)dt.
t
t
t
2Σ11
2Σ22
t
t
By using the covariations among the Wishart elements we have
11 2
1
2
12
22
2
2
Σ
Q
+
Q
+
2Σ
(Q
Q
+
Q
Q
)
+
Σ
Q
+
Q
11
12
21
22
t
12
22
t
t
11
21
22
Σ11
t Σt
2 Q211 + Q221 Q212 + Q222
Σ12
t
+ Σ12
+
+
2
(Q
Q
+
Q
Q
)
11 12
21 22
t
22
Σ11
Σ22
Σ11
t
t
t Σt
Σ12
t
12
Σ11
Q211 + Q221
− 2 11
t (Q11 Q12 + Q21 Q22 ) + Σt
Σt
Σ12
t
12
2
2
22
−2 22 Σt Q12 + Q22 + Σt (Q11 Q12 + Q21 Q22 ) dt,
Σt
d ρ12 t =
which leads to:
2 d ρ12 t = 1 − ρ12
t
ρ12
Q212 + Q222 Q211 + Q221
t (Q11 Q12 + Q21 Q22 )
p
+
−
2
22
11
22
Σt
Σt
Σ11
t Σt
Now let us compute the drift of the process ρ12
t .
28
!
dt.
12
Σt
We differentiate both sides of the equality ρ12
t = √ 11
Σt Σ22
t
and we consider the finite
variation terms:
dρ12
t
=
1
p
dΣ12
t
11
22
Σt Σt
+
Σ12
t d
!
1
p
22
Σ11
t Σt
1
+ d Σ12 , √
Σ11 Σ22 t
1
=p
Ω11 Ω21 + Ω12 Ω22 + M21 Σ11
+ M12 Σ22
+ (M11 + M22 ) Σ12
dt
t
t
t
22
Σ11
Σ
t t


1
1
p
− q
 Ω211 + Ω212 + 2M11 Σ11
+ Σ12
+ 2M12 Σ12
t
t
t
3
Σ22
t
2 Σ11
t


1
1 
22
 Ω221 + Ω222 + 2M21 Σ12
− q
+
2M
Σ
+p
22
t
t
3
Σ11
t
2 Σ22
t
11 22 3
3
+ p
2 d Σ t + p 11 22 22 2 d Σ t
11
22
11
8 Σt Σt Σt
8 Σt Σt Σt



1
1 
1
 d Σ11 , Σ12
+ q
d Σ11 , Σ22 t  dt + p
− q
t
22
3
Σt
22 3
4 Σ11
2 Σ11
t Σt
t


1
1 
 d Σ12 , Σ22 + Diffusions.
− q
+p
t
3
Σ11
t
2 Σ22
t
Now we use the formulas of the covariations of the Wishart elements and we arrive to an
expression which can be written as follows:
12 2
12
dρ12
=
A
ρ
+
B
ρ
+
C
dt + Diffusions,
t
t
t
t
t
t
where7 :
s
From the definition of Ω =
√
Σ22
t
M12 −
Σ11
t
s
Σ11
t
At = p
(Q11 Q12 + Q21 Q22 ) −
M21
11
22
Σ22
Σt Σt
t
Ω2 + Ω 2
Q2 + Q2
Q2 + Q2
Ω2 + Ω 2
Bt = − 11 11 12 − 21 22 22 + 11 11 21 + 12 22 22 h0
2Σt
2Σt
2Σt
2Σt
1
(Ω11 Ω21 + Ω12 Ω22 − 2 (Q11 Q12 + Q21 Q22 ))
Ct = p
22
Σ11
t Σt
s
s
Σ22
Σ11
t
t
+
M12 +
M21 .
11
Σt
Σ22
t
1
βQ> and the Gindikin condition we deduce that Bt is
negative. As a by-product, we easily deduce the instantaneous covariation between the
7
Notice that the diffusion term and both the expressions for Bt and Ct are different from the ones
obtained by Buraschi et al. (2006).
29
1.5 2.0 2.5
0.5 1.0
Volatility
Time Varying Volatilities
0
200
400
600
800
1000
800
1000
Index
0.4
0.0
Correlation
0.8
Time Varying Correlation
0
200
400
600
Index
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 thepMontep
Carlo
11 ,
experiments. Given Σt the dynamic covariancematrix, the volatilities
are
Σ
Σ22
t
t .
p
p
12
11
22
The correlation is obtained by computing Σt /
Σt + Σt .
Wishart element Σ11
t and the correlation process:
1
Σ12
Σ12
d ρ12 , Σ11 t = p
dhΣ11 , Σ12 it − t11 dhΣ11 , Σ11 it − t22 dhΣ22 , Σ22 it
2Σt
2Σt
Σ11 Σ22
st t
Σ11
12 2
t
=2
(Q11 Q12 + Q21 Q22 ) dt.
1
−
ρ
t
Σ22
t
Using the fact that Q ∈ GL(n, R)8 there exists a unique couple (K, Q̃) ∈ O(n) × Pn 9 such
that Q = K Q̃. 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
s
12 11 Σ11
12 2
t
d ρ ,Σ t = 2
1
−
ρ
Q̃
Q̃
+
Q̃
12
11
22 dt.
t
Σ22
t
8
GL(n, R) is the linear group, the set of invertible matrices.
9
O(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
0.10
0.08
Objective
0.06
0.04
0.02
−300
−200
−100
300
200
100
0
w
0
100
−100
200
w
−200
300 −300
Figure 2: Integrand of the C-GMM estimation criterion.
The figure displays the characteristic function with the integrated volatility presented in
equation (??). The parameters used for to compute this characteristic function are those
used in the Monte Carlo experiments.
31
M11
M12
M21
M22
Q11
Q12
Q21
Q22
β
ρ1
ρ2
Number of obs.
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Bias
RMSE
Daily
500
1500
0.0133 0.0382
1.5126 1.5213
-0.04
-0.014
1.0136 1.0341
-0.056
-0.04
0.981
1.0205
0.1667 0.1784
1.5378 1.5502
0.1077 0.1069
0.088
0.0872
0.0309 0.0287
0.0664 0.0675
-3E-04 -6E-04
0.0876 0.0862
0.0792 0.0772
0.0678 0.0661
0.0632 -0.066
4.335
4.3472
0.0055 -0.007
0.4617 0.4913
-0.004 0.0265
0.5231 0.4734
Weekly
500
1500
0.227
0.0706
1.529
1.6011
-0.048 -0.036
1.098
1.0896
-0.085 -0.036
1.104
1.1101
0.070
0.0657
1.511
1.5778
0.132
0.1166
0.476
0.3568
0.022
0.0208
0.554
0.4467
-0.059 -0.008
0.553
0.371
0.057
0.0574
0.455
0.4424
0.044
-0.107
4.546
4.4364
0.035
0.0268
0.641
0.8031
0.063
0.0445
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:
0.0225
−0.0054
−5 −3
Σ0 =
.M =
, ρ = 0.3 0.4 ,
(50)
−0.0054
0.0144
−3 −5
0.1133137 0.03335871
Q=
, β = 15.
(51)
0.0000000 0.07954368
Two different types of simulations are presented: one of the sample includes 500 daily observations and a
second one uses 1500 daily observations, as in Carrasco et al. (2007).
SP500
Min. :-0.128129
1st Qu.:-0.009940
Median : 0.002491
Mean : 0.001535
3rd Qu.: 0.013506
Max. : 0.123746
FTSE
Min. :-0.141420
1st Qu.:-0.010858
Median : 0.002101
Mean : 0.001014
3rd Qu.: 0.013424
Max. : 0.135879
DAX
Min. :-0.197775
1st Qu.:-0.014283
Median : 0.003727
Mean : 0.001523
3rd Qu.: 0.019416
Max. : 0.171546
CAC 40
Min. :-0.149149
1st Qu.:-0.015173
Median : 0.002117
Mean : 0.001095
3rd Qu.: 0.018025
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 the
SP500, FTSE, DAX and CAC time series, on a weekly sampling frequency. The dataset starts on January
2nd 1990 and ends on June 30th 2007.
32
33
SP500/CAC
-2.94
(0.217)
0.33
(0.267)
0.96
(0.372)
-3.41
(0.261)
-0.07
(0.005)
0.01
(0.007)
0.02
(0.005)
0.12
(0.005)
12.25
(0.776)
0.6
(0.103)
-0.4
(0.063)
SP500/FTSE
-2.65
(0.309)
0.74
(0.519)
1.79
(0.491)
-3.04
(0.384)
-0.01
(0.008)
-0.08
(0.007)
-0.08
(0.007)
-0.03
(0.01)
12.7
(1.236)
0.4
(0.098)
0.28
(0.081)
DAX/CAC
-3.22
(0.15)
1.51
(0.266)
1.51
(0.168)
-3.49
(0.193)
0.06
(0.004)
-0.04
(0.004)
-0.08
(0.007)
-0.1
(0.004)
11.34
(0.664)
0.16
(0.076)
0.4
(0.06)
FTSE/DAX
-2.62
(0.158)
0.04
(0.358)
0.74
(0.153)
-3.53
(0.215)
0.06
(0.005)
-0.04
(0.004)
0.09
(0.009)
0.08
(0.005)
10.74
(0.738)
-0.2
(0.081)
-0.26
(0.076)
FTSE/CAC
-2.61
(0.181)
0.5
(0.236)
0.76
(0.162)
-2.48
(0.165)
0.12
(0.003)
0.04
(0.003)
0.01
(0.004)
0.06
(0.003)
10.87
(0.638)
-0.3
(0.069)
-0.81
(0.11)
The table presents the parameters estimated using the C-GMM method developed in Carrasco et al. (2007), using a simple index procedure. The integration grid
is ω = [−300; 300]. The datasets are made of daily observations, from 01/01/1990 until 09/12/2007, using active days closing prices. Standard deviations are given
between brackets.
Table 3: Historical parameter estimates for the WASC model using the daily dataset.
M11
Std. Dev.
M12
Std. Dev.
M21
Std. Dev.
M22
Std. Dev.
Q11
Std. Dev.
Q12
Std. Dev.
Q21
Std. Dev.
Q22
Std. Dev.
β
Std. Dev.
ρ1
Std. Dev.
ρ2
Std. Dev.
SP500/DAX
-3.44
(0.239)
1.17
(0.211)
0.22
(0.469)
-2.87
(0.21)
-0.05
(0.01)
-0.1
(0.012)
0.07
(0.007)
0.07
(0.014)
10.65
(0.733)
0.3
(0.067)
-0.2
(0.071)
34
SP500/CAC
-2.67
(1.077)
1.28
(0.566)
1.54
(1.661)
-3.79
(0.393)
-0.05
(0.012)
0.02
(0.014)
0.03
(0.004)
0.13
(0.006)
13.58
(1.376)
0.01
(0.041)
-0.62
(0.039)
SP500/FTSE
-3.43
(0.671)
1.13
(0.538)
0.82
(0.72)
-3.21
(0.291)
0.07
(0.006)
0.08
(0.005)
-0.02
(0.004)
-0.05
(0.004)
13.17
(1.929)
-0.4
(0.031)
0.64
(0.058)
DAX/CAC
-3.14
(0.326)
0.44
(0.343)
0.63
(0.38)
-3.9
(0.264)
0.01
(0.003)
-0.15
(0.002)
-0.11
(0.007)
-0.1
(0.008)
14.85
(0.392)
0.39
(0.038)
0.26
(0.076)
FTSE/DAX
-2.89
(0.279)
0.83
(0.564)
0.89
(0.264)
-1.27
(0.326)
0.04
(0.011)
0.06
(0.007)
0.09
(0.006)
0.01
(0.005)
11.56
(1.847)
-0.23
(0.078)
-0.49
(0.065)
FTSE/CAC
-3.42
(0.267)
0.01
(0.527)
1.09
(0.26)
-3.76
(0.312)
-0.12
(0.006)
-0.05
(0.005)
-0.02
(0.006)
-0.07
(0.004)
14.37
(1.29)
0.36
(0.062)
0.11
(0.054)
The table presents the parameters estimated using the C-GMM method developed in Carrasco et al. (2007), using a simple index procedure. The integration grid is
ω = [−300; 300]. The datasets are made of weekly observations, from 01/01/1990 until 09/12/2007, using active days closing prices. Standard deviations are given
between brackets.
Table 4: Historical parameter estimates for the WASC model using the weekly dataset.
M11
Std. Dev.
M12
Std. Dev.
M21
Std. Dev.
M22
Std. Dev.
Q11
Std. Dev.
Q12
Std. Dev.
Q21
Std. Dev.
Q22
Std. Dev.
β
Std. Dev.
ρ1
Std. Dev.
ρ2
Std. Dev.
SP500/DAX
-2.83
(0.671)
1.87
(0.376)
0.42
(1.002)
-3.29
(0.305)
0.04
(0.003)
0.05
(0.005)
0
(0.005)
-0.12
(0.007)
12.44
(1.205)
-0.75
(0.067)
0.35
(0.035)
35
Vol
Vol
Vol
Vol
vol
vol
vol
vol
Asset
Asset
Asset
Asset
1
2
1
2
SP500/CAC
0.146
0.241
0.117
0.263
SP500/FTSE
0.161
0.171
0.146
0.189
DAX/CAC
0.2
0.215
0.221
0.361
FTSE/DAX
0.216
0.179
0.197
0.122
FTSE/CAC
0.241
0.144
0.243
0.172
Asset
Asset
Asset
Asset
1
2
1
2
SP500/FTSE
-0.327
-0.468
-0.56
-0.678
DAX/CAC
-0.224
-0.431
-0.219
-0.466
FTSE/DAX
-0.327
-0.143
-0.541
-0.307
Table 6: Skew estimated from the different datasets, with different sample frequency.
Skew
Skew
Skew
Skew
SP500/CAC
-0.687
-0.349
-0.328
-0.611
FTSE/CAC
-0.366
-0.84
-0.373
-0.299
Q11 ρ1 + Q21 ρ2
Corr dYt1 , dΣ11
= p
t
Q211 + Q221
Q12 ρ1 + Q22 ρ2
.
Corr dYt2 , dΣ22
= p
t
Q212 + Q222
In the table, the first line corresponds to the skew associated to the first asset. For example, for the column SP500/CAC, the first line corresponds to the skew of
the SP500. The second line displays the one obtained for the CAC. The skew is obtained when computing:
Weekly
Daily
SP500/DAX
-0.337
-0.36
-0.75
-0.612
In the table, the first line corresponds to the volatility of volatility of the first asset. For example, for the column SP500/CAC, the first line corresponds to the
volatility of volatility of the SP500. The second line displays the one obtained for the CAC. The volatility of volatility is obtained when computing:
q
Vol/vol Asset 1 = 2 Q211 + Q221
q
Vol/vol Asset 2 = 2 Q212 + Q222 .
Table 5: Volatility of volatility estimated from the different datasets, with different sample frequency.
Weekly
Daily
SP500/DAX
0.172
0.244
0.08
0.26
36
κ1
κ2
κ
κ1
κ2
κ
SP500/CAC
Asset 1 Asset 2
5.88
6.82
-0.363
-0.208
5.517
6.612
5.34
7.58
-2.288
-1.308
3.052
6.272
SP500/FTSE
Asset 1 Asset 2
5.3
6.08
-0.612
-0.586
4.688
5.494
20.86
6.42
-1.509
-1.536
19.351
4.884
DAX/CAC
Asset 1 Asset 2
6.44
6.98
-2.149
-2.489
4.291
4.491
6.28
7.8
-0.672
-0.796
5.608
7.004
FTSE/DAX
Asset 1 Asset 2
5.24
7.06
-0.038
-0.074
5.202
6.986
5.78
2.54
-0.844
-1.781
4.936
0.759
−κ
FTSE/CAC
Asset 1 Asset 2
5.22
4.96
-0.589
-0.983
4.631
3.977
6.84
7.52
-0.012
-0.021
6.828
7.499
κ1 is the idiosyncratic part of the mean reverting parameter. κ2 is the cross-component in the parameter and κ is the standard Heston-like
mean reverting parameter. (ρ12 ) = Corr(dlogS1 , dlogS2 ) is the long term correlation between the asset log returns. The drift is evaluated
at its stationary value, using the values presented in table 7.
dΣ11
t




s


22


Σt
11 
12 
(ρt ) dt + ...
= ... + Σt 2M11 + 2M12
11
{z }
Σt
|−κ

|
{z
}
 1


−κ2
|
{z
}
Table 7: Mean reverting parameter associated to Σt .
This table displays the mean reverting parameter associated to the volatilities. For the first asset, we have:


Weekly
Daily
SP500/DAX
Asset 1 Asset 2
6.88
5.74
-1.493
-0.74
5.387
5
5.66
6.58
-3.739
-1.804
1.921
4.776
37
SP500/CAC
0.158
0.206
0.417
0.18
0.238
0.676
SP500/FTSE
0.158
0.158
0.405
0.18
0.18
0.674
DAX/CAC
0.221
0.206
0.766
0.26
0.238
0.831
FTSE/DAX
0.221
0.158
0.669
0.26
0.18
0.739
Table 8: Long term covariance matrix implied by the estimates.
SP500/DAX
0.158
0.221
0.449
0.18
0.26
0.694
FTSE/CAC
0.206
0.158
0.761
0.238
0.18
0.781
A(t)
B(t)
C(t)
A(t)
B(t)
C(t)
SP500/CAC
-1.108
-3.246
1.722
-2.76
-3.723
3.988
SP500/FTSE
-2.376
-3.345
3.929
-1.056
-3.89
4.664
DAX/CAC
-2.902
-2.544
4.21
-0.868
-7.545
3.945
FTSE/DAX
-0.918
-2.779
2.283
-1.761
-1.986
2.849
FTSE/CAC
-1.199
-2.782
2.874
-1.21
-4.761
4.571
The computation of this drift is presented in the appendices. The drift is computed at its stationary value, using the values presented in
table 7.
12 2
12
dρ12
t = (At (ρt ) + Bt (ρt ) + Ct )dt + diffusion part.
Table 9: Mean reverting parameters associated to ρ12
t .
12
This table displays the drift part of the correlation coefficient (ρt ) in the two-asset case, for the value estimated in the previous tables.
The drift is given by:
Weekly
Daily
SP500/DAX
-1.522
-3.022
4.371
-2.922
-2.485
3.629
22
12
The long term covariance matrix is estimated through the usual moment estimators. Its diagonal elements are Σ11
t and Σt . (ρt ) is the long term correlation
computed from the estimated long term covariance matrix.
Weekly
Daily
√
√Σ11
Σ22
ρ12
√
√Σ11
Σ22
ρ12
drho
0.70
0.8
0.80
0.9
rho
DAX
CAC
rho
1.0
SP500
DAX
0.90
1.1
1.00
0.4
0.2
0.0
0.55
0.60
0.40
rho
0.65
0.70
DAX
FTSE
rho
0.45
SP500
CAC
0.75
0.50
0.80
0.55
2.0
1.5
1.0
0.60
0.70
0.80
CAC
FTSE
rho
0.75
0.85
0.90
rho
0.65 0.70 0.75 0.80 0.85 0.90
0.65
SP500
FTSE
12
The blue line is the linear term of the latter relation. The red one is the total expected variation of d(ρ12
t ) depending upon the (ρt ) values.
The green line signals the long term correlation implied by the estimated WASC model. This long term correlation is the only positive root
of the polynomial defined by the stationary drift of the correlation, using the values presented in table 7.
12 2
12
dρ12
t = (At (ρt ) + Bt (ρt ) + Ct )dt + diffusion part.
12
Figure 3: d(ρ12
t ) as a function of (ρt ) using the daily dataset.
12
This figure displays the drift term associated to d(ρ ) as a function of (ρ12 ), using daily estimation results. This relation is specified in the
following equation:
drho
0.0 0.5 1.0 1.5 2.0
−1.0
2
1
0
−1
drho
drho
−0.2
−0.4
0.6
0.2
−0.2
−0.6
drho
drho
0.5
0.0
−1.0
1.0
0.5
0.0
−0.5
38
drho
0.40
0.45
rho
0.50
DAX
CAC
rho
0.55
0.60
0.65 0.70 0.75 0.80 0.85 0.90
SP500
DAX
2.0
1.5
1.0
0.5
0.65
0.60
0.75
0.65
0.75
rho
0.85
DAX
FTSE
rho
0.70
SP500
CAC
0.95
0.80
0.85
1.5
1.0
0.9
CAC
FTSE
rho
1.0
1.1
rho
0.65 0.70 0.75 0.80 0.85 0.90 0.95
0.8
SP500
FTSE
12
The blue line is the linear term of the latter relation. The red one is the total expected variation of d(ρ12
t ) depending upon the (ρt ) values.
The green line signals the long term correlation implied by the estimated WASC model. This long term correlation is the only positive root
of the polynomial defined by the stationary drift of the correlation, using the values presented in table 7.
2
12
dρ12 = (At (ρ12
t ) + Bt (ρt ) + Ct )dt + diffusion part.
12
Figure 4: d(ρ12
t ) as a function of (ρt ) using the weekly dataset.
12
This figure displays the drift term associated to d(ρt ) as a function of (ρ12
t ), using weekly estimation results. This relation is specified in
the following equation:
drho
0.0 0.5 1.0 1.5 2.0
−1.0
1.0
0.5
0.0
−0.5
drho
drho
0.0
−1.0
1.5
1.0
0.5
0.0
−0.5
drho
drho
0.5
0.0
−1.0
1.5
1.0
0.5
0.0
−1.0 −0.5
39
40
SP500/FTSE
0.484
0.755
SP500/DAX
0.058
0.012
SP500/CAC
0.009
0.016
SP500/FTSE
0.02
0.049
DAX/CAC
0.028
0.034
FTSE/DAX
0.025
0.022
FTSE/DAX
0.328
0.541
FTSE/CAC
0.03
0.038
FTSE/CAC
0.864
0.376
decomposition of Q is unique and is refered to as polar decomposition. We refer to Faraut (2006) for further results on matrix analysis.
O(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, such that Q = K Q̃. This
Using the fact that Q ∈ GL(n, R), where GL(n, R) is the linear group, i.e. the set of invertible matrices, there exists a unique couple (K, Q̃) ∈ O(n) × Pn , where
Table 11: Off diagonal term of the polar decomposition of Q estimated from the different datasets, with different sample
frequency.
Daily
Weekly
DAX/CAC
0.431
0.466
Table 10: Norm of the ρ vector.
SP500/CAC
0.721
0.620
p
The norm of ρ is computed as ||ρ|| = (ρ> ρ), which is the Euclidian norm.
Daily
Weekly
SP500/DAX
0.361
0.828