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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.

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(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.

Y1, . . . ,Yniid∼ F .

FY (y) := n−1n∑

I(Yi > y).

P(Y > y) = 1− FY (Mn + ε) = 0, ∀ ε > 0,

Y1, . . . ,Yniid∼ F .

FY (y) := n−1n∑

I(Yi > y).

P(Y > y) = 1− FY (Mn + ε) = 0, ∀ ε > 0,

Y1, . . . ,Yniid∼ F .

FY (y) := n−1n∑

I(Yi > y).

P(Y > y) = 1− FY (Mn + ε) = 0, ∀ ε > 0,

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

rare and extreme rare rare and extreme

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

rare and extreme rare rare and extreme

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.

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Gθ(z) = exp[−

1 + ξ

(z − µσ

)−1/ξ], θ = (µ, σ, ξ),

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Gθ(z) = exp[−

1 + ξ

(z − µσ

)−1/ξ], θ = (µ, σ, ξ),

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Gθ(z) = exp[−

1 + ξ

(z − µσ

)−1/ξ], θ = (µ, σ, ξ),

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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).

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Notes:

i: ξ → 0: light-tail (Gumbel);ii: ξ > 0: heavy-tail (Fréchet);

iii: ξ < 0: short-tail (negative Weibull).

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Notes:

i: ξ → 0: light-tail (Gumbel);ii: ξ > 0: heavy-tail (Fréchet);iii: ξ < 0: short-tail (negative Weibull).

Extreme 6= Rare

Notes:

Extreme 6= Rare

Notes:

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

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.

Extreme 6= Rare

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

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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.

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)

Extreme 6= Rare

Gθ(z) = exp[−

1 + ξ

(z − µσ

)−1/ξ

], θ = (µ, σ, ξ),

Remark

[0,1]wH(dw) =

12. (1)

Extreme 6= Rare

Gθ(z) = exp[−

1 + ξ

(z − µσ

)−1/ξ

], θ = (µ, σ, ξ),

Remark

[0,1]wH(dw) =

12. (1)

Extreme 6= Rare

Gθ(z) = exp[−

1 + ξ

(z − µσ

)−1/ξ

], θ = (µ, σ, ξ),

Remark

[0,1]wH(dw) =

12. (1)

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]

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.

Extreme 6= Rare

Theorem

If P(M?N,2 6 z1,M?

GH(0, z1)× (0, z2) := GH (z1, z2) = exp− 2

∫[0,1]

1− wz2

)H(dw)

, z1, z2 > 0,

for some H ∈ H .

Remark

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.

Extreme 6= Rare

Theorem

If P(M?N,2 6 z1,M?

GH(0, z1)× (0, z2) := GH (z1, z2) = exp− 2

∫[0,1]

1− wz2

)H(dw)

, z1, z2 > 0,

for some H ∈ H .

Remark

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.

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 + 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.

Extreme 6= Rare

(R,W ) =

(Z1 + Z2,

Z1 + Z2

We then have:

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(R,W ) =

(Z1 + Z2,

Z1 + Z2

We then have:

Extreme 6= Rare

(R,W ) =

(Z1 + Z2,

Z1 + Z2

We then have:

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(R,W ) =

(Z1 + Z2,

Z1 + Z2

We then have:

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).

Extreme 6= Rare

.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

Extreme 6= Rare

StoryboardAgenda

4 The End

Extreme 6= Rare

StoryboardAgenda

4 The End

Extreme 6= Rare

StoryboardAgenda

4 The End

Extreme 6= Rare

StoryboardAgenda

4 The End

Extreme 6= Rare

Time-Varying Joint LossesDynamics of Extremes

−0.10 −0.05 0.00 0.05 0.10

CAC 40

−0.10 −0.05 0.00 0.05 0.10−0.

FTSE 100

−0.10 −0.05 0.00 0.05 0.10−0.

DAX 30

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.

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

dH0(w) = expαk + βk c(w), k = 1, . . . ,K .

H0 is the so-called baseline spectral measure.

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

dH0(w) = expαk + βk c(w), k = 1, . . . ,K .

H0 is the so-called baseline spectral measure.

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.’

0.81.0

1.21.4

1.00.0

1.01.2

1.00.0

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.’

0.81.0

1.21.4

1.00.0

1.01.2

1.00.0

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

udo−

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

udo−

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

0.00.2

0.40.6

0.81.0

Extreme 6= Rare

h → hx

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Extreme 6= Rare

h → hx

0.0 0.2 0.4 0.6 0.8 1.0

0.00.2

0.40.6

0.81.0

Extreme 6= Rare

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− wz2

)dHx (w)

And how to estimate Hx if one must?

Extreme 6= Rare

GHx (z1, z2) = exp− 2

1− wz2

)dHx (w)

Extreme 6= Rare

GHx (z1, z2) = exp− 2

1− wz2

)dHx (w)

Extreme 6= Rare

GHx (z1, z2) = exp− 2

1− wz2

)dHx (w)

Extreme 6= Rare

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∑

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.

Extreme 6= Rare

, i = 1, . . . ,n.

hx(w) =n∑

θb(x) =1/2∑n

i=1 πb,i (x)wi.

Extreme 6= Rare

, i = 1, . . . ,n.

hx(w) =n∑

θb(x) =1/2∑n

i=1 πb,i (x)wi.

Extreme 6= Rare

There are three parameters involved in the estimator:

b: smooths in the x-direction.

hx(w) =n∑

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.

Extreme 6= Rare

hx(w) =n∑

πb ,i

0whx (w) dw =

=ν/2 + τ

ν + 2τ= 1/2.

Extreme 6= Rare

hx(w) =n∑

πb ,i

0whx (w) dw =

=ν/2 + τ

ν + 2τ= 1/2.

Extreme 6= Rare

hx(w) =n∑

πb ,i

0whx (w) dw =

=ν/2 + τ

ν + 2τ= 1/2.

Extreme 6= Rare

hx(w) =n∑

πb ,i

0whx (w) dw =

=ν/2 + τ

ν + 2τ= 1/2.

Extreme 6= Rare

hx(w) =n∑

πb ,i

0whx (w) dw =

=ν/2 + τ

ν + 2τ= 1/2.

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.

Extreme 6= Rare

Time-Varying Joint LossesDynamics of Extremes

−0.10 −0.05 0.00 0.05 0.10

CAC 40

−0.10 −0.05 0.00 0.05 0.10−0.

FTSE 100

−0.10 −0.05 0.00 0.05 0.10−0.

DAX 30

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.

Extreme 6= Rare

Time-Varying Joint LossesDynamics of Extremes (Zoom: CAC 40 vs. DAX 30)

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Time-Varying Extremal DependenceCAC 40 vs. DAX 30 and FTSE 100 vs. DAX 30

19901995

20002005

20100.0

0.20.4

0.60.8

19901995

20002005

20100.0

0.20.4

0.60.8

CAC 40 − DAX 30CAC 40 − DAX 30CAC 40 − DAX 30

19901995

20002005

20100.0

0.20.4

0.60.8

19901995

20002005

20100.0

0.20.4

0.60.8

FTSE 100 − DAX 30FTSE 100 − DAX 30FTSE 100 − DAX 30

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.

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.

Extreme 6= Rare

Other Related WorkProfiles of Husler–Reiss and Asymmetric Logistic Spectral Surfaces

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Ongoing Work: ICovariate-adjusted Pairwise Beta Spectral Density Estimate

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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.

Extreme 6= Rare

Thanks!

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