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We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of-fit procedure based on Cramér-von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN.
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The Canadian Journal of Statistics
Vol. xx, No. yy, 2019, Pages 1–25
La revue canadienne de statistique
1
Goodness-of-fit for regime-switching copula
models with application to option pricing
Bouchra R. Nasri 1*, Bruno N. R ´
emillard2and Mamadou Yamar Thioub2
1McGill University, Department of Mathematics and Statistics, 805 Rue Sherbrooke O, Montr´
eal
(Qu´
ebec), QC H3A 0B9
2HEC Montr´
eal, 3000 chemin de la Cˆ
ote Sainte-Catherine, Montr´
eal (Qu´
ebec), Canada H3T 2A7
Key words and phrases: Goodness-of-fit; time series; copulas ; regime-switching models ; generalized error
models.
MSC 2010: Primary 62M10; secondary 62P05
Abstract:
We consider several time series and for each of them, we fit an appropriate dynamic parametric model.
This produces serially independent error terms for each time series. The dependence between these error
terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the pa-
rameters and a sequential goodness-of-fit procedure based on Cram´
er-von Mises statistics is proposed to
select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the
proposed methodology. As an example of application, we evaluate a European put-on-max option on the
returns of two assets. In order to facilitate the use of our methodology, we have built a R package HMMcop-
ula available on CRAN. The Canadian Journal of Statistics xx: 1–25; 2019 c
2019 Statistical Society
of Canada
R´
esum´
e: Nous consid´
erons plusieurs s´
eries temporelles univari´
ees, et pour chacune nous trouvons un
mod`
ele dynamique param´
etrique appropri´
e. Nous obtenons alors des termes d’erreur ind´
ependants pour
chaque s´
erie. La d´
ependance entre ces termes d’erreur est ensuite mod´
elis´
ee par une copule avec change-
ment de r´
egime. L’algorithme EM est utilis´
e pour estimer les param`
etres et une proc´
edure s´
equentielle de
tests d’ad´
equation bas´
es sur la statistique de Cram´
er-von Mises est propos´
ee pour s´
electionner le nombre
appropri´
e de r´
egimes. Nous r´
ealisons une s´
erie d’exp´
eriences num´
eriques afin d’´
evaluer la validit´
e et la per-
formance de la m´
ethodologie propos´
ee. Comme exemple d’application, nous ´
evaluons le prix d’une option
de vente europ´
eenne sur le rendement maximal de deux titres en utilisant un mod`
ele de copule `
a change-
ment de r´
egime. Finalement, afin de faciliter l’utilisation future de la m´
ethodologie propos´
ee, nous avons
construit une librairie de fonctions bas´
ee sur le progiciel R, qui s’intitule HMMcopula, et qui est disponible
gratuitement sur CRAN. La revue canadienne de statistique xx: 1–25; 2019 c
2019 Soci´
et´
e statistique
du Canada
1. INTRODUCTION
In finance, many instruments are based on several risky assets and their evaluation rest on the
joint distribution of these assets. In fact, to determine this joint distribution, we must take into
account the serial dependence in each asset, as well as the dependence between the assets. Under-
estimating the latter can have devastating financial and economic consequences, as exemplified
by the 2008 financial crisis. We must also consider that the dependence may vary with time, po-
tentially increasing in crisis periods. Some ways to take into account time-varying dependence
have been proposed. Recently, Adams et al. (2017) fitted DCC-GARCH models (Engle, 2002)
*Author to whom correspondence may be addressed.
E-mail: bouchra.nasri@gmail.com
c
2019 Statistical Society of Canada / Soci´
et´
e statistique du Canada
CJS ???
2 Vol. xx, No. yy
to multivariate time series, which is a bit restrictive in terms of dependence since it is based on
the multivariate Gaussian distribution. To overcome this limitation, and because copulas are spe-
cially designed to model dependence, it is no wonder that many time-varying dependence models
are based on copulas.
To our knowledge, the first papers involving time-dependent copulas were Patton (2004) and
van den Goorbergh et al. (2005).InPatton (2004), the author fitted a Gaussian copula on monthly
returns (assumed independent), where the correlation parameter was a function of covariates. In
van den Goorbergh et al. (2005), the authors, in order to evaluate call-on-max options, fitted
a copula family to the residuals of two GARCH time series, with a parameter expressed as a
function of the volatilities. Note that both are special cases of what is now known as single-
index copula (Fermanian and Lopez, 2018). One can also use the methodology proposed in Nasri
and R´
emillard (2019), where generalized error models are fitted to each time series, and the
underlying copula has time-dependent parameters. In order to be able to take into account abrupt
changes in the dependence, it can be appropriate to use regime-switching copulas.
This approach has been proposed recently for vines in St¨
ober and Czado (2014), Fink et al.
(2017) and for hierarchical Archimedean models in H¨
ardle et al. (2015). In all cases, the dynamic
models for the marginal distributions were ARMA-GARCH, and there was no formal test of
goodness-of-fit. The selection of the number of regimes was based on comparisons of likelihoods,
using also rolling windows. There is not yet a theory supporting this method in our setting but
results from Capp´
e et al. (2005)[Chapter 15] shows that the BIC selection criterion works for
HMM with a discrete finite state space for the observations; unfortunately, here this hypothesis
is not met.
In this article, we propose a formal goodness-of-fit test for regime-switching copulas, which
was not done before. As a by-product, we obtain another way to select the number of regimes
based on P-values. More precisely, in Section 2, we describe the model for the time series and we
define regime-switching copulas. In Section 3, we detail the estimation procedure, the goodness-
of-fit test, and the selection of the number of regimes. Numerical experiments are performed in
Section 4to assess the validity of the procedures to choose the number of regimes. In Section 5,
we give an example of application for option pricing, along the same lines as van den Goorbergh
et al. (2005) but with different data. Note that we have built a R package for regime switching
copula models, HMMcopula available at CRAN (Thioub et al., 2018).
2. LINKING MULTIVARIATE TIME SERIES WITH REGIME-SWITCHING COPULAS
To introduce copula-based models, we proceed in two steps: first, for each univariate time se-
ries, we use a “generalized error model” (Du, 2016) to produce iid univariate series; second,
regime-switching copulas are fitted to these series. To fix ideas, let Xt= (X1t, . . . , Xdt), be
a multivariate time series. For each j∈ {1, . . . , d}, let Fj,t1contains information from the
past of Xj1,...Xj,t1, and possibly information from exogenous variables as well. Further
set Ft=d
j=1Fj,t . Assume that for each j∈ {1, . . . , d}, there exist continuous, increasing,
and Fj,t1-measurable functions Gα,jt so that εjt =Gα,jt(Xjt)are iid with continuous dis-
tribution function Fjand density fj, for some unknown parameter α∈ A. Note that stochas-
tic volatility models and Hidden Markov models (HMM) are particular cases of generalized
error models. Next, to introduce the dependence between the time series, we choose a se-
quence of Ft1-measurable copulas Ct, so that the joint conditional distribution function Kt
of εt= (ε1t, . . . , εdt)given Ft1is Kt(x) = Ct{F(x)}, with F(x)=(F1(x1), . . . , Fd(xd))>,
for any x= (x1, . . . , xd)>Rd. In particular, Ut=F(εt)Ct, for every t∈ {1, . . . , n}.
This way of modeling dependence between several time series is usually applied to innova-
tions of stochastic volatility models (van den Goorbergh et al., 2005; Chen and Fan, 2006; Pat-
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 3
ton, 2006; R´
emillard, 2017). For example, suppose that X1t=µ1t(α) + σ1t(α)ε1t,ε1tF1,
where µ1tand σ1tare F1,t1-measurable, and the innovations ε1tare independent of F1,t1. In
this case, one could take Gα,1t(x1) = x1µ1t(α)
σ1t(α), and then ε1t=Gα,1t(X1t)F1. We can also
consider Gaussian HMM models for some univariate time series. For example, suppose that there
exists a Markov chain ston {1, . . . , m}with transition matrix Qso that given s1=i1, . . . , sn=
in,X11, . . . , X1nare independent, and X1tN(µit, σ2
it). If ηt1(k)is the probability of being
in regime k∈ {1, . . . , m}at time t1given the past observations X11, . . . , X1,t1, then the
conditional distribution G1tof X1tgiven the past is G1t(x) = Pm
k=1 Wt1(k)F(k)(x), where
Wt1(k) = Pm
j=1 ηt1(j)Qjk is the probability of being in regime kat time tgiven the past
observations, and F(k)is the cdf of a Gaussian distribution with mean µkand variance σ2
k. It
then follows that the sequence U1t=G1t(X1t)are iid uniform random variables.
After having chosen the generalized error models for each univariate time series, we need
to choose the regime-switching copula model Ctfor the multivariate series Ut. This means that
there exists a finite Markov chain τton {1, . . . , `}with transition matrix P, so that given τ1=
i1, . . . , τn=in,U1,...,Unare independent, and UtCβit,t∈ {1, . . . , n}, where {Cβ;β
B} is a given parametric copula family. Also we assume the usual smoothness conditions on the
associated densities cβso that the pseudo-maximum likelihood estimator exists. Note that for
a given j∈ {1, . . . , d}, one needs that the values Ujt,t∈ {1, . . . , n}, are iid uniform. This is
indeed true as proven in the following theorem.
Theorem 1. Suppose that the multivariate time series Uthas distribution function Ctgiven
Ft1. Then for any given j∈ {1, . . . , d}, the values Ujt ,t∈ {1, . . . , n}, are iid uniform.
Proof of Theorem 1. F. or simplicity, suppose that j= 1. By hypothesis, P(U1t
u1, . . . , Udt ud|Ft1) = Ct(u1, . . . , ud). From the properties of copulas, one gets that
P(U1tu1|Ft1) = Ct(u1,1,...,1) = u1. As a result, one may conclude that U11, . . . , U1n
are iid uniform.
Since the generalized errors εtare not observable, αbeing unknown, the latter must be
estimated by a consistent estimator αn. One can then compute the pseudo-observations
en,t = (en,1t, . . . , en,dt)>=Gαn,t (Xt), where en,jt =Gαn,jt (Xjt),j∈ {1, . . . , d}and t
{1, . . . , n}. Using these pseudo-observations might be a problem, but in Nasri and R´
emillard
(2019), it was shown that using the normalized ranks of these pseudo-observations, one can esti-
mate the parameters β1,...,β`and P, as if one was observing U1,...,Un. The same applies
to the goodness-of-fit test that will be defined in Section 3.2.
Based on Theorem 2, note that in order to simulate the multivariate time series, it suf-
fices to generate Ut= (U1t, . . . , Udt)according to the regime-switching copula model, set
εjt =F1
j(Ujt ), and then compute Xjt =G1
α,jt (εjt),j∈ {1, . . . , d}, and t∈ {1, . . . , n}.
3. ESTIMATION AND GOODNESS-OF-FIT TEST
We first present general regime-switching models which can be applied to univariate time series
or copula. Then, we describe an estimation procedure and a goodness-of fit test for regime-
switching copula models. Finally, we propose a sequential procedure for selecting the optimal
number of regimes.
3.1. General regime-switching models
Let τtbe a homogeneous discrete-time Markov chain on S={1, . . . , `}, with transition proba-
bility matrix Pon S×S. Given τ1=k1, . . . , τn=kn, the observations Y= (Y1, . . . , Yn)are
independent with densities gβkt,t∈ {1, . . . , n}. Set θ= (β1,...,β`, P ). Then the joint density
DOI: The Canadian Journal of Statistics / La revue canadienne de statistique
4 Vol. xx, No. yy
of τ= (τ1, . . . , τn)and Yis
fθ(τ, Y) = n
Y
t=1
Pτt1t!×
n
Y
t=1
gβτt(Yt),(1)
so one can write
log fθ(τ, Y) =
n
X
t=1
log Pτt1t+
n
X
t=1
log gβτt(Yt).(2)
Because the regimes τtare not observable, an easy way to estimate the parameters is to use
the EM algorithm (Dempster et al., 1977), which proceeds in two steps: expectation (E step),
where Qy(˜
θ,θ) = Eθ{log f˜
θ(τ, Y)|Y=y}is computed, and maximization (M step), where
one computes
θ(k+1) = arg max
θQyθ,θ(k),
starting from an initial value θ(0). As k→ ∞,θ(k)converges to the maximum likelihood es-
timator of the density of Y. The formulas for the EM steps are given in Appendix 6. As a
particular case of regime-switching models, if Pij =νj, then one gets mixture models. In this
case τ1, . . . , τnare iid. The simplified formulas for the EM steps are given in Appendix . For
application to copulas, the density gβis the density of a parametric family of copulas Cβ, with
β∈ B. However Y1, . . . , Ynare not observable so they must be replaced by the normalized ranks
of the pseudo-observations en,t, i.e., Yjt = rank(en,j t)/(n+ 1).
3.2. Goodness-of-fit
In this section, we propose a methodology to perform a goodness-of-fit test on a multivariate
time series, by using the Rosenblatt’s transform. First, following R´
emillard (2013), under the
general regime-switching model described in Section 3.1, the conditional density ftof Ytgiven
Y1, . . . , Yt1can be expressed as a mixture viz.
ft(yt|y1, . . . , yt1) =
`
X
i=1
f(i)(yt)
`
X
j=1
ηt1(j)Pji =
`
X
i=1
f(i)(yt)Wt1(i),(3)
where f(i)=gβiand
Wt1(i) =
`
X
j=1
ηt1(j)Pji , i ∈ {1. . . `},(4)
ηt(j) = f(j)(yt)
Zt|t1
`
X
i=1
ηt1(i)Pij , j ∈ {1, . . . , `},(5)
Zt|t1=
`
X
j=1
f(j)(yt)
`
X
i=1
ηt1(i)Pij .(6)
Note that formulas (3)–(6) also hold for univariate Gaussian HMM; in this case, f(j)is the
Gaussian density with mean µjand variance σ2
j. Next, let i∈ {1, . . . , `}be fixed and suppose
that Z= (Z1, . . . , Zd)has density f(i). For any q∈ {1, . . . , d}, denote by f(i)
1:qthe density of
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 5
(Z1, . . . , Zq). Also, let f(i)
qbe the conditional density of Zqgiven Z1, . . . , Zq1. Further denote
by F(i)
qthe distribution function corresponding to density f(i)
q. The Rosenblatt’s transform Ψt
corresponding to the density (3) conditional on y1, . . . , yt1Rdis given by
Ψ(1)
t(y1t) =
`
X
i=1
Wt1(i)F(i)
1(y1t),(7)
and for q∈ {2, . . . , d},
Ψ(q)
t(y1t, . . . , yqt) = P`
i=1 Wt1(i)f(i)
1:q1(y1t, . . . , yq1,t)F(i)
q(ytq)
P`
i=1 Wt1(i)f(i)
1:q1(y1t, . . . , yq1,t).(8)
Suppose now that U1,...,Unis a random sample of size nof d-dimensional vectors drawn
from a joint continuous distribution Pbelonging to a parametric family of regime-switching
copula models with `regimes. Formally, the hypothesis to be tested is
H0:P∈ P ={Pθ;θ∈ O} vs H1:P/∈ P.
Under H0, it follows that V1= Ψ1(U1,θ),V2= Ψ2(U1,U2,θ),...,Vn=
Ψn(U1,...,Un,θ)are iid uniform over (0,1)d, where Ψ1(·,θ),...,Ψn(·,θ)are the
Rosenblatt’s transforms for the true parameters θ∈ O. However, θmust be estimated, say by
θn. Also, the random vectors U1,...,Unare not observable, so they must be replaced by the
normalized ranks un,t of the pseudo-observations en,t,t∈ {1, . . . , n}. Then, define the pseudo-
observations Vn,t = Ψt(un,t,θn),t∈ {1, . . . , n}, and for any u= (u1, . . . , ud)[0,1]d,
define the empirical process Dn(u) = 1
nPn
t=1 Qd
j=1 1(Vn,jt uj). To test H0against H1,
Genest et al. (2009) suggest to use the Cram´
er-von Mises type statistic Sndefined by
Sn=Sn(Vn,1,...,Vn,n) = nZ[0,1]d
Dn(u)
d
Y
j=1
uj
2
du
=1
n
n
X
t=1
n
X
i=1
d
Y
q=1
{1max (Vn,qt, Vn,q i)} − 1
2d1
n
X
t=1
d
Y
q=1 1V2
n,qt+n
3d.
We can interpret Snas the distance of our empirical distribution and the independence copula.
Since Vn,t ,t∈ {1, . . . , n}are almost uniformly distributed over (0,1)dunder H0, large values
of Snlead to the rejection of the null hypothesis. Unfortunately, the limiting distribution of the
test statistic will depend on the unknown parameter θ, but it does not depend on the estimated
parameters of the univariate time series (Nasri and R´
emillard, 2019). Therefore, we will use the
parametric bootstrap described in Algorithm 1to estimate P-values.
Algorithm 1 For a given number of regimes `, get estimator θnof θusing the EM algorithm
described in Section 3.1, applied to the pseudo-observations un,t,t∈ {1, . . . , n}. Then compute
the statistic Sn=Sn(Vn,1,...,Vn,n), using the pseudo-observations Vn,t = Ψt(un,t ,θn),
t∈ {1, . . . , n}. Then for k= 1, . . . , B,Blarge enough, repeat the following steps:
Generate a random sample U
1,...,U
nfrom distribution Pθn, i.e., from a regime-switching
copula model with parameter θn.
Get the estimator θ
nfrom U
1,...,U
n.
DOI: The Canadian Journal of Statistics / La revue canadienne de statistique
6 Vol. xx, No. yy
Compute the normalized ranks u
n,1,...,u
n,n from U
1,...,U
n.
Compute the pseudo-observations V
n,t =Ψtu
n,t,θ
n,t∈ {1, . . . , n}and calculate S(k)
n=
SnV
n,1,...,V
n,n.
Then, an approximate P-value for the test based on the Cram´
er-von Mises statistic Snis
given by
1
B
B
X
k=1
1S(k)
n> Sn.
3.3. Selecting the number of regimes
There are many ways one could select the copula model and the number of regimes. In the lit-
erature on regime-switching copulas, see, e.g., St¨
ober and Czado (2014), Fink et al. (2017), it is
often suggested to choose the model with the smallest AIC/BIC. However, there is no empirical
study backing up this idea. Note also that model selection based on AIC or BIC does not guaran-
tee that the model is correct. This is why one could also rely on the goodness-of-fit test described
in the previous section. In the case of a Gaussian HMM, R´
emillard (2013) suggested to choose
the number of regimes `as the first `for which the P-value is larger than 5%. We will also use
the same idea here. The consistency of all these procedures is investigated numerically in Section
4.
4. NUMERICAL EXPERIMENTS
In this section we consider Monte Carlo experiments for assessing the power of the proposed
goodness-of-fit test and the validity of the procedures proposed in Section 3.3. To this end, we
generated random samples of size n∈ {250,500,1000}from four regime-switching bivariate
copula families: Clayton, Frank, Gaussian, and Gumbel with one, two, and three regimes. For
the 1-regime model, all copulas have a Kendall’s τ=.5, while for the 2-regime copula, we
took τ=.25 for regime 1 and τ= 0.75 for regime 2, with transition matrix P= 0.25 0.75
0.50 0.50 !.
Finally, for the 3-regime copula, we took τ=.25 for regime 1, τ= 0.5for regime 2, and τ=
0.75 for regime 3, with transition matrix P=
0.5 0.25 0.25
0.3 0.5 0.2
0.1 0.3 0.6
.
For each sample size and for each model with a given number of regimes, we performed
1000 replications and in each replication, when needed, B= 100 bootstrap samples were used
to compute the P-value of the test statistic Sn.
In the first set of experiments, we assess the power of the proposed goodness-of-fit test for
different copula families. To this end, we fix the number of regimes `∈ {2,3}and vary the
copula family. The results displayed in Table 1show that for two regimes, the empirical levels
are not significantly different from the target value of 5%. For three regimes, for all but the
Frank copula with n= 250, the empirical levels are not significantly different from 5%. Next,
for two and three regimes, the estimated power is quite good, and as expected, it increases with
the sample size. From these results, we may conclude that the goodness-of-fit test can distinguish
between copula families when the number of regimes is fixed.
In the second set of experiments, we assess the power of the goodness-of-fit test for different
regimes. To this end, we fix the copula family and we vary the number of regimes `∈ {1,2,3}.
We observe from Table 2that for all models, the empirical levels are not significantly different
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 7
TABL E 1: Percentage of rejection of H0at the 5% level for copula models with `∈ {2,3}regimes, with
N= 1000 replications and B= 100 bootstrap samples.
Copula family under H0
`= 2 `= 3
H1Clayton Frank Gaussian Gumbel Clayton Frank Gaussian Gumbel
n= 250
Clayton 7.6 45.8 54.9 89.8 3.9 65.1 51.9 92.5
Frank 24.4 4.8 14.5 36.1 21.7 10.5 10.2 36.2
Gaussian 20.1 8.2 4.8 18.6 23.2 19.9 3.3 16.1
Gumbel 41.3 16.7 8.5 4.4 36.0 29.2 14.7 5.4
n= 500
Clayton 6.5 77.2 92.5 100 5.20 94.6 93.6 100
Frank 65.3 5.5 31.6 80.3 75.4 8.0 20.6 85.8
Gaussian 60.2 12.3 5.5 39.4 70.3 25.6 4.6 43.7
Gumbel 89.4 16.7 18.2 6.3 90.3 47.3 28.7 4.7
n= 1000
Clayton 5.3 99.5 100 100 5.4 100.0 100.0 100.0
Frank 99.0 5.3 67.8 100 99.9 7.0 58.7 99.8
Gaussian 97.3 29.0 5.8 77.3 70.3 24.8 3.6 43.6
Gumbel 100 43.3 41.5 5.4 99.9 82.0 66.0 6.1
from the target value of 5%. Also, when the true number of regimes is one, the percentage of
rejection of the null hypothesis of two or three regimes is also about 5%. Next, when the true
number of regimes is two or three, then the null hypothesis of one regime is easily rejected, while
generally, the percentage of rejection of the hypothesis of three (resp. two) regimes is about 5%
when there are in fact two (resp. three) regimes. We can conclude that when there are more than
one regime, the goodness-of-fit test rejects easily the null hypothesis of one regime. However,
when the true number of regimes is ` > 2, to get a good power for rejecting kregimes, with
1< k < `, one needs a large sample size. Furthermore, it seems that the goodness-of-fit test is
likely to accept a model with more regimes than necessary. This justifies that we should select
the least number of regimes with a P-value larger than 5%. This procedure is investigated next
when there are one or two regimes. The results, displayed in Table 3, show that the proposed
methodology works fine, especially when the sample size is large enough. We also tried this
method of selection when there are three regimes but the results were not satisfactory, because
most of the time, two regimes were selected, in agreement with the results in Table 2.
Finally, we repeated the second set of experiments using the AIC and BIC criteria instead
of the goodness-of-fit test for a sample size of n= 1000 only. The results are given in Table 4
and they show that when the true model has one or two regimes, both criteria are quite good,
the better one being the BIC. However, when the model has three regimes, the true number of
regimes is almost never discovered with the BIC, while the percentage is a bit better with the
AIC. These results are similar to those obtained in Table 2by using our goodness-of-fit test.
From all these results, trying to distinguish between two and three regimes seems illusory
when n1000. We checked the results of the estimation procedure when there are `3
regimes and the estimation errors are too large, because in this case, the number of parame-
ters to estimate is at least `2. We believe that in increasing the sample size enough, the estimation
errors would be smaller and one could then distinguish between two or three regimes. However,
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8 Vol. xx, No. yy
TABL E 2: Percentage of rejection of H0at the 5% level for the regime-switching Clayton, Frank,
Gaussian, and Gumbel copula models with one, two, and three regimes, using N= 1000 replications and
B= 100 bootstrap samples.
Copula family under H0
Clayton Frank
H11 regime 2 regimes 3 regimes 1 regime 2 regimes 3 regimes
n= 250
1 regime 4.5 3.8 4.0 5.8 5.3 6.4
2 regimes 99.6 7.0 4.5 75.4 5.0 10.2
3 regimes 79.7 5.6 4.2 36.3 7.0 11.6
n= 500
1 regime 4.2 3.9 3.4 5.4 6.0 6.1
2 regimes 100.0 7.0 5.4 96.4 5.2 6.9
3 regimes 98.3 6.2 5.6 66.9 5.2 7.9
n= 1000
1 regime 4.4 4.5 4.5 5.5 5.1 5.3
2 regimes 100.0 7.3 6.1 100.0 5.7 4.8
3 regimes 100.0 6.4 5.3 92.3 5.5 6.1
Gaussian Gumbel
H11 regime 2 regimes 3 regimes 1 regime 2 regimes 3 regimes
n= 250
1 regime 5.0 6.2 6.7 6.7 6.0 4.9
2 regimes 94.4 5.1 4.6 59.2 4.7 4.5
3 regimes 57.6 4.8 4.4 26.1 4.8 4.6
n= 500
1 regime 5.4 5.2 5.3 5.0 4.3 4.6
2 regimes 99.9 5.4 4.9 92.3 5.2 4.2
3 regimes 87.9 5.9 4.5 46.4 5.4 4.9
n= 1000
1 regime 4.9 6.1 5.0 5.6 5.8 6.0
2 regimes 100.0 4.4 4.5 99.8 7.1 5.1
3 regimes 99.9 4.3 4.4 80.3 4.6 4.2
numerical experiments to prove this would require months of computations. In the end, we rec-
ommend to combine goodness-of-fit tests and the AIC/BIC criteria, to ensure at least that the
chosen model is valid.
Note that one should also expect better results for the power of the goodness-of-fit test by
taking a larger number of bootstrap samples. Here, in order to build the tables in a reasonable
time, we restricted ourselves to bootstrap samples of size B= 100, which is quite small. In real
life, we do not repeat the experiments N= 1000 times, so we may use at least B= 1000, espe-
cially when the P-value is around 5%. Furthermore, we did not consider the regime-switching
Student copula since it has more parameters and, according to Table 6, the computation time is
approximately 10 times longer for the Student family than for the Gumbel family, which has the
longest computation time amongst the four other families.
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 9
TABL E 3: Estimation of the number of regimes `for N= 1000 replications, using B= 100 bootstrap
samples. Boldface values indicate the percentage of the correct choice of the number of regimes.
Copula family
Clayton Frank Gaussian Gumbel
Number of regimes Number of regimes Number of regimes Number of regimes
`?1 2 1 2 1 2 1 2
n= 250
194.4 0.8 94.8 25.1 95.3 5.4 95.2 37.5
22.3 91.8 1.5 57.0 1.6 88.7 1.5 58.0
30.5 2.9 0.7 2.3 0.4 2.3 0.4 1.2
42.8 4.5 3.0 15.6 2.7 3.6 3.6 3.3
n= 500
193.8 094.8 2.4 93.6 0.1 95.2 8.6
22.4 92.3 1.6 76.4 1.9 95.4 1.2 86.7
30.4 1.8 0.7 2.9 0.5 1.6 0.6 1.0
43.4 5.9 2.9 18.3 4 2.9 3.0 3.7
n= 1000
195.0 0.0 94.1 0.0 95.6 0.0 94.9 0.1
22.2 94.3 1.7 79.5 0.8 95.2 1.2 93.7
30.6 1.7 0.3 2.1 0.7 1.0 0.4 1.0
42.2 4.0 3.9 18.4 2.9 3.8 3.5 5.2
5. APPLICATION TO OPTION PRICING
In this application, we want to evaluate a European put-on-max option on Amazon (amzn)
and Apple (aapl) stocks. The payoff of this option is given by Φ(s1, s2) = max{K
max(s1, s2),0}, where s1and s2are the values of the stocks at the maturity of the option,
normalized to start at 1$, and Kis the strike price. An investor would be interested in such an
option to protect the returns of his assets, since he will exercise the option if both returns are
lower than log K. Also this option is cheaper than a put-on-min. In order to evaluate this option,
we need first to find the joint distribution of both assets. Next, we will choose an appropriate risk
neutral probability measure.
5.1. Joint distribution
The first step is to fit dynamic models for the univariate time series. To this end, we used the
adjusted prices of Amazon and Apple from January 1, 2015 to June 29, 2018. The sample size is
880 observations for each time series. The 879 daily log-returns of the stocks are shown in Figure
1. Since van den Goorbergh et al. (2005) used GARCH models for the log-returns of the assets
they considered, we also tried to fit GARCH(p,q) models with Gaussian innovations, but we
rejected the null hypothesis for p, q 3. We next tried to fit Gaussian HMMs to the log-returns.
Using the selection procedure described in Section 3.2, we obtained a Gaussian HMM with three
regimes for the daily log-returns of Amazon as well as for the daily log-returns of Apple. Here,
the P-values are 38.8% and 15.1% respectively, computed using B= 1000 bootstrap samples.
The estimated parameters for both time series are given in Table 5. Note that the regimes are
ordered by their stationary distribution (ν), meaning that the least frequent regime is 1, and the
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10 Vol. xx, No. yy
TABL E 4: Percentage of selection of the number of regimes `∈ {1,2,3}for a sample size n= 1000 and
N= 1000 replications, based on the AIC and BIC criteria. Boldface values indicate the percentage of the
correct choice of the number of regimes.
Copula family under H0
Clayton
AIC BIC
H11 regime 2 regimes 3 regimes 1 regime 2 regimes 3 regimes
1 regime 97.4 2.6 0 100 0 0
2 regimes 0 96.4 3.6 0 99.9 0.1
3 regimes 0 90.2 9.8 0 99.9 0.1
Frank
AIC BIC
H11 regime 2 regimes 3 regimes 1 regime 2 regimes 3 regimes
1 regime 97.8 2.2 0 100 0 0
2 regimes 0 97.2 2.8 0 100 0
3 regimes 0 98.2 1.8 4.4 95.6 0
Gaussian
AIC BIC
H11 regime 2 regimes 3 regimes 1 regime 2 regimes 3 regimes
1 regime 98.4 1.6 0 100 0 0
2 regimes 0 98.1 1.9 0 100 0
3 regimes 0 95.3 4.7 0 100 0
Gumbel
AIC BIC
H11 regime 2 regimes 3 regimes 1 regime 2 regimes 3 regimes
1 regime 97.6 2.4 0 100 0 0
2 regimes 0 96.2 3.8 0 100 0
3 regimes 0 96.4 3.6 0.9 99.1 0
most frequent regime is 3. As can be seen from Figure 1, for Amazon, regime 1 corresponds to
large positive returns with a frequency of 2%, while for Apple, regime 1 consists in large negative
values with a frequency of 10%. So in both cases, regime 1 is not a persistent state, while the two
other regimes are much more persistent. For the two stocks, since regime 2 has always a negative
mean, it can be interpreted as a bear market, while for regime 3, the mean µ3is positive, as well
as µ3σ2
3/2, so this regime can be interpreted as a bull market.
From now on, let X1tdenotes the log-returns of Amazon and let X2tdenotes the log-returns
of Apple, and let F1tand F2tbe the conditional distributions of X1tand X2tgiven the past
observations, corresponding to the densities defined by Equation (3). Further set U1t=F1t(X1t)
and U2t=F2t(X2t). As defined in Section 3, let en,jt =Fn,j t(Xjt ),j= 1,2, be the pseudo-
observations, where Fn,jt is the conditional distribution function computed with the parameters
of Table 5. The graph of the normalized ranks of un,t = (un,1t, un,2t)is displayed in Figure 2.
Next, in order to select the appropriate regime-switching copula model, we performed goodness-
of-fit tests using B= 1000 bootstrap samples to select the copula family and the number of
regimes amongst the Clayton, Frank, Gaussian, Gumbel and Student families. Note that for the 1-
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 11
FIGURE 1: Daily log returns and predicted regimes for Amazon and Apple.
TABL E 5: Estimated parameters for the log-returns of Amazon and Apple, using Gaussian HMM. Here, ν
is the stationary distribution of the regimes, and Qis the transition matrix.
Amazon Apple
Regime Regime
Parameter 1 2 3 1 2 3
µ×1024.4122 -0.1179 0.1892 -0.2777 -0.0433 0.2215
σ×10357.2765 22.9170 10.1334 2.4846 20.6456 8.8176
ν0.0199 0.2574 0.7227 0.1015 0.3894 0.5091
Q
0.1572 0.3978 0.4450
0.0545 0.8849 0.0606
0.0038 0.0300 0.9662
0.0674 0.4154 0.5172
0.0000 0.8788 0.1212
0.1859 0.0098 0.8042
regime and 2-regime Student copulas, and for the 2-regime Gaussian copula, we took B= 10000
in order to get more precise results. The corresponding P-values are given in Table 6, together
with the computation time in seconds for B= 1000 bootstrap samples,
and the BIC values. From this table, based on the P-values, we can see that the 2-regime
Gaussian and the 2-regime Student copula models are valid, while the 1-regime Student copula
model is almost acceptable. However, the estimated degrees of freedom of the 2-regime Student
copula model are very large, indicating that it is indeed a 2-regime Gaussian copula. We then
DOI: The Canadian Journal of Statistics / La revue canadienne de statistique
12 Vol. xx, No. yy
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Apple
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Amazon
FIGURE 2: Scatter plot of the normalized ranks of the pseudo-observations un,t for Apple and Amazon.
restricted the degrees of freedom to be less than 25, and we obtained the upper bounded as their
estimations. Looking at the BIC values, we see that the smallest one is for the 2-regime Gaussian
copula. Based on these results, we choose the 2-regime Gaussian copula as the best model. Its
estimated parameters appear in Table 7.
TABL E 6: P-values (in percentage) of the different regime-switching copula families, together with the
computation time in seconds and the BIC criterion.
Copula family
Clayton Frank Gaussian Gumbel Student
Number of regimes Number of regimes Number of regimes Number of regimes Number of regimes
1 2 1 2 1 2 1 2 1 2
P-value 0.0 1.0 0.6 0.8 0.0 9.8 0.0 0.0 4.4 10.1
Sec. 195 1272 128 490 175 712 198 1342 1715 15386
BIC -174.0 -181.3 -188.8 -190.8 -179.6 -205.3 -168.7 -182.3 -201.8 -191.8
TABL E 7: Estimated parameters for the 2-regime Gaussian copula. Here, τis Kendall’s tau,
ρ= sin(πτ /2) is the correlation coefficient of the copula, νis the stationary distribution of the regimes,
and Pis the transition matrix.
Parameter Regime 1 Regime 2
τ0.0859 0.5816
ρ0.1346 0.7917
ν0.5209 0.4791
P 0.7414 0.2586
0.2812 0.7188 !
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 13
5.2. Bivariate option pricing
In order to price an option with payoff Φ(S1n, S2n)over ntrading days, we perform a Monte
Carlo simulation under a risk neutral measure. First, as in van den Goorbergh et al. (2005), we
assume that the selected regime-switching copula model with parameters appearing in Table 7
is also valid under the risk neutral measure. Next, for the dynamic models of both time series,
we assume that we still have Gaussian HMM, but with new parameters, namely ˜µjk =rσ2
jk
2
,˜σjk =σjk, and ˜
Q(j)=Q(j), where ris the risk free daily interest rate. This way, under the
risk neutral measure, the discounted prices ertSj t =ePt
i=1(Xj ir)form a martingale, for each
j= 1,2.
The following steps illustrate the procedure to evaluate a European option with payoff Φin
the case of a general regime-switching copula with `regimes, where each univariate time series
is modeled by a Gaussian HMM with mjregimes and parameters ˜µj1,...,˜µj`j,˜σj1,...,˜σj`j,
˜
Q(j):
1. Generate Ut,t∈ {1, . . . , n}, from the regime-switching copula model.
2. For t∈ {1, . . . , n}, and j= 1,2, compute the conditional distribution function Fjt under
the risk neutral measure, and set Xjt =F1
jt (Ujt).
3. For j= 1,2, compute Sjn =ePn
t=1 Xji .
4. Repeat Ntimes steps 13, in order to get Nindependent values of (S1n, S2n).
The value of the option is then approximated by the average of the discounted values
ernΦ(S1n, S2n).
To evaluate the put-on-max option, we used N= 10000 simulations, with a maturity of n=
20 trading days and a risk free rate r= 4%.
Figure 3displays the price of the option as a function of the strike Kfor the best model,
i.e., the 2-regime Gaussian copula, versus the other four 2-regime copula families. As expected,
the prices given by the 2-regime Gaussian copula and the 2-regime Student are almost identical.
Note that the prices of the 2-regime Frank and Gumbel copulas are always lower than those of
the 2-regime Gaussian copula, while those of the 2-regime Clayton copula are higher than those
of the 2-regime Gaussian copula when the strike value is lower than 1.0005, and are lower when
the strike is larger than 1.0005.
6. CONCLUSION
In this paper, for a regime-switching copula model, we proposed a methodology based on a
goodness-of-fit test to select the copula family and the number of regimes. This methodology
can also be used for mixtures of copula models, as well as for univariate HMM. We performed
Monte Carlo simulations with a sample size n∈ {250,500,1000}, and we showed that the level
of the goodness-of-fit test is correct and that it is powerful enough to distinguish between regime-
switching copula families and also to detect if there is more than one regime. The proposed
procedure for selecting the number of regimes works when the sample size is large enough and
there are less than three regimes. For three regimes or more, the sample size must be larger than
1000. As an example of application, we showed how to evaluate a European put-on-max option,
but the proposed methodology can also be applied to a wide range of options on multivariate
assets. The empirical results emphasize the importance of choosing the correct copula family.
DOI: The Canadian Journal of Statistics / La revue canadienne de statistique
14 Vol. xx, No. yy
FIGURE 3: Comparison of put-on-max prices for n= 20 trading days maturity, as a function of the strike,
between a 2-regime Gaussian copula and 2-regime Clayton, Frank, Gumbel and Student copula models.
Acknowledgments
The authors are grateful to the Guest Editor, Cody Hyndman and two anonymous referees for
their comments and suggestions. Partial funding in support of this work was provided by the
Natural Sciences and Engineering Research Council of Canada (Grant 04430–2014), the Cana-
dian Statistical Sciences Institute (postdoctoral fellowship), the Fonds de recherche du Qu´
ebec –
Nature et technologies (2015–PR–183236 and postdoctoral fellowship 259667), and the Groupe
d’´
etudes et de recherche en analyse des d´
ecisions (postdoctoral fellowship).
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 15
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APPENDIX
DOI: The Canadian Journal of Statistics / La revue canadienne de statistique
16 Vol. xx, No. yy
Estimation for general regime-switching models
E-Step
Set ˜
θ= ( ˜
β1,..., ˜
β`,˜
P). Then, according to (R´
emillard, 2013, Appendix 10.A),
Qy(˜
θ,θ) = Eθ{log f˜
θ(τ, Y )|Y=y}
=
n
X
t=1 X
jSX
kS
Pθ(τt1=j, τt=k|Y=y) log ˜
Pjk +
n
X
t=1 X
jS
Pθ(τt=j|Y=y) log g˜
βj(yt)
=
n
X
t=1 X
jSX
kS
Λθ,t(j, k) log ˜
Pjk +
n
X
t=1 X
jS
λθ,t(j) log g˜
βj(yt),
where λθ,t(j) = P(τt=j|Y=y)and Λθ,t(j, k ) = P(τt1=j, τt=k|Y=y), for all t
{1, . . . , n}and j, k S. Next, define for all jS,¯ηθ,n(j)=1/`,ηθ,0(j)=1/`,
¯ηθ,t(j) = P rob(τt=j|yt+1 , . . . , yn), t = 1, . . . , n 1,
ηθ,t(j) = P r ob(τt=j|y1, . . . , yt), t = 1, . . . , n.
It follows easily that for t= 1, . . . , n,ηt(j) = gβj(yt)
Zt|t1P`
i=1 ηt1(i)Pij , where
Zt|t1=
`
X
j=1
gβj(yt)
`
X
i=1
ηt1(i)Pij .
Next, for all i∈ {1, . . . , l}, and for all t= 0, . . . , n 1,
¯ηθ,t(i) = P`
β=1 ¯ηθ,t+1(β)Pgββ(yt+1)
P`
k=1 P`
β=1 ¯ηθ,t+1(β)Pgββ(yt+1 ),
λθ,t(i) = ηθ,t (iηθ,t(i)
Pl
k=1 ηθ,t(k)¯ηθ,t(k).
Hence, for all i, j ∈ {1, . . . , l}, and for all t= 1, . . . , n,
Λθ,t(i, j ) = Pij ηθ,t1(i) ¯ηθ,t(j)gβj(yt)
Pl
k=1 Pl
β=1 Pηθ,t1(k)¯ηθ,t(β)gββ(yt).
As a result, for all i∈ {1, . . . , l}, and for every t= 1, . . . , n,Pl
j=1 Λθ,t(i, j ) = λθ,t1(i).
M-Step
For this step, given θ(k),θ(k+1) is defined as θ(k+1) = arg maxθQyθ,θ(k). Setting λ(k)
t(i) =
λθ(k),t(i)and Λ(k)
t(i, j)=Λθ(k),t (i, j), it follows from Section that
θ(k+1) = arg max
θ
n
X
t=1 X
i,jS
Λ(k)
t(i, j) log Pij +
n
X
t=1 X
iS
λ(k)
t(i) log gβi(yt).
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
2019 GOODNESS-OF-FIT FOR REGIME-SWITCHING COPULA MODELS 17
Using Lagrange multipliers, the function to maximize is h(θ, ψ), where ψ= (ψ1, . . . , ψ`), and
h(θ, ψ) =
n
X
t=1 X
i,jS
Λ(k)
t(i, j) log Pij +
n
X
t=1 X
iS
λ(k)
t(i) log gβi(yt) +
l
X
i=1
ψi
1
`
X
j=1
Pij
.
For i, j Swe have ∂h
∂Pi,j =Pn
t=1 Λ(k)
t(i, j)1
Pij ψi. As a result, for any i, j S, the partial
derivative of hwith respect to Pij is zero if and only if ψiPij =Pn
t=1 Λ(k)
t(i, j). Summing over
jyields that
ψi=
`
X
j=1
ψiPij =
`
X
j=1
n
X
t=1
Λ(k)
t(i, j) =
n
X
t=1
λ(k)
t1(i) =
n
X
t=1
λθ(k),t1(i).
Hence P(k+1)
ij =Pn
t=1 Λ(k)
t(i, j).Pn
t=1 λ(k)
t1(i). Also, maximizing hwith respect to
β1,...,β`amounts to maximize Pn
t=1 P`
i=1 λ(k)
t(i) log gβi(yt)with respect to βi, for all iS.
Estimation for general mixture models
This model is a particular case of regime-switching where Pij =νj,j∈ {1, . . . , `}. So, under
this model, τtis a sequence of iid observations with distribution ν= (ν1, . . . , ν`). The algo-
rithm described previously can then be simplified. To this end, set θ= (β1,...,β`, ν ). The joint
density of τ= (τ1, . . . , τn)and Yis fθ(τ, Y )=(Qn
t=1 ντt)×Qn
t=1 gβτt(Yt), yielding
log fθ(τ, Y ) =
n
X
t=1
log Pτt1t+
n
X
t=1
log gβτt(Yt).
E-Step
Set ˜
θ= ( ˜
β1,..., ˜
β`,˜ν). Then, according to the previous computations,
Qy(˜
θ,θ) = Eθ{log f˜
θ(τ, Y )|Y=y}=
n
X
t=1 X
jS
λθ,t(j)log ˜νj+ log g˜
βj(Yt),(1)
where λθ,t(j) = Pθ(τt=j|Y=y) = ˜νjg˜
βτt(yt)
P`
k=1 ˜νkg˜
βk(yt)for all t∈ {1, . . . , n}and jS.
M-Step
For this step, given θ(k),θ(k+1) is defined as θ(k+1) = arg maxθQyθ,θ(k). Setting λ(k)
t(i) =
λθ(k),t(i), one obtains
Qy(˜
θ,θ) =
n
X
t=1
`
X
j=1
λ(k)
t(j)log ˜νj+ log g˜
βj(Yt)
=
n
X
t=1
`
X
j=1
λ(k)
t(j) log ˜νj+
n
X
t=1
`
X
j=1
λ(k)
t(j) log g˜
βj(Yt).
DOI: The Canadian Journal of Statistics / La revue canadienne de statistique
18 Vol. xx, No. yy
For jSwe have, Qy
˜νj=n
˜νjPn
t=1 λ(k)
t(j). Hence ˜νj(k+1) =Pn
t=1 λ(k)
t(j)
n,j∈ {1, . . . , l}, and
˜
βj
(k+1) = arg max
˜
βj
n
X
t=1
`
X
j=1
λ(k)
t(j) log g˜
βj(yt).
Received 10 March 2019
Accepted 18 October 2019
The Canadian Journal of Statistics / La revue canadienne de statistique DOI:
... We note, however, that it is also possible to fit HMMs with state-dependent copulae (e.g. Nasri et al., 2020;Ötting et al., 2021). In the future, we plan to further compare these various approaches to modelling correlation in movement data. ...
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... Bucio-Pacheco 等 [142] 将 Patton [34] 的时变 copula 模型应用于以 墨西哥股票指数和标普 500 指数为标的的二元期权定价. Nasri 等 [143] 将机制转换 copula 模型应用于 两资产的欧式最大认沽期权 (put-on-max option) 的定价. ...
... However, as expected, detecting more than one regime requires longer data sets, which is true for any stochastic model. The power of the goodness of fit test has been studied in a similar context (Nasri et al., 2020), and the authors showed that the selection procedure based on P values is valid and efficient. ...
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