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On the Likelihood of Black Swan Events
Miguel de Carvalho
School of Mathematics, University of Edinburgh
Global Portuguese Mathematicians
June 25, 2019
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 1 / 31
Examples of Extreme Events
Some Examples of Extreme EventsIn Cauda Venenum (The poison in the tail)
Figure: Floods in the UK (Storms Desmond and Eva), earthquakes in New Zealandand Japan, hurricane Katrina, mudslides in Rio de Janeiro, February 2010 Chileanearthquake.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 2 / 31
Motivating the Need for Extrapolating into the Tails
Introduction and MotivationBlack–Swans
Consider a random sample
Y1, . . . ,Yniid∼ F .
The empirical distribution function is
FY (y) := n−1n∑
i=1
I(Yi > y).
Obviously, the empirical distribution functionhas no ability to extrapolate into the tails,beyond observed data; i.e.,
P(Y > y) = 1− FY (Mn + ε) = 0, ∀ ε > 0,
where Mn ≡ maxY1, . . . ,Yn.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 3 / 31
Motivating the Need for Extrapolating into the Tails
Introduction and MotivationBlack–Swans
Consider a random sample
Y1, . . . ,Yniid∼ F .
The empirical distribution function is
FY (y) := n−1n∑
i=1
I(Yi > y).
Obviously, the empirical distribution functionhas no ability to extrapolate into the tails,beyond observed data; i.e.,
P(Y > y) = 1− FY (Mn + ε) = 0, ∀ ε > 0,
where Mn ≡ maxY1, . . . ,Yn.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 3 / 31
Motivating the Need for Extrapolating into the Tails
Introduction and MotivationBlack–Swans
Consider a random sample
Y1, . . . ,Yniid∼ F .
The empirical distribution function is
FY (y) := n−1n∑
i=1
I(Yi > y).
Obviously, the empirical distribution functionhas no ability to extrapolate into the tails,beyond observed data; i.e.,
P(Y > y) = 1− FY (Mn + ε) = 0, ∀ ε > 0,
where Mn ≡ maxY1, . . . ,Yn.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 3 / 31
Motivating the Need for Extrapolating into the Tails
Introduction and MotivationBlack–Swans
Consider a random sample
Y1, . . . ,Yniid∼ F .
The empirical distribution function is
FY (y) := n−1n∑
i=1
I(Yi > y).
Obviously, the empirical distribution functionhas no ability to extrapolate into the tails,beyond observed data; i.e.,
P(Y > y) = 1− FY (Mn + ε) = 0, ∀ ε > 0,
where Mn ≡ maxY1, . . . ,Yn.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 3 / 31
Extreme 6= Rare
Small Probability EventsIn Cauda Venenum (The poison in the tail)
Statistics of Extremes is thus about small probability events, linked to the tail of adistribution.
But beware, rare is not the same as extreme.
−15 −10 −5 0 5 10 15
0.00
0.05
0.10
0.15
0.20
x
Den
sity
rare and extreme rare rare and extreme
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 4 / 31
Extreme 6= Rare
Small Probability EventsIn Cauda Venenum (The poison in the tail)
Statistics of Extremes is thus about small probability events, linked to the tail of adistribution.
But beware, rare is not the same as extreme.
−15 −10 −5 0 5 10 15
0.00
0.05
0.10
0.15
0.20
x
Den
sity
rare and extreme rare rare and extreme
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 4 / 31
Extreme 6= Rare
Statistical Modeling of Univariate Extremes
Definition (Statistical Model)
Let G be a probability measure on (Ω,A), and let Θ be a parameter space. The familyGθ : θ ∈ Θ is a statistical model.
Obviously not every statistical model is appropriate for modeling risk.
Candidate statistical models should possess the ability to extrapolate into the tails of adistribution, beyond existing data.
Theorem (Extremal Types Theorem)
If there exist sequences an > 0 and bn such that P(Mn − bn)/an 6 z → Gθ(z), asn→∞, for some non-degenerate distribution Gθ , then
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ], θ = (µ, σ, ξ),
defined on z : 1 + ξ(z − µ)/σ > 0 where µ ∈ R, σ ∈ (0,∞), and ξ ∈ R.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 5 / 31
Extreme 6= Rare
Statistical Modeling of Univariate Extremes
Definition (Statistical Model)
Let G be a probability measure on (Ω,A), and let Θ be a parameter space. The familyGθ : θ ∈ Θ is a statistical model.
Obviously not every statistical model is appropriate for modeling risk.
Candidate statistical models should possess the ability to extrapolate into the tails of adistribution, beyond existing data.
Theorem (Extremal Types Theorem)
If there exist sequences an > 0 and bn such that P(Mn − bn)/an 6 z → Gθ(z), asn→∞, for some non-degenerate distribution Gθ , then
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ], θ = (µ, σ, ξ),
defined on z : 1 + ξ(z − µ)/σ > 0 where µ ∈ R, σ ∈ (0,∞), and ξ ∈ R.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 5 / 31
Extreme 6= Rare
Statistical Modeling of Univariate Extremes
Definition (Statistical Model)
Let G be a probability measure on (Ω,A), and let Θ be a parameter space. The familyGθ : θ ∈ Θ is a statistical model.
Obviously not every statistical model is appropriate for modeling risk.
Candidate statistical models should possess the ability to extrapolate into the tails of adistribution, beyond existing data.
Theorem (Extremal Types Theorem)
If there exist sequences an > 0 and bn such that P(Mn − bn)/an 6 z → Gθ(z), asn→∞, for some non-degenerate distribution Gθ , then
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ], θ = (µ, σ, ξ),
defined on z : 1 + ξ(z − µ)/σ > 0 where µ ∈ R, σ ∈ (0,∞), and ξ ∈ R.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 5 / 31
Extreme 6= Rare
Statistical Modeling of Univariate Extremes
Definition (Statistical Model)
Let G be a probability measure on (Ω,A), and let Θ be a parameter space. The familyGθ : θ ∈ Θ is a statistical model.
Obviously not every statistical model is appropriate for modeling risk.
Candidate statistical models should possess the ability to extrapolate into the tails of adistribution, beyond existing data.
Theorem (Extremal Types Theorem)
If there exist sequences an > 0 and bn such that P(Mn − bn)/an 6 z → Gθ(z), asn→∞, for some non-degenerate distribution Gθ , then
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ], θ = (µ, σ, ξ),
defined on z : 1 + ξ(z − µ)/σ > 0 where µ ∈ R, σ ∈ (0,∞), and ξ ∈ R.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 5 / 31
Extreme 6= Rare
Univariate Extremes—AsymptoticsTrinity of limiting distributions
R. A. Fisher B. Gnedenko M. Fréchet E. Gumbel
Notes:
µ and σ are location and scale parameters; ξ is a shape parameter that determines the ratedecay of the tail:
i: ξ → 0: light-tail (Gumbel);
ii: ξ > 0: heavy-tail (Fréchet);iii: ξ < 0: short-tail (negative Weibull).
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 6 / 31
Extreme 6= Rare
Univariate Extremes—AsymptoticsTrinity of limiting distributions
R. A. Fisher B. Gnedenko M. Fréchet E. Gumbel
Notes:
µ and σ are location and scale parameters; ξ is a shape parameter that determines the ratedecay of the tail:
i: ξ → 0: light-tail (Gumbel);ii: ξ > 0: heavy-tail (Fréchet);
iii: ξ < 0: short-tail (negative Weibull).
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 6 / 31
Extreme 6= Rare
Univariate Extremes—AsymptoticsTrinity of limiting distributions
R. A. Fisher B. Gnedenko M. Fréchet E. Gumbel
Notes:
µ and σ are location and scale parameters; ξ is a shape parameter that determines the ratedecay of the tail:
i: ξ → 0: light-tail (Gumbel);ii: ξ > 0: heavy-tail (Fréchet);iii: ξ < 0: short-tail (negative Weibull).
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 6 / 31
Extreme 6= Rare
Univariate Extremes—AsymptoticsTrinity of limiting distributions
R. A. Fisher B. Gnedenko M. Fréchet E. Gumbel
Notes:
µ and σ are location and scale parameters; ξ is a shape parameter that determines the ratedecay of the tail:
i: ξ → 0: light-tail (Gumbel);ii: ξ > 0: heavy-tail (Fréchet);iii: ξ < 0: short-tail (negative Weibull).
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 6 / 31
Extreme 6= Rare
Univariate Extremes—AsymptoticsTrinity of limiting distributions
R. A. Fisher B. Gnedenko M. Fréchet E. Gumbel
Notes:
µ and σ are location and scale parameters; ξ is a shape parameter that determines the ratedecay of the tail:
i: ξ → 0: light-tail (Gumbel);ii: ξ > 0: heavy-tail (Fréchet);iii: ξ < 0: short-tail (negative Weibull).
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 6 / 31
Extreme 6= Rare
Introduction, Motivation, and BackgroundWhat is this Talk About?
Relevant models for the multivariate setting should allows us to understand the dependencebetween extremes (extremal dependence) of two random variables, and its dynamics (say,over time).
A special type of statistical model which plays a key role on multivariate extreme valuemodeling:
Definition (Measure-Dependent Measure)
Let F be the space of all probability measures that can be defined over (Ω0,A0). If GH is aprobability measure on (Ω1,A1), for all H ∈ H ⊆ F , then we say that
GH
is a measure-dependent measure. The family GH : H ∈ H is said to be a set ofmeasure-dependent measures, if GH is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 7 / 31
Extreme 6= Rare
Introduction, Motivation, and BackgroundWhat is this Talk About?
Relevant models for the multivariate setting should allows us to understand the dependencebetween extremes (extremal dependence) of two random variables, and its dynamics (say,over time).
A special type of statistical model which plays a key role on multivariate extreme valuemodeling:
Definition (Measure-Dependent Measure)
Let F be the space of all probability measures that can be defined over (Ω0,A0). If GH is aprobability measure on (Ω1,A1), for all H ∈ H ⊆ F , then we say that
GH
is a measure-dependent measure. The family GH : H ∈ H is said to be a set ofmeasure-dependent measures, if GH is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 7 / 31
Extreme 6= Rare
Introduction, Motivation, and BackgroundWhat is this Talk About?
Relevant models for the multivariate setting should allows us to understand the dependencebetween extremes (extremal dependence) of two random variables, and its dynamics (say,over time).
A special type of statistical model which plays a key role on multivariate extreme valuemodeling:
Definition (Measure-Dependent Measure)
Let F be the space of all probability measures that can be defined over (Ω0,A0). If GH is aprobability measure on (Ω1,A1), for all H ∈ H ⊆ F , then we say that
GH
is a measure-dependent measure. The family GH : H ∈ H is said to be a set ofmeasure-dependent measures, if GH is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 7 / 31
Extreme 6= Rare
Introduction, Motivation, and BackgroundWhat is this Talk About?
Relevant models for the multivariate setting should allows us to understand the dependencebetween extremes (extremal dependence) of two random variables, and its dynamics (say,over time).
A special type of statistical model which plays a key role on multivariate extreme valuemodeling:
Definition (Measure-Dependent Measure)
Let F be the space of all probability measures that can be defined over (Ω0,A0).
If GH is aprobability measure on (Ω1,A1), for all H ∈ H ⊆ F , then we say that
GH
is a measure-dependent measure. The family GH : H ∈ H is said to be a set ofmeasure-dependent measures, if GH is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 7 / 31
Extreme 6= Rare
Introduction, Motivation, and BackgroundWhat is this Talk About?
Relevant models for the multivariate setting should allows us to understand the dependencebetween extremes (extremal dependence) of two random variables, and its dynamics (say,over time).
A special type of statistical model which plays a key role on multivariate extreme valuemodeling:
Definition (Measure-Dependent Measure)
Let F be the space of all probability measures that can be defined over (Ω0,A0). If GH is aprobability measure on (Ω1,A1), for all H ∈ H ⊆ F , then we say that
GH
is a measure-dependent measure.
The family GH : H ∈ H is said to be a set ofmeasure-dependent measures, if GH is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 7 / 31
Extreme 6= Rare
Introduction, Motivation, and BackgroundWhat is this Talk About?
Relevant models for the multivariate setting should allows us to understand the dependencebetween extremes (extremal dependence) of two random variables, and its dynamics (say,over time).
A special type of statistical model which plays a key role on multivariate extreme valuemodeling:
Definition (Measure-Dependent Measure)
Let F be the space of all probability measures that can be defined over (Ω0,A0). If GH is aprobability measure on (Ω1,A1), for all H ∈ H ⊆ F , then we say that
GH
is a measure-dependent measure. The family GH : H ∈ H is said to be a set ofmeasure-dependent measures, if GH is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 7 / 31
Extreme 6= Rare
From Univariate to Bivariate Extremes
The generalized EVD is a three parameter family
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ
+
], θ = (µ, σ, ξ),
which plays an important role in univariate extreme value theory.
However, often the interest is in the extremes of more than one variable, but moving to ahigher number of dimensions increases sharply the complexity of the models in terms ofnumber of parameters.
The first challenge one faces when modeling multivariate extremes is that the estimationobjects of interest are often infinite-dimensional, contrary to what happens in the univariatecase where only three parameters are needed.
Remark
Below, H denotes the space of all probability measures H which can be defined over([0, 1],B[0,1]), where B[0,1] is the Borel sigma-algebra on [0, 1], and which obey the meanconstraint ∫
[0,1]wH(dw) =
12. (1)
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 8 / 31
Extreme 6= Rare
From Univariate to Bivariate Extremes
The generalized EVD is a three parameter family
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ
+
], θ = (µ, σ, ξ),
which plays an important role in univariate extreme value theory.
However, often the interest is in the extremes of more than one variable, but moving to ahigher number of dimensions increases sharply the complexity of the models in terms ofnumber of parameters.
The first challenge one faces when modeling multivariate extremes is that the estimationobjects of interest are often infinite-dimensional, contrary to what happens in the univariatecase where only three parameters are needed.
Remark
Below, H denotes the space of all probability measures H which can be defined over([0, 1],B[0,1]), where B[0,1] is the Borel sigma-algebra on [0, 1], and which obey the meanconstraint ∫
[0,1]wH(dw) =
12. (1)
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 8 / 31
Extreme 6= Rare
From Univariate to Bivariate Extremes
The generalized EVD is a three parameter family
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ
+
], θ = (µ, σ, ξ),
which plays an important role in univariate extreme value theory.
However, often the interest is in the extremes of more than one variable, but moving to ahigher number of dimensions increases sharply the complexity of the models in terms ofnumber of parameters.
The first challenge one faces when modeling multivariate extremes is that the estimationobjects of interest are often infinite-dimensional, contrary to what happens in the univariatecase where only three parameters are needed.
Remark
Below, H denotes the space of all probability measures H which can be defined over([0, 1],B[0,1]), where B[0,1] is the Borel sigma-algebra on [0, 1], and which obey the meanconstraint ∫
[0,1]wH(dw) =
12. (1)
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 8 / 31
Extreme 6= Rare
From Univariate to Bivariate Extremes
The generalized EVD is a three parameter family
Gθ(z) = exp[−
1 + ξ
(z − µσ
)−1/ξ
+
], θ = (µ, σ, ξ),
which plays an important role in univariate extreme value theory.
However, often the interest is in the extremes of more than one variable, but moving to ahigher number of dimensions increases sharply the complexity of the models in terms ofnumber of parameters.
The first challenge one faces when modeling multivariate extremes is that the estimationobjects of interest are often infinite-dimensional, contrary to what happens in the univariatecase where only three parameters are needed.
Remark
Below, H denotes the space of all probability measures H which can be defined over([0, 1],B[0,1]), where B[0,1] is the Borel sigma-algebra on [0, 1], and which obey the meanconstraint ∫
[0,1]wH(dw) =
12. (1)
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 8 / 31
Extreme 6= Rare
Statistical Modeling of Bivariate Extremes
What are relevant statistical models for statistics of bivariate extremes? Is there anextension of the generalized EVD for the bivariate setting?
Theorem
If P(M?N,2 6 z1,M?
N,1 6 z2)→ GH (z1, z2), as n→∞, with G being a non-degenerate distributionfunction, then
GH(0, z1)× (0, z2) := GH (z1, z2) = exp− 2
∫[0,1]
max(
wz1,
1− wz2
)H(dw)
, z1, z2 > 0,
for some H ∈ H .
Remark
1 Similarities between generalized EVD and bivariate EVD: both start with an ‘exp,’ but forbivariate EVD Θ = H .
2 Since (1) is the only constraint on H, neither H nor G can have a finite parameterization.3 A bivariate extreme value distribution is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 9 / 31
Extreme 6= Rare
Statistical Modeling of Bivariate Extremes
What are relevant statistical models for statistics of bivariate extremes? Is there anextension of the generalized EVD for the bivariate setting?
Theorem
If P(M?N,2 6 z1,M?
N,1 6 z2)→ GH (z1, z2), as n→∞, with G being a non-degenerate distributionfunction, then
GH(0, z1)× (0, z2) := GH (z1, z2) = exp− 2
∫[0,1]
max(
wz1,
1− wz2
)H(dw)
, z1, z2 > 0,
for some H ∈ H .
Remark
1 Similarities between generalized EVD and bivariate EVD: both start with an ‘exp,’ but forbivariate EVD Θ = H .
2 Since (1) is the only constraint on H, neither H nor G can have a finite parameterization.
3 A bivariate extreme value distribution is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 9 / 31
Extreme 6= Rare
Statistical Modeling of Bivariate Extremes
What are relevant statistical models for statistics of bivariate extremes? Is there anextension of the generalized EVD for the bivariate setting?
Theorem
If P(M?N,2 6 z1,M?
N,1 6 z2)→ GH (z1, z2), as n→∞, with G being a non-degenerate distributionfunction, then
GH(0, z1)× (0, z2) := GH (z1, z2) = exp− 2
∫[0,1]
max(
wz1,
1− wz2
)H(dw)
, z1, z2 > 0,
for some H ∈ H .
Remark
1 Similarities between generalized EVD and bivariate EVD: both start with an ‘exp,’ but forbivariate EVD Θ = H .
2 Since (1) is the only constraint on H, neither H nor G can have a finite parameterization.3 A bivariate extreme value distribution is a measure-dependent measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 9 / 31
Extreme 6= Rare
Bivariate Extremes—GeometryPseudo-Polar Transformation
A pseudo-polar transformation is useful for better understanding the role of H. Let
(R,W ) =
(Z1 + Z2,
Z1
Z1 + Z2
),
and denote R and W as the pseudo-radius and pseudo-angles, respectively. If:
Z1 is relatively large: W ≈ 1.Z2 is relatively large: W ≈ 0.
Z1 and Z2 are both extreme: W ≈ 1/2.
The spectral probability measure H determines the ‘synchronization’ between jointextremes, and is thus an estimating target of interest.
We then have:
statistical problem: lack of knowledge on H;
inference challenge: obtaining estimates which obey the marginal momentconstraints, and which define a density on the simplex.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 10 / 31
Extreme 6= Rare
Bivariate Extremes—GeometryPseudo-Polar Transformation
A pseudo-polar transformation is useful for better understanding the role of H. Let
(R,W ) =
(Z1 + Z2,
Z1
Z1 + Z2
),
and denote R and W as the pseudo-radius and pseudo-angles, respectively. If:
Z1 is relatively large: W ≈ 1.Z2 is relatively large: W ≈ 0.
Z1 and Z2 are both extreme: W ≈ 1/2.
The spectral probability measure H determines the ‘synchronization’ between jointextremes, and is thus an estimating target of interest.
We then have:
statistical problem: lack of knowledge on H;
inference challenge: obtaining estimates which obey the marginal momentconstraints, and which define a density on the simplex.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 10 / 31
Extreme 6= Rare
Bivariate Extremes—GeometryPseudo-Polar Transformation
A pseudo-polar transformation is useful for better understanding the role of H. Let
(R,W ) =
(Z1 + Z2,
Z1
Z1 + Z2
),
and denote R and W as the pseudo-radius and pseudo-angles, respectively. If:
Z1 is relatively large: W ≈ 1.Z2 is relatively large: W ≈ 0.
Z1 and Z2 are both extreme: W ≈ 1/2.
The spectral probability measure H determines the ‘synchronization’ between jointextremes, and is thus an estimating target of interest.
We then have:
statistical problem: lack of knowledge on H;
inference challenge: obtaining estimates which obey the marginal momentconstraints, and which define a density on the simplex.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 10 / 31
Extreme 6= Rare
Bivariate Extremes—GeometryPseudo-Polar Transformation
A pseudo-polar transformation is useful for better understanding the role of H. Let
(R,W ) =
(Z1 + Z2,
Z1
Z1 + Z2
),
and denote R and W as the pseudo-radius and pseudo-angles, respectively. If:
Z1 is relatively large: W ≈ 1.Z2 is relatively large: W ≈ 0.
Z1 and Z2 are both extreme: W ≈ 1/2.
The spectral probability measure H determines the ‘synchronization’ between jointextremes, and is thus an estimating target of interest.
We then have:
statistical problem: lack of knowledge on H;
inference challenge: obtaining estimates which obey the marginal momentconstraints, and which define a density on the simplex.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 10 / 31
Extreme 6= Rare
Bivariate Extremes—GeometryPseudo-Polar Transformation
A pseudo-polar transformation is useful for better understanding the role of H. Let
(R,W ) =
(Z1 + Z2,
Z1
Z1 + Z2
),
and denote R and W as the pseudo-radius and pseudo-angles, respectively. If:
Z1 is relatively large: W ≈ 1.Z2 is relatively large: W ≈ 0.
Z1 and Z2 are both extreme: W ≈ 1/2.
The spectral probability measure H determines the ‘synchronization’ between jointextremes, and is thus an estimating target of interest.
We then have:
statistical problem: lack of knowledge on H;
inference challenge: obtaining estimates which obey the marginal momentconstraints, and which define a density on the simplex.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 10 / 31
Extreme 6= Rare
Opening CreditsSome references:
Mhalla, L., de Carvalho, M., Chavez-Demoulin, V. (2019, in press) “Regression Type Modelsfor Extremal Dependence,”Scandinavian Journal of Statistics.
Castro, D., de Carvalho, M., and Wadsworth, J. (2018),“Time-Varying Extreme Value Dependence with Application to Leading European StockMarkets,”Annals of Applied Statistics, 12, 283–309.
Castro, D., and de Carvalho, M. (2017),“Spectral Density Regression for Bivariate Extremes,”Stochastic Environmental Research and Risk Assessment , 73, 1279–1288.
de Carvalho, M., and Davison, A. C. (2014),“Spectral Density Ratio Models for Multivariate Extremes,”Journal of the American Statistical Association, 109, 764–776.
†Thanks to collaborators:
V. Chavez-Demoulin (UNIL), D. Castro (KAUST), A. C. Davison (EPFL), L. Mhalla(U. Genève), J. Wadsworth (U. Lancaster).
.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 11 / 31
Extreme 6= Rare
Opening CreditsSome references:
Mhalla, L., de Carvalho, M., Chavez-Demoulin, V. (2019, in press) “Regression Type Modelsfor Extremal Dependence,”Scandinavian Journal of Statistics.
Castro, D., de Carvalho, M., and Wadsworth, J. (2018),“Time-Varying Extreme Value Dependence with Application to Leading European StockMarkets,”Annals of Applied Statistics, 12, 283–309.
Castro, D., and de Carvalho, M. (2017),“Spectral Density Regression for Bivariate Extremes,”Stochastic Environmental Research and Risk Assessment , 73, 1279–1288.
de Carvalho, M., and Davison, A. C. (2014),“Spectral Density Ratio Models for Multivariate Extremes,”Journal of the American Statistical Association, 109, 764–776.
†Thanks to collaborators:
V. Chavez-Demoulin (UNIL), D. Castro (KAUST), A. C. Davison (EPFL), L. Mhalla(U. Genève), J. Wadsworth (U. Lancaster).
.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 11 / 31
Extreme 6= Rare
Opening CreditsSome references:
Mhalla, L., de Carvalho, M., Chavez-Demoulin, V. (2019, in press) “Regression Type Modelsfor Extremal Dependence,”Scandinavian Journal of Statistics.
Castro, D., de Carvalho, M., and Wadsworth, J. (2018),“Time-Varying Extreme Value Dependence with Application to Leading European StockMarkets,”Annals of Applied Statistics, 12, 283–309.
Castro, D., and de Carvalho, M. (2017),“Spectral Density Regression for Bivariate Extremes,”Stochastic Environmental Research and Risk Assessment , 73, 1279–1288.
de Carvalho, M., and Davison, A. C. (2014),“Spectral Density Ratio Models for Multivariate Extremes,”Journal of the American Statistical Association, 109, 764–776.
†Thanks to collaborators:
V. Chavez-Demoulin (UNIL), D. Castro (KAUST), A. C. Davison (EPFL), L. Mhalla(U. Genève), J. Wadsworth (U. Lancaster).
.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 11 / 31
Extreme 6= Rare
Opening CreditsSome references:
Mhalla, L., de Carvalho, M., Chavez-Demoulin, V. (2019, in press) “Regression Type Modelsfor Extremal Dependence,”Scandinavian Journal of Statistics.
Castro, D., de Carvalho, M., and Wadsworth, J. (2018),“Time-Varying Extreme Value Dependence with Application to Leading European StockMarkets,”Annals of Applied Statistics, 12, 283–309.
Castro, D., and de Carvalho, M. (2017),“Spectral Density Regression for Bivariate Extremes,”Stochastic Environmental Research and Risk Assessment , 73, 1279–1288.
de Carvalho, M., and Davison, A. C. (2014),“Spectral Density Ratio Models for Multivariate Extremes,”Journal of the American Statistical Association, 109, 764–776.
†Thanks to collaborators:
V. Chavez-Demoulin (UNIL), D. Castro (KAUST), A. C. Davison (EPFL), L. Mhalla(U. Genève), J. Wadsworth (U. Lancaster).
.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 11 / 31
Extreme 6= Rare
Opening CreditsSome references:
Mhalla, L., de Carvalho, M., Chavez-Demoulin, V. (2019, in press) “Regression Type Modelsfor Extremal Dependence,”Scandinavian Journal of Statistics.
Castro, D., de Carvalho, M., and Wadsworth, J. (2018),“Time-Varying Extreme Value Dependence with Application to Leading European StockMarkets,”Annals of Applied Statistics, 12, 283–309.
Castro, D., and de Carvalho, M. (2017),“Spectral Density Regression for Bivariate Extremes,”Stochastic Environmental Research and Risk Assessment , 73, 1279–1288.
de Carvalho, M., and Davison, A. C. (2014),“Spectral Density Ratio Models for Multivariate Extremes,”Journal of the American Statistical Association, 109, 764–776.
†Thanks to collaborators:
V. Chavez-Demoulin (UNIL), D. Castro (KAUST), A. C. Davison (EPFL), L. Mhalla(U. Genève), J. Wadsworth (U. Lancaster).
.M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 11 / 31
Extreme 6= Rare
StoryboardAgenda
1 Introduction, Motivation, and Background (Done!)
2 Opening Credits (Done!)
3 Nonstationary Multivariate Extremes (Next)
4 The End
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 12 / 31
Extreme 6= Rare
StoryboardAgenda
1 Introduction, Motivation, and Background (Done!)
2 Opening Credits (Done!)
3 Nonstationary Multivariate Extremes (Next)
4 The End
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 12 / 31
Extreme 6= Rare
StoryboardAgenda
1 Introduction, Motivation, and Background (Done!)
2 Opening Credits (Done!)
3 Nonstationary Multivariate Extremes (Next)
4 The End
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 12 / 31
Extreme 6= Rare
StoryboardAgenda
1 Introduction, Motivation, and Background (Done!)
2 Opening Credits (Done!)
3 Nonstationary Multivariate Extremes (Next)
4 The End
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 12 / 31
Extreme 6= Rare
StoryboardAgenda
1 Introduction, Motivation, and Background (Done!)
2 Opening Credits (Done!)
3 Nonstationary Multivariate Extremes (Next)
4 The End
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 12 / 31
Extreme 6= Rare
Time-Varying Joint LossesDynamics of Extremes
−0.10 −0.05 0.00 0.05 0.10
−0.
10−
0.05
0.00
0.05
0.10
CAC 40
DA
X 3
0
−0.10 −0.05 0.00 0.05 0.10−0.
10−
0.05
0.00
0.05
0.10
FTSE 100
CA
C 4
0
−0.10 −0.05 0.00 0.05 0.10−0.
10−
0.05
0.00
0.05
0.10
DAX 30
FT
SE
100
Figure: Scatterplots using a time-varying color palette for daily log-returns for CAC 40 (FR) andDAX 30 (GR) spanning the period from January 1, 1988 to January 1, 2014.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 13 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremesde Carvalho & Davison (2014)
In a recent paper de Carvalho and Davison (2014) proposed a model for a family of spectralmeasures H0, . . . ,HK .
Definition
Let Hk ∈ H be absolutely continuous, for k = 1, . . . ,K . The family H1, . . . ,HK is a spectraldensity ratio family, if there exists an absolutely continuous H0 ∈ H , tilting parameters(αk , βk ) ∈ R2, and c : [0, 1] 7→ R such that
dHk
dH0(w) = expαk + βk c(w), k = 1, . . . ,K .
H0 is the so-called baseline spectral measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 14 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremesde Carvalho & Davison (2014)
In a recent paper de Carvalho and Davison (2014) proposed a model for a family of spectralmeasures H0, . . . ,HK .
Definition
Let Hk ∈ H be absolutely continuous, for k = 1, . . . ,K . The family H1, . . . ,HK is a spectraldensity ratio family, if there exists an absolutely continuous H0 ∈ H , tilting parameters(αk , βk ) ∈ R2, and c : [0, 1] 7→ R such that
dHk
dH0(w) = expαk + βk c(w), k = 1, . . . ,K .
H0 is the so-called baseline spectral measure.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 14 / 31
Extreme 6= Rare
Nonstationary Bivariate ExtremesCastro and de Carvalho (2017)
Castro and de Carvalho (2017) proposed a method which is able to ‘interpolate in between’cross sections of a ‘angular surface.’
x
0.81.0
1.21.4
1.6
1.8
2.0
w
0.0
0.2
0.4
0.6
0.8
1.00.0
0.5
1.0
1.5
2.0
x
1.01.2
1.4
1.6
1.8
w
0.0
0.2
0.4
0.6
0.8
1.00.0
0.5
1.0
1.5
2.0
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 15 / 31
Extreme 6= Rare
Nonstationary Bivariate ExtremesCastro and de Carvalho (2017)
Castro and de Carvalho (2017) proposed a method which is able to ‘interpolate in between’cross sections of a ‘angular surface.’
x
0.81.0
1.21.4
1.6
1.8
2.0
w
0.0
0.2
0.4
0.6
0.8
1.00.0
0.5
1.0
1.5
2.0
x
1.01.2
1.4
1.6
1.8
w
0.0
0.2
0.4
0.6
0.8
1.00.0
0.5
1.0
1.5
2.0
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 15 / 31
Extreme 6= Rare
Nonstationary Bivariate ExtremesAllowing for the Dependence Structure to Evolve Over the Predictor
Such models allow for modeling extremal dependence in settings such as...
10 15 20 25 30 35
0.0
0.2
0.4
0.6
0.8
1.0
x
Pse
udo−
angl
e
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 16 / 31
Extreme 6= Rare
Nonstationary Bivariate ExtremesAllowing for the Dependence Structure to Evolve Over the Predictor
But they do not allow for...
0 1 2 3 4 5 6 7
0.0
0.2
0.4
0.6
0.8
1.0
x
Pse
udo−
angl
e
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 17 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Definition (Angular Surface)
Suppose Hx ∈ H is absolutely continuous for all x ∈ X . The pd spectral density is defined ashx = dHx/dw , and we refer to the set hx (w) : w ∈ [0, 1], x ∈ X as the angular surface.
The target is to assess how extremal dependence evolves over a certain covariate x . In otherwords, we want to incorporate nonstationarity into the extremal dependence structure.
h → hx
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
w
Spe
ctra
l Den
sity
1020
3040
50
0.00.2
0.40.6
0.81.0
0
2
4
6
8
xw
Spe
ctra
l Sur
face
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 18 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Definition (Angular Surface)
Suppose Hx ∈ H is absolutely continuous for all x ∈ X . The pd spectral density is defined ashx = dHx/dw , and we refer to the set hx (w) : w ∈ [0, 1], x ∈ X as the angular surface.
The target is to assess how extremal dependence evolves over a certain covariate x . In otherwords, we want to incorporate nonstationarity into the extremal dependence structure.
h → hx
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
w
Spe
ctra
l Den
sity
1020
3040
50
0.00.2
0.40.6
0.81.0
0
2
4
6
8
xw
Spe
ctra
l Sur
face
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 18 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Definition (Angular Surface)
Suppose Hx ∈ H is absolutely continuous for all x ∈ X . The pd spectral density is defined ashx = dHx/dw , and we refer to the set hx (w) : w ∈ [0, 1], x ∈ X as the angular surface.
The target is to assess how extremal dependence evolves over a certain covariate x . In otherwords, we want to incorporate nonstationarity into the extremal dependence structure.
h → hx
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
w
Spe
ctra
l Den
sity
1020
3040
50
0.00.2
0.40.6
0.81.0
0
2
4
6
8
xw
Spe
ctra
l Sur
face
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 18 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Let’s now regard the subject of predictor-dependent bivariate extremes from anotherviewpoint.
The simplest approach for modeling nonstationary extremes was popularized long ago byDavison and Smith (1990), and it is based on indexing the location and scale parameters ofthe generalized EVD by a predictor, x ∈ X ,
G(µx ,σx ,ξ)(z) = exp[−1 + ξ(z − µx )/σx−1/ξ].
What would be the ‘analogue’ of this approach to the bivariate setting?
How to model ‘nonstationary extremal dependence structures’?
The natural approach is based on indexing the parameter of the bivariate extreme valuedistribution (H) with a covariate, i.e. considering Hx : x ∈ X
GHx (z1, z2) = exp− 2
∫ 1
0max
(wz1,
1− wz2
)dHx (w)
.
And how to estimate Hx if one must?
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 19 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Let’s now regard the subject of predictor-dependent bivariate extremes from anotherviewpoint.
The simplest approach for modeling nonstationary extremes was popularized long ago byDavison and Smith (1990), and it is based on indexing the location and scale parameters ofthe generalized EVD by a predictor, x ∈ X ,
G(µx ,σx ,ξ)(z) = exp[−1 + ξ(z − µx )/σx−1/ξ].
What would be the ‘analogue’ of this approach to the bivariate setting?
How to model ‘nonstationary extremal dependence structures’?
The natural approach is based on indexing the parameter of the bivariate extreme valuedistribution (H) with a covariate, i.e. considering Hx : x ∈ X
GHx (z1, z2) = exp− 2
∫ 1
0max
(wz1,
1− wz2
)dHx (w)
.
And how to estimate Hx if one must?
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 19 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Let’s now regard the subject of predictor-dependent bivariate extremes from anotherviewpoint.
The simplest approach for modeling nonstationary extremes was popularized long ago byDavison and Smith (1990), and it is based on indexing the location and scale parameters ofthe generalized EVD by a predictor, x ∈ X ,
G(µx ,σx ,ξ)(z) = exp[−1 + ξ(z − µx )/σx−1/ξ].
What would be the ‘analogue’ of this approach to the bivariate setting?
How to model ‘nonstationary extremal dependence structures’?
The natural approach is based on indexing the parameter of the bivariate extreme valuedistribution (H) with a covariate, i.e. considering Hx : x ∈ X
GHx (z1, z2) = exp− 2
∫ 1
0max
(wz1,
1− wz2
)dHx (w)
.
And how to estimate Hx if one must?
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 19 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Let’s now regard the subject of predictor-dependent bivariate extremes from anotherviewpoint.
The simplest approach for modeling nonstationary extremes was popularized long ago byDavison and Smith (1990), and it is based on indexing the location and scale parameters ofthe generalized EVD by a predictor, x ∈ X ,
G(µx ,σx ,ξ)(z) = exp[−1 + ξ(z − µx )/σx−1/ξ].
What would be the ‘analogue’ of this approach to the bivariate setting?
How to model ‘nonstationary extremal dependence structures’?
The natural approach is based on indexing the parameter of the bivariate extreme valuedistribution (H) with a covariate, i.e. considering Hx : x ∈ X
GHx (z1, z2) = exp− 2
∫ 1
0max
(wz1,
1− wz2
)dHx (w)
.
And how to estimate Hx if one must?
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 19 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Assume observations (Wi , xi )ni=1. For any x ∈ X , we define the estimator
πb,i(x) =Kb(x−xi )∑nj=1 Kb(x−xj )
, i = 1, . . . ,n.
hx(w) =n∑
i=1
πb,i(x) × β(w ; νwi θb(x) + τ, ν1− wi θb(x) + τ)
θb(x) =1/2∑n
i=1 πb,i (x)wi.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 20 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Assume observations (Wi , xi )ni=1. For any x ∈ X , we define the estimator
πb,i(x) =Kb(x−xi )∑nj=1 Kb(x−xj )
, i = 1, . . . ,n.
hx(w) =n∑
i=1
πb,i(x) × β(w ; νwi θb(x) + τ, ν1− wi θb(x) + τ)
θb(x) =1/2∑n
i=1 πb,i (x)wi.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 20 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
Assume observations (Wi , xi )ni=1. For any x ∈ X , we define the estimator
πb,i(x) =Kb(x−xi )∑nj=1 Kb(x−xj )
, i = 1, . . . ,n.
hx(w) =n∑
i=1
πb,i(x) × β(w ; νwi θb(x) + τ, ν1− wi θb(x) + τ)
θb(x) =1/2∑n
i=1 πb,i (x)wi.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 20 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
There are three parameters involved in the estimator:
b: smooths in the x-direction.
hx(w) =n∑
i=1
πb ,i
(x)× β(w ; ν wiθb(x) + τ , ν 1− wiθ b(x)+ τ )
ν: smooths in the w-direction.
τ : slightly adjusts the centering of the kernel.
The moment constraint is satisfied, since∫ 1
0whx (w) dw =
∑ni=1 Kb(x − xi )νWiθb(x) + τ(ν + 2τ)
∑ni=1 Kb(x − xi )
=ν/2 + τ
ν + 2τ= 1/2.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 21 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
There are three parameters involved in the estimator:
b: smooths in the x-direction.
hx(w) =n∑
i=1
πb ,i
(x)× β(w ; ν wiθb(x) + τ , ν 1− wiθ b(x)+ τ )
ν: smooths in the w-direction.
τ : slightly adjusts the centering of the kernel.
The moment constraint is satisfied, since∫ 1
0whx (w) dw =
∑ni=1 Kb(x − xi )νWiθb(x) + τ(ν + 2τ)
∑ni=1 Kb(x − xi )
=ν/2 + τ
ν + 2τ= 1/2.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 21 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
There are three parameters involved in the estimator:
b: smooths in the x-direction.
hx(w) =n∑
i=1
πb ,i
(x)× β(w ; ν wiθb(x) + τ , ν 1− wiθ b(x)+ τ )
ν: smooths in the w-direction.
τ : slightly adjusts the centering of the kernel.
The moment constraint is satisfied, since∫ 1
0whx (w) dw =
∑ni=1 Kb(x − xi )νWiθb(x) + τ(ν + 2τ)
∑ni=1 Kb(x − xi )
=ν/2 + τ
ν + 2τ= 1/2.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 21 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
There are three parameters involved in the estimator:
b: smooths in the x-direction.
hx(w) =n∑
i=1
πb ,i
(x)× β(w ; ν wiθb(x) + τ , ν 1− wiθ b(x)+ τ )
ν: smooths in the w-direction.
τ : slightly adjusts the centering of the kernel.
The moment constraint is satisfied, since∫ 1
0whx (w) dw =
∑ni=1 Kb(x − xi )νWiθb(x) + τ(ν + 2τ)
∑ni=1 Kb(x − xi )
=ν/2 + τ
ν + 2τ= 1/2.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 21 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
There are three parameters involved in the estimator:
b: smooths in the x-direction.
hx(w) =n∑
i=1
πb ,i
(x)× β(w ; ν wiθb(x) + τ , ν 1− wiθ b(x)+ τ )
ν: smooths in the w-direction.
τ : slightly adjusts the centering of the kernel.
The moment constraint is satisfied, since∫ 1
0whx (w) dw =
∑ni=1 Kb(x − xi )νWiθb(x) + τ(ν + 2τ)
∑ni=1 Kb(x − xi )
=ν/2 + τ
ν + 2τ= 1/2.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 21 / 31
Extreme 6= Rare
Nonstationary Bivariate Extremes
There are three parameters involved in the estimator:
b: smooths in the x-direction.
hx(w) =n∑
i=1
πb ,i
(x)× β(w ; ν wiθb(x) + τ , ν 1− wiθ b(x)+ τ )
ν: smooths in the w-direction.
τ : slightly adjusts the centering of the kernel.
The moment constraint is satisfied, since∫ 1
0whx (w) dw =
∑ni=1 Kb(x − xi )νWiθb(x) + τ(ν + 2τ)
∑ni=1 Kb(x − xi )
=ν/2 + τ
ν + 2τ= 1/2.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 21 / 31
Extreme 6= Rare
Time-Varying Joint LossesData Description
Our data consist of daily closing stock index levels of three leading European stock markets:CAC 40, DAX 30, and FTSE 100. Index values were gathered from Datastream in terms oflocal currency.
Our sample period spans from January 1, 1988 to January 1, 2014 (6784 observations).
As a unit of analysis we use daily negative returns, which can be used as proxies for lossesin these markets.
Our interest relies in the joint extremal behavior of three stock markets through time: FTSE100, CAC 40 and DAX 30.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 22 / 31
Extreme 6= Rare
Time-Varying Joint LossesData Description
Our data consist of daily closing stock index levels of three leading European stock markets:CAC 40, DAX 30, and FTSE 100. Index values were gathered from Datastream in terms oflocal currency.
Our sample period spans from January 1, 1988 to January 1, 2014 (6784 observations).
As a unit of analysis we use daily negative returns, which can be used as proxies for lossesin these markets.
Our interest relies in the joint extremal behavior of three stock markets through time: FTSE100, CAC 40 and DAX 30.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 22 / 31
Extreme 6= Rare
Time-Varying Joint LossesData Description
Our data consist of daily closing stock index levels of three leading European stock markets:CAC 40, DAX 30, and FTSE 100. Index values were gathered from Datastream in terms oflocal currency.
Our sample period spans from January 1, 1988 to January 1, 2014 (6784 observations).
As a unit of analysis we use daily negative returns, which can be used as proxies for lossesin these markets.
Our interest relies in the joint extremal behavior of three stock markets through time: FTSE100, CAC 40 and DAX 30.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 22 / 31
Extreme 6= Rare
Time-Varying Joint LossesData Description
Our data consist of daily closing stock index levels of three leading European stock markets:CAC 40, DAX 30, and FTSE 100. Index values were gathered from Datastream in terms oflocal currency.
Our sample period spans from January 1, 1988 to January 1, 2014 (6784 observations).
As a unit of analysis we use daily negative returns, which can be used as proxies for lossesin these markets.
Our interest relies in the joint extremal behavior of three stock markets through time: FTSE100, CAC 40 and DAX 30.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 22 / 31
Extreme 6= Rare
Time-Varying Joint LossesDynamics of Extremes
−0.10 −0.05 0.00 0.05 0.10
−0.
10−
0.05
0.00
0.05
0.10
CAC 40
DA
X 3
0
−0.10 −0.05 0.00 0.05 0.10−0.
10−
0.05
0.00
0.05
0.10
FTSE 100
CA
C 4
0
−0.10 −0.05 0.00 0.05 0.10−0.
10−
0.05
0.00
0.05
0.10
DAX 30
FT
SE
100
Figure: Scatterplots using a time-varying color palette for daily log-returns for CAC 40 (FR) andDAX 30 (GR) spanning the period from January 1, 1988 to January 1, 2014.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 23 / 31
Extreme 6= Rare
Time-Varying Joint LossesDynamics of Extremes (Zoom: CAC 40 vs. DAX 30)
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 24 / 31
Extreme 6= Rare
Time-Varying Extremal DependenceCAC 40 vs. DAX 30 and FTSE 100 vs. DAX 30
19901995
20002005
20100.0
0.20.4
0.60.8
1.0
0
0.5
1
1.5
2
Timew
19901995
20002005
20100.0
0.20.4
0.60.8
1.0
0
0.5
1
1.5
2
Timew
CAC 40 − DAX 30CAC 40 − DAX 30CAC 40 − DAX 30
Spe
ctra
l Sur
face
19901995
20002005
20100.0
0.20.4
0.60.8
1.0
0
0.5
1
1.5
2
Timew
19901995
20002005
20100.0
0.20.4
0.60.8
1.0
0
0.5
1
1.5
2
Timew
FTSE 100 − DAX 30FTSE 100 − DAX 30FTSE 100 − DAX 30
Spe
ctra
l Sur
face
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 25 / 31
Extreme 6= Rare
Ongoing Work: IGeneralized Additive Modeling
Two things that the previous strategy does not tackle:
It seems difficult to extend it to the D-dimensional setting.
Since the approach is nonparametric a huge amount of data is required.
We have recently proposed a model where we set in advance an extreme value copula, andthen model the parameters of the corresponding copula through generalized additive model.
Inference is conducted by a maximum penalized likelihood with maximization of penalizedlog-likelihood being based on a Newton–Raphson algorithm.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 26 / 31
Extreme 6= Rare
Ongoing Work: IGeneralized Additive Modeling
Two things that the previous strategy does not tackle:
It seems difficult to extend it to the D-dimensional setting.
Since the approach is nonparametric a huge amount of data is required.
We have recently proposed a model where we set in advance an extreme value copula, andthen model the parameters of the corresponding copula through generalized additive model.
Inference is conducted by a maximum penalized likelihood with maximization of penalizedlog-likelihood being based on a Newton–Raphson algorithm.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 26 / 31
Extreme 6= Rare
Ongoing Work: IGeneralized Additive Modeling
Two things that the previous strategy does not tackle:
It seems difficult to extend it to the D-dimensional setting.
Since the approach is nonparametric a huge amount of data is required.
We have recently proposed a model where we set in advance an extreme value copula, andthen model the parameters of the corresponding copula through generalized additive model.
Inference is conducted by a maximum penalized likelihood with maximization of penalizedlog-likelihood being based on a Newton–Raphson algorithm.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 26 / 31
Extreme 6= Rare
Ongoing Work: IGeneralized Additive Modeling
Two things that the previous strategy does not tackle:
It seems difficult to extend it to the D-dimensional setting.
Since the approach is nonparametric a huge amount of data is required.
We have recently proposed a model where we set in advance an extreme value copula, andthen model the parameters of the corresponding copula through generalized additive model.
Inference is conducted by a maximum penalized likelihood with maximization of penalizedlog-likelihood being based on a Newton–Raphson algorithm.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 26 / 31
Extreme 6= Rare
Ongoing Work: IGeneralized Additive Modeling
Two things that the previous strategy does not tackle:
It seems difficult to extend it to the D-dimensional setting.
Since the approach is nonparametric a huge amount of data is required.
We have recently proposed a model where we set in advance an extreme value copula, andthen model the parameters of the corresponding copula through generalized additive model.
Inference is conducted by a maximum penalized likelihood with maximization of penalizedlog-likelihood being based on a Newton–Raphson algorithm.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 26 / 31
Extreme 6= Rare
Other Related WorkGeneralized Additive Modeling
Advantages of using a GAM approach:
It does not suffer from the above-mentioned shortcomings.
Disadvantages:
If one chooses the ‘wrong’ dependence structure the fits may not be the most sensibleones.
Here are some proofs of concept with examples on simulated data.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 27 / 31
Extreme 6= Rare
Other Related WorkProfiles of Husler–Reiss and Asymmetric Logistic Spectral Surfaces
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 28 / 31
Extreme 6= Rare
Ongoing Work: ICovariate-adjusted Pairwise Beta Spectral Density Estimate
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 29 / 31
Extreme 6= Rare
The EndFinal Remarks
Modeling time-changing dependence structures is important in practice, but moremethodological work needs to be developed in this direction.
Experiments with real data may suggest that in some cases the ‘true’ data generatingprocess of interest, could transition smoothly from asymptotic dependence to asymptoticindependence.
The need for developing models able to switch from asymptotic dependence to asymptoticindependence over time is of utmost importance.
We applied one of the proposed models to assess the dynamics governing the extremaldependence of some leading European stock markets over the last decades, and findevidence of an increase in extremal dependence over recent years.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 30 / 31
Extreme 6= Rare
The EndFinal Remarks
Modeling time-changing dependence structures is important in practice, but moremethodological work needs to be developed in this direction.
Experiments with real data may suggest that in some cases the ‘true’ data generatingprocess of interest, could transition smoothly from asymptotic dependence to asymptoticindependence.
The need for developing models able to switch from asymptotic dependence to asymptoticindependence over time is of utmost importance.
We applied one of the proposed models to assess the dynamics governing the extremaldependence of some leading European stock markets over the last decades, and findevidence of an increase in extremal dependence over recent years.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 30 / 31
Extreme 6= Rare
The EndFinal Remarks
Modeling time-changing dependence structures is important in practice, but moremethodological work needs to be developed in this direction.
Experiments with real data may suggest that in some cases the ‘true’ data generatingprocess of interest, could transition smoothly from asymptotic dependence to asymptoticindependence.
The need for developing models able to switch from asymptotic dependence to asymptoticindependence over time is of utmost importance.
We applied one of the proposed models to assess the dynamics governing the extremaldependence of some leading European stock markets over the last decades, and findevidence of an increase in extremal dependence over recent years.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 30 / 31
Extreme 6= Rare
The EndFinal Remarks
Modeling time-changing dependence structures is important in practice, but moremethodological work needs to be developed in this direction.
Experiments with real data may suggest that in some cases the ‘true’ data generatingprocess of interest, could transition smoothly from asymptotic dependence to asymptoticindependence.
The need for developing models able to switch from asymptotic dependence to asymptoticindependence over time is of utmost importance.
We applied one of the proposed models to assess the dynamics governing the extremaldependence of some leading European stock markets over the last decades, and findevidence of an increase in extremal dependence over recent years.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 30 / 31
Extreme 6= Rare
The EndFinal Remarks
Modeling time-changing dependence structures is important in practice, but moremethodological work needs to be developed in this direction.
Experiments with real data may suggest that in some cases the ‘true’ data generatingprocess of interest, could transition smoothly from asymptotic dependence to asymptoticindependence.
The need for developing models able to switch from asymptotic dependence to asymptoticindependence over time is of utmost importance.
We applied one of the proposed models to assess the dynamics governing the extremaldependence of some leading European stock markets over the last decades, and findevidence of an increase in extremal dependence over recent years.
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 30 / 31
Extreme 6= Rare
Thanks!
M. de Carvalho On the Likelihood of Black Swan Events June 25, 2019 31 / 31