foundations and applications of modern nonparametric...

W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly s is u n d S to c h a stik

Vladimir Spokoiny

Foundations and Applications of ModernNonparametric Statistics

Mohrenstr. 39, 10117 Berlin [email protected]/spokoiny October 10, 2009

Robust Risk Management

Outline

1 Robust Risk ManagementMotivation. Market RiskAdaptive univariate volatility estimationAccounting for heavy tailsICA: dimension reductionConclusion and OutlookReferences

,Modern Nonparametric Statistics October 10, 2009 2 (50)

Robust Risk Management

Outline

1. Motivation

2. Nonstationarity: adaptive volatility (AV) estimation, procedure andsome theoretical properties

3. Heavy tails: generalized hyperbolic (GH) distribution and AV

4. Dimension reduction: Independent component analysis (ICA)

5. Conclusion


Robust Risk Management Motivation. Market Risk

Stock market crash

October 19 1987 Dow Jones industrial dropped by over 500 pointsand the consequent economic depression



About market risk

Market risk: uncertainty due to changes in market prices and rates, thecorrelations among them and their levels of volatility, Jorion (2001).

. Regulatory: risk charge w.r.t. 1% risk level over the last 250days.

. ensure the adequacy of capital

. restrict the happening of large losses

. Internal supervisory: measuring and controlling risk level ofholding portfolios

Target: estimate distribution (quantile) of returns.



Measuring risk exposures

xt = Σ1/2t εt

xt ∈ IRd : asset returns with Var(xt|Ft−1) = Σt , εt standardizedstochastic innovations.

Target: estimate distribution (quantile) of returns.



Critical gaps

Standard assumptions Stylized facts

vola. model stationary nonstationary

(Black-Scholes/ GARCH)innovations Gaussian heavy tails

(e.g. RiskMetrics)dimension low-dimensional high-dimensional



Illustration of nonstationarity

The realized variances, the sum of squared returns sampled at 15 minutestick-by-tick, of Dow Jones Euro StoXX 50 Index futures.

2004/12/08 2005/02/18 2005/05/020

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6



Illustration of heavy tails

Estimated (log) density of the daily devolatilized DEM/USD returns. Timeinterval: 1979/12/01 to 1994/04/01 (3719 observations).

−5 −4 −3 −2 −1 0 1 2 3 4−8

−7

−6

−5

−4

−3

−2

−1

0Nonparametric kernelGaussianGH(1,1.74,−0.02,0.78,0.01)


Robust Risk Management Adaptive univariate volatility estimation

(Classical) Volatility estimation

Model: Rt =√θtεt, where Rt ∈ IR, εt|Ft−1 ∼ N(0, 1)

Suppose θt ≡ θ∗ (homogeneity) for t = 1, · · · , T .Maximum Likelihood Estimate (MLE):

θ̃ = argmaxθ∈Θ

L(θ) = argmaxθ∈Θ

∑t

`(Rt, θ)

with `(Rt, θ) = −(1/2) log(2πθ)−R2t /(2θ) . Leads to

θ̃ =1T

∑tR2t .



Standard ways of accounting for nonstationarity

. Reestimate parameters using a time varying window, e.g.

θ̃t =1M

M∑i=1

R2t−i, M = 250.

. Assign weights with decreasing importance to the historicalobservations, e.g.

θ̃t =∞∑m=0

ηmR2t−m−1

/ ∞∑m=0

ηm, η = 0.94.

Goal: identify the weighting scheme for every time point t .



Idea: Identification of weighting scheme

At time point t , choose a weighting scheme from

{W (1)t ,W

(2)t , · · · ,W (K)

t }

which leads to the best possible accuracy of estimation.

“Oracle” choice W(k∗)t : the largest weighting scheme for which the

approximation θt ≈ θ∗ still holds.

Aim: mimic the “oracle” choice.



Local ML Estimation

Given a weighting scheme Wt = {wst}, s ≤ t , the weighted (local)maximum likelihood estimate (MLE) is:

θ̃t = argmaxθ

L(W, θ) =∑s

wstR2s

/∑s

wst =∑s

wstR2s

/Nt.

Fitted local likelihood:

L(W, θ̃) = maxθL(W, θ).



Accuracy of local est. under homogeneity

Define L(W, θ, θ′) = L(W, θ)− L(W, θ′) .

Theorem (Polzehl and Sp (2006))

It holds for any θ

L(Wt, θ̃t, θ) = maxθ′

L(Wt, θ′, θ) = NtK(θ̃t, θ)

where Nt =∑

swst and K(θ, θ′) = −0.5{log(θ/θ′) + 1− θ/θ′

}is the

Kullback-Leibler information.

Moreover, if θt ≡ θ∗ , then for any z ≥ 0

IPθ∗(L(W, θ̃, θ∗) ≥ z

)≤ 2e−z.



Risk bound and confidence set

In the local homogeneous case ( θt ≡ θ∗ for wst > 0 ),. the estimation loss L(Wt, θ̃t, θ

∗) is stochastically bounded:

IEθ∗∣∣L(Wt, θ̃t, θ

∗)∣∣r ≡ IEθ∗∣∣NtK(θ̃t, θ∗)

∣∣r ≤ rr ,

where rr = 2r∫z≥0 zr−1e−zdz = 2rΓ (r) ;

. leads to the confidence set:

Et(z) = {θ : NtK(θ̃t, θ) ≤ z}

in the sense that

IPθ∗(Et(z) 63 θ∗

)≤ α.



“Small modeling bias” (SMB) condition

Local parametric assumption (LPA): θt ≈ θ .

Applying a parametric assumption θs ≡ θ in the nonparametricsituation leads to modeling bias measured by

∆(Wt, θ) =∑s

K(θs, θ

)1(wst > 0).

“SMB” ⇔ “∆(Wt, θ) is small for some θ ”

(with a high probability).



Accuracy of local estimation under SMB

TheoremLet θ be such that

IE∆(Wt, θ) ≤ ∆

for some ∆ ≥ 0 “(SMB)”. Then for any r > 0

IE log(

1 +

∣∣NtK(θ̃t, θ)∣∣r

rr

)≤ 1 +∆



Inference under SMB

Under (local) homogeneity θt ≡ θ∗ , the fitted log-likelihoodL(Wt, θ̃t, θ

∗) = NtK(θ̃t, θ∗) has bounded first moment and yields theconfidence set Et(z) = {θ : NtK(θ̃t, θ) ≤ z} .

Under SMB IE∆(W, θ) ≤ ∆ :

. The “loss” L(Wt, θ̃t, θ) = NtK(θ̃t, θ) is stochastically bounded,yielding the accuracy θ̃t − θ � 1/

√Nt .

. the “parametric” confidence set Et(z) still applies with a slightlylarger covering error.

“Oracle” choice: the largest scheme for which the SMB holds.



Setup of adaptive local exponential smoothing

Given a finite set {ηk, k = 1, . . . ,K} , define w(k)st = ηt−sk for s ≤ t :

η1 = 0.60 η2 = 0.68 · · · ηK = 0.98↓ ↓ ↓ ↓θ̃(1)t θ̃

(2)t · · · θ̃

(K)t

N1 = 2.48 ≤ N2 = 3.09 ≤ · · · NK = 56.28

where the local MLEs for k = 1, · · · ,K are:

θ̃(k)t = N−1

k

∑m≤0

ηmk R2t−m−1



Spatial stagewise aggregation procedure

. Initialization: θ̂(1)t = θ̃

(1)t .

. Loop: for k ≥ 2

θ̂(k)t =

(γk

θ̃(k)t

+1− γkθ̂(k−1)t

)−1

where the aggregating parameter γk is computed as:

γk = Kag

(NkK(θ̃(k)

t , θ̂(k−1)t )/zk−1

)If γk = 0 then terminate with θ̂

(k)t = . . . = θ̂

(K)t = θ̂

(k−1)t .

. Final estimate: θ̂t = θ̂(K)t .



Parameters

. Kag(u) aggregation kernel, default choiceKag(u) = {1− (u− 1/6)+}+

. ES localizing schemes W(k)t for η1 < η2 < . . . < ηK . Default

η1 = 0.60 , ηk+1 = 1.25 ∗ ηk , ηK = 0.98 providing a smoothingwindow of length below 500 .



Choice of critical values by “propagation” condition

In the parametric case θt ≡ θ∗ : IEθ∗∣∣NkK(θ̃

(k)

t ,θ∗)∣∣r ≤ rr .

supθ∗∈Θ

IEθ∗∣∣NKK(θ̃

(K)

t , θ̂(K)

t )∣∣r ≤ ρrr

supθ∗∈Θ

IEθ∗∣∣NkK(θ̃

(k)

t , θ̂(k)

t )∣∣r ≤ (k − 1)ρrr

K − 1, k = 2, . . . ,K.

. α is similar to testing level, default choice 0.5 ;

. r is the power of polynomial losses, default choice 1/2 ;



Sequential choice of critical values

. Choice of z1 leading to the aggregated estimate θ̂(k)t (z1) by

setting z2 = . . . = zK−1 =∞ :

IEθ∗∣∣NkK

(θ̃(k)t , θ̂

(k)t (z1)

)∣∣r ≤ ρrrK − 1

, k = 2, . . . ,K.

. Choice of z2 leading to the aggregated estimate θ̂(k)t (z1, z2) by

setting z3 = . . . = zK−1 =∞ :

IEθ∗∣∣NkK

(θ̃(k)t , θ̂

(k)t (z1, z2)

)∣∣r ≤ 2ρrrK − 1

, k = 3, . . . ,K.

. Choice of zk leading to the aggregated estimate θ̂(`)t (z1, · · · , zk)

by setting zk+1 = . . . = zK−1 =∞ :

IEθ∗∣∣N`K

(θ̃(`)t , θ̂

(`)t (z1, . . . , zk)

)∣∣r ≤ kρrrK − 1

, ` = k + 1, . . . ,K.



Critical values w.r.t. different ρ

2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1alpha = 1.0 (default)alpha = 0.5alpha = 0.7alpha = 1.5



“Oracle” result

TheoremLet maxk≤k◦ IE∆

(k)t ≤ ∆ for some k◦ , θ and ∆ . Then

IE log(

1 +N

1/2k◦ K1/2

(θ̂t, θ

)ar1/2

)≤ log

(1 + cur

−11/2

√zk◦)

+∆+ α+ 1

where cu is constant.

� �� 1 2 k◦ k̂ K

Propagation under SMB

θ̃(k)t ≈ θ̂(k)

t

Stability

Nk◦K(θ̂(k)t , θ̂

(k◦)t

)≤ zk◦



Simulation with Gaussian innovations

300 400 500 600 700 800 900 10000

0.5

1

1.5generated volSSALMSES (η = 0.94)

Estimated volatility process based on one realized simulation datawith εt ∼ N(0, 1) .



Simulation with Gaussian innovations

Let θ̃1/2t is the local MLE with η = 0.94 and

RAE =( T∑t=301

∣∣θ̂1/2t − θ1/2

t

∣∣)/( T∑t=301

∣∣θ̃1/2t − θ1/2

t

∣∣) = 0.84

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 LMS SSA

0.75

1.50

2.25

3.00

Results are based on 1000 simulated data.,

Modern Nonparametric Statistics October 10, 2009 27 (50)


Outline

1. Motivation X

2. Nonstationarity: adaptive volatility (AV) estimation X

3. Heavy tails: generalized hyperbolic (GH) distribution and AV


5. Conclusion


Robust Risk Management Accounting for heavy tails

CH model with heavy tailed innovations

Conditional heteroscedasticity model:

yt = θtεt, with IE(ε2t |Ft−1

)= 1.

Local homogeneity assumption: θt ≈ θ for t ∈ I .

Problem: big returns can be caused by heavy tails rather than by thechanges in the volatility parameters.



Generalized Hyperbolic (GH) distribution

X ∼ GH with density:

fGH(ε;λ, α, β, δ, µ) =(ι/δ)λ√2πKλ(δι)

Kλ−1/2

{α√δ2 + (ε− µ)2

}{√

δ2 + (ε− µ)2/α}1/2−λ · e

β(ε−µ)

Where ι2 = α2 − β2 , Kλ(·) is the modified Bessel function of the thirdkind with index λ : Kλ(ε) = 1

2

∫∞0 yλ−1 exp{− ε

2(y + y−1)} dyFurthermore, the following conditions must be fulfilled:

. δ ≥ 0 , |β| < α if λ > 0

. δ > 0 , |β| < α if λ = 0

. δ > 0 , |β| ≤ α if λ < 0



Performance of GH distribution

Estimated (log) density of the daily devolatilized DEM/USD returns. Timeinterval: 1979/12/01 to 1994/04/01 (3719 observations).

−5 −4 −3 −2 −1 0 1 2 3 4−8

−7

−6

−5

−4

−3

−2

−1

0Nonparametric kernelGaussianGH(1,1.74,−0.02,0.78,0.01)



Subclass of GH distribution

The parameters (µ, δ, β, α)> can be interpreted as trend, riskiness,asymmetry and the likeliness of extreme events.

Normal-inverse Gaussian (NIG) distribution: λ = −1/2,

fNIG(ε;α, β, δ, µ) =αδ

π

K1

{α√δ2 + (ε− µ)2

}√δ2 + (ε− µ)2

e{δι+β(ε−µ)}.

where ε, µ ∈ IR , 0 < δ and |β| ≤ α .



Adaptive estimation with NIG innovations

Model: Rt =√θtεt , where εt|Ft−1 ∼ NIG .

Theoretical problem: exponential moment IE{exp(λε2t )} does notexist. Therefore the risk function NtK(θ̃t, θ∗) does not haveexponential moments.

Power transformation: yt,p =(R2t

)p , 0 ≤ p < 1/2 :

IE{yt,p | Ft−1} = θpt IE|εt|2p = θptCp = ϑt,p

Approach: apply the adaptive estimation after the power transformationto estimate ϑt,p .



Procedure

1. Do power transformation to the squared returns Yt = R2t :

Yt,p = Y pt .

2. Compute the estimate ϑ̂t,p of the parameter ϑt,p from Yt,papplying the critical values zk obtained for the Gaussian case.

3. Estimate the value Cp s.t. the innovations are standardized.

4. Compute the estimates θ̂t = (ϑ̂t,p/Cp)1/p and identify the NIG

distributional parameters from ε̃t = Rtθ̂t−1/2

.

5. (Optional) Calculate critical values zk with the identified NIGparameters using Monte Carlo simulation. Repeat the aboveprocedure to estimate θt .



Outline

1. Motivation X

2. Nonstationarity: adaptive volatility (AV) estimation X

3. Heavy tails: GH distribution and AV X


5. Conclusion


Robust Risk Management ICA: dimension reduction

High-dimensional analysis

GHICA: (Chen, Haerdle and Spokoiny 2009)

rt = b>t xt = b>t W−1yt

= b>t W−1D

1/2t εt

where rt is the portfolio return, xt ∈ IRd are asset returns, bt istrading strategy, yt ∈ IRd is an independent vector, Dt is thecovariance matrix of yt and W a nonsingular matrix.



How to find ICs? - Minimize mutual information

I(W, y) =d∑j=1

H(yj)−H(y)

=d∑j=1

H(yj)−H(x)− log |det(W )|

mind∑j=1

H(yj) ≥d∑j=1

minH(yj)

where H(·) is the entropy, see Hyvaerinen, Karhunen and Oja (2001)



Procedure: GHICA

1. Implement ICA to get ICs.

2. Estimate variance of each IC by using the local exponentialsmoothing approach

3. Identify GH distributional parameters of the innovations of each IC

4. Estimate the density of portfolio returns using the FFT technique

5. Calculate risk measures



Foreign exchange rate portfolio

. Data: 7 FX rate 1997/01/02 to 2006/01/05 (2332 observations).

. Dynamic trading strategies: b(3)(t) = x(t−1)∑dj=1 xj(t−1)

, where

x(t) = {x1(t), · · · , xd(t)}>. EUR/USD and EUR/SGD rates aremost correlated with the coefficient 0.6745

. Goal: GHICA versus DCCN (DCC with the Gaussian distributionalassumption)



Risk analysis of the dynamic exchange rate portfolio

The best results to fulfill the regulatory requirement are marked by r . Therecommended method to the investor is marked by i . For the internal supervisory,we recommend the method marked by s .

ICA: dimension reduction 4-3

Risk analysis of the dynamic exchange rate portfolio. The bestresults to fulfill the regulatory requirement are marked by r . Therecommended method to the investor is marked by i . For theinternal supervisory, we recommend the method marked by s .6.4. Risk management with real data 143

GHICA DCCN

h b(t) pr p̂r RC ES p̂r RC ES

1 b(3)(t) 1% 1.28%s 0.0453r 0.0778 1.59% 0.0494 0.0254i

b(3)(t) 0.5% 0.59%s 0.0493 0.1944i 0.94% 0.0547 0.0289

5 b(3)(t) 1% 1.53%s 0.0806r 0.2630i 4.17% 0.0993 0.1735b(3)(t) 0.5% 0.79%s 0.1092 0.2801i 3.44% 0.1100 0.1389

Table 6.2: Risk analysis of the dynamic exchange rate portfolio. The best results to fulfillthe regulatory requirement are marked by r. The recommended method to the investor ismarked by i. For the internal supervisory, we recommend the method marked by s.

standardized DCCN returns are theoretically cross independent and the Gaussian quantilesof the portfolio can be easily calculated. The dynamic mean, variance of the portfolio’sreturns have values of:

IE{r(t)} = b(t)>Σ(1/2)x (t)IE{εx(t)}

Var{r(t)} = b(t)>Σ(1/2)x (t)Var{εx(t)}Σ(1/2)>

x (t)b(t)

The GHICA method in general presents better results than the DCCN. Except the valueof ES at 1% level, the GHICA fulfills the requirements of regulatory, internal supervisoryand investors, see Table 6.2. For h = 1 day forecasts, the DCCN gives although a closerVaR value to 1.6%, i.e. the ideal probability for regulatory, its risk charge with a valueof 0.0494 is larger than that based on the GHICA, 0.0453. Therefore the GHICA is morefavored in fulfilling the minimal regulatory requirement.

The two real data studies show that the GHICA method fulfills the minimal regulatoryrequirement by controlling the risk inside 1.6% level and requiring small risk charge, inparticular satisfies the internal supervisory requirement by precisely measuring risk level asexpected and favors the investors’ requirement by delivering small size of loss. In summary,the GHICA method is not only a fast procedure given either static or dynamic portfoliosbut also produces better results than several alternative risk management methods.

GHICA logosmall


Robust Risk Management Conclusion and Outlook

Conclusion

. Propose approach to account for nonstationarity and heavy tailsand deal with high dimensional financial data

. Provide realistic and fast risk management methods whichoutperform standard methods


Robust Risk Management Conclusion and Outlook

Outlook

. Time-varying ICA

. Online estimation based on high-frequency data


Robust Risk Management References

Reference

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Reference

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Reference

Chen, Y., Haerdle, W. and Jeong, S.Nonparametric risk management with generalized hyperbolicdistributions.SFB 649, discussion paper 2005-001, 2005, submitted.

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Chen, Y. and Spokoiny, V.Local exponential smoothing with applications to volatilityestimation and risk management.working paper, 2006.



Reference

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Reference

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Reference

Hyvaerinen, A. and Oja, E.Independent component analysis: Algorithms and applicationsNeural Networks, 13: 411–430, 1999.

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Reference

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Reference

Polzehl, J. and Spokoiny, V.Propagation-Separation Approach for Local Likelihood EstimationProbability Theory and Related Fields, 135: 335–362, 2006.

Belomestny, D. and Spokoiny, V.Spatial aggregation of local likelihood estimates with applicationsto classificationWIAS Preprint, 2006.


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