Electronic copy available at: http://ssrn.com/abstract=1054721 Estimating the Wishart Affine Stochastic Correlation Model using the Empirical Characteristic Function * Jos´ e 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 em- pirical 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, Fran¸cois-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, Z¨ urich (2008), Mathematical and Statistical Methods for Insurance and Finance, Venice, Italy (2008), the 2nd International Workshop on Computa- tional and Financial Econometrics, Neuchˆ atel, Switzerland (2008), the First PhD Quantitative Finance Day, Swiss Banking Institute, Z¨ urich (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 Sup´ erieure d’Ing´ enieurs L´ eonard de Vinci, D´ epartement Math´ ematiques et Ing´ enierie Financi` ere, 92916 Paris La D´ efense, France. Email: jose.da [email protected] and Zeliade Systems, 56, Rue Jean- Jacques Rousseau, 75001 Paris. ‡ Universit` a 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 Gen` eve 73. E-mail: fl[email protected]. 1
40
Embed
Estimating the Wishart Affine Stochastic Correlation Model using the empirical characteristic function
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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.
∗ 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 [email protected] and Zeliade Systems, 56, Rue Jean-Jacques Rousseau, 75001 Paris.‡Universita degli Studi di Padova , Dipartimento di Matematica Pura ed Applicata, Via Trieste 63,
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
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
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.
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
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.