The Cumulative Sums of Squares for Retrospective Detection of Changes in Variance

Mathematical Finance: The CumulativeSums of Squares for RetrospectiveDetection of Changes in Variance

Jean-François Emmenegger, Anna Mankevych

Department of Quantitative Economics, University of Fribourg,[email protected], [email protected]

Schweizer Statistiktage, 28.-30. Oktober 2009, Genève, Suisse

J.-F. Emmenegger Mathematical Finance

The programme : systemic risk (1)

On March 2009, the Chairman of the Board ofGovernors of the Federal Reserve System, Dr. Ben S.Bernanke : important statement concerning FinancialReforms.

Creation of an authority specifically charged withmonitoring and addressing systemic risks

President Barack Obama : his Wall Street speech onSeptember 14, 2009

KEYWORD : learning the lessons of Lehman and thecrisis from which we are still recovering


Market risk (2)

The firm RiskMetrics Group :

In the beginning, we defined the language of risk.Today, we are redefining it.

Originally founded upon a measurement of market riskin a portfolio,

RiskMetrics Group is now the recognized standard infinancial risk management.

Market Risk is risk that the value of an investment willdecrease.


Basics : Returns of time series ! (3)

The SMI and SPI stock market index and two blue chipsSMI: Period of 1989:01:03-2008:03:28

Ind

ex v

alu

es

1989 1991 1993 1995 1997 1999 2001 2003 2005 20070

2500

5000

7500

10000SMI_SAVE

SPI: Period of 1989:01:03-2008:03:28

Ind

ex v

alu

es

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 20070

2500

5000

7500

10000SPI_SAVE

UBSN: Period of 1997:12:15-2008:03:28

CH

F

1998 1999 2000 2001 2002 2003 2004 2005 2006 20070

25

50

75

100UBSN_SAVE

CSGN: Period of 1997:12:16-2008:03:28

CH

F

1998 1999 2000 2001 2002 2003 2004 2005 2006 20070

25

50

75

100CSGN_SAVE

Examples : SMI, SPI, UBSN, CCSGN !The calculation of returns or log returns !

rt = P(t + 1)− P(t)P(t) ∼ ln(P(t + 1)

P(t) ) (1)


Observations (4)

Returns: SMI and SPI stock market index and two blue chipsSMI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1989 1991 1993 1995 1997 1999 2001 2003 2005 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSMI

SPI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSPI

UBSN: Period of 1997:12:15-2008:03:28

dai

ly r

etu

rns

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GUBS

CSGN: Period of 1997:12:16-2008:03:28d

aily

ret

urn

s

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GCSG

Log-Returns: SMI and SPI stock market index and two blue chipsSMI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1989 1991 1993 1995 1997 1999 2001 2003 2005 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSMILN

SPI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSPILN

UBSN: Period of 1997:12:15-2008:03:28

dai

ly r

etu

rns

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GUBSLN

CSGN: Period of 1997:12:16-2008:03:28

dai

ly r

etu

rns

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GCSGLN

What do we observe ?No constant variance, ev. piecewise constantvariances ?Peaks showing structural changes !What is the distribution ?


Heteroskedasticity (5)

Returns: SMI and SPI stock market index and two blue chipsSMI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1989 1991 1993 1995 1997 1999 2001 2003 2005 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSMI

SPI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSPI

UBSN: Period of 1997:12:15-2008:03:28

dai

ly r

etu

rns

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GUBS

CSGN: Period of 1997:12:16-2008:03:28

dai

ly r

etu

rns

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GCSG

Log-Returns: SMI and SPI stock market index and two blue chipsSMI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1989 1991 1993 1995 1997 1999 2001 2003 2005 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSMILN

SPI: Period of 1989:01:03-2008:03:28

dai

ly r

etu

rns

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.100

-0.075

-0.050

-0.025

-0.000

0.025

0.050

0.075

0.100GSPILN

UBSN: Period of 1997:12:15-2008:03:28

dai

ly r

etu

rns

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GUBSLN

CSGN: Period of 1997:12:16-2008:03:28

dai

ly r

etu

rns

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15GCSGLN

Hsu, Miller, Wichern model(1974) : piecewise constantvariancesEngle (1982) : ARCH modelsBollerslev(1986) : GARCH models

εt |ψt−1 ∼ N (0, ht), ht = α0 +p∑

i=1βiht−i +

q∑i=1

αje2t−i (2)


Empirical distribution of logged returns (6)

−0.10 −0.05 0.00 0.05 0.10

10

20

30

40

50 Density

Sk=0.618Ku=8.601,JB=15’192

Returns of SMI

−0.075 −0.05 −0.025 0 0.025 0.05 0.075 0.1

10

20

30

40

50

Density

Sk=0.758,Ku=10.130JB=13’455

Returns of SPI

−0.1 0.0 0.1 0.2

10

20

30

40

50 Density

Sk=0.567,12.494,JB=16’966

Returns of UBSN

−0.10 −0.05 0.00 0.05 0.10 0.15

10

20

Density

Sk=0.267,Ku=5.301,JB=3’059

Returns of CSGN

The distribution is far from normal

Sk = µ3σ3 Ku = µ4

σ4 JB = n6 (S2

k + (Ku − 3)3

4 ) (3)


Theories on the distribution of returns (7)

Bachelier (1900) : normal distribution

Mandelbrot and Fama (1963) : heavy tails, high peaks,fractals

Mandelbrot (1963) and Rachev (2000) : stabledistributions

Propositions : Pareto-, Cauchy-, Lévy-, t-distributions,α-stable distribution

And what about Value at Risk (VaR) ?)


Measuring of market risk : Value at Risk (8)

Value at Risk (VaR) is the maximum loss not exceededwith a given probability defined as the confidencelevel, over a given period of time.On the background : the normal distribution

FIGURE: The 10%-value of risk of a normally distributed portfolio


Other suggested distributions (9)

Lévy f (x; c) =√

( c2π )e−

c2x

x32

(4)

Cauchy f (x;µ, c) = 1π

[ c(x − µ)2 + c2 ] (5)

Pareto f (x; k, xm) = k xkm

xk+1 for x > xm (6)

the family of α-stable distributions

ϕ(t) = EeitX ={

exp{−σα|t|α(1− iβ t|t| tan(πα2 )) + iµt}, α 6= 1

exp{−σ|t|(1 + iβ 2π

t|t| ln(|t|)) + iµt}, α = 1,

(7)


Fourier Transforms and the charac. function (10)

The characteristic function ΦX (u) of a random variable X

ΦX (u) = E(eiuX ) =∫ ∞−∞

eiuxdFX (x) =∫ ∞−∞

eiux fX (x)dx. (8)

fX (x) = 12π

∫ ∞−∞

e−iuxΦX (u)du. (9)

(8), (9) are Fourier Transforms, Bracewell (1986)

ΦX (2πw) = IFT (fX ) =∫ ∞−∞

e2πiwx fX (x)dx. (10)

fX (x) = FT (ΦX ) =∫ ∞−∞

e−2πiuxΦX (u)du. (11)


Example : Normal Distribution N (0, 1) (11)

-3 -2 -1 1 2 3x

0.1

0.2

0.3

0.4

0.5Re�fX�x��

-3 -2 -1 1 2 3x

0.1

0.2

0.3

0.4

0.5Im�fX�x��

-3 -2 -1 1 2 3u

0.2

0.4

0.6

0.8

1Re��X�u��

-3 -2 -1 1 2 3u

0.2

0.4

0.6

0.8

1Im��X�u��

µ = 0, σ = 1

fX (x) = 12π

∫ ∞−∞

exp(−iux)exp(iuµ− u2σ2

2 )du. (12)

fX (x) = 1√2πσ

exp(−(x − µ)2

2σ2 ). (13)


Stability : α-stable distributions (12)

Mandelbrot (1963), Rachev (2000) and Olszewski (2005)suggest applying alpha stable distributions to modelthe series of returns with non-normal distributions.

Stable distributions are a family of distributions whicharises from the Generalized Central Limit Theorem.

Stable non-Gaussian distributions possess severalproperties that make them attractive in theapplications, namely heavy tails (schwere Flanken),excess kurtosis (Wölbung), asymmetries (Schiefe)


Examples : α-stable distribution (13)

Symmetric and non-symmetric α-stable distributions

ϕ(t) = EeitX ={

exp{−σα|t|α(1− iβ t|t| tan(πα2 )) + iµt}, α 6= 1

exp{−σ|t|(1 + iβ 2π

t|t| ln(|t|)) + iµt}, α = 1,

(14)


Eastern European indexes and FoRex (14)

Returns of Eastern European indexes and FoRex ratesThe distributions are not normalheavy tails, highly peaked (kurtosis) and skewness


Iterated cumulative sum of squares (ICSS) Dk (15)

Inclan and Tiao (1994) : Iterative Cumulative Sums ofSquares (ICSS) algorithm to detect change points

Subject : detection of multiple changes of variance

independent observations ak , k = 1, ..,Tbuild an iterated cumulative sums of squares (ICSS)algorithm to detect changes of variance

E(at) = 0 and variances Var(at) = σ2t .

Define a centered cumulated sum of squares Dk :

Ck =∑k

t=1 a2t

Dk = CkCT− k

T , k = 1, ...,T , D0 = DT = 0 (15)


Hsu, Miller, Wichern models and CSS Dk (16)

0 100 200 300 400 500 600 700

−2

0

2

Distribution of white noise N(0,1)

0 100 200 300 400 500 600 700

−2

0

2

Distribution of heteroskedastic noise

0 100 200 300 400 500 600 700−2

−1

0

1

2Distribution of Dk

0 100 200 300 400 500 600 700

−1

0

1

2Distribution of Dk

A white noise series of N (0, 1) with constant varianceA series with two changes of variance : t = 391, t = 518σ2

t =τ20 = 1, t = 1, ..., 390 ; σ2

t =τ21 =0.365, t = 391, ..., 517 and

σ2t =τ2

2 =1.033, t = 518, .., 700.


Approximate expected value of√

(T/2)Dk (17)

√T2 E(D25) =

√50(E( C25

C100)− 25

100) =√

50( 251× 25 + 3× 75 −

14)

=√

50( 25250 −

14) =

√50(−3

20 ) = −1.0606.(16)


Asymptotic behaviour of Dk (18)

Let W represent a Brownian motion process,E [Wt ] = 0, E [WtWs] = s, 0 ≤ s < t. Let W o denote aBrownian bridge, W o

t = Wt − tWt , E [W ot ] = 0,

E [W ot W o

s ] = s(1− t), 0 ≤ s < t, and W o0 = W o

1 = 0, withprobability 1.

Theorem 1 : Let a1, a2, .... be a sequence ofindependent, identically distributed normal randomvariables ak ∼ N (0, σ2

a). Let Dk = Ck/CT − k/T , whereCk =

∑ki=1 a2

i . Then√

T/2Dk →W o.


The Brownian Bridge (19)

Monte Carlo simulation

Settings : M = 1

Sample sizes : T=2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.1

0.2

0.3

0.4

0.5

0.6distribution of |Dk|, formulae (1)


A set of Brownian Bridges (20)

Monte Carlo simulation

Settings : M = 20

Sample sizes : T=500

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.5

1.0

1.5

DensityDistribution of the max|Dk|, formulae (1) N(s=0.309)

0 50 100 150 200 250 300 350 400 450 500

0.5

1.0

1.5 Distribution of |Dk|, formulae (1)


Distributions of max(√

T/2|Dk |) (21)

Monte Carlo simulationSettings : M = 10’000Sample sizes : T=100, 200, 300, 400, 500

0.5 1.0 1.5 2.0 2.5

0.5

1.0

1.5

2.0 DensityDistribution of the max|Dk|, formulae (1)

0.5 1.0 1.5 2.0

0.5

1.0

1.5


0.5 1.0 1.5 2.0

0.5

1.0

1.5


0.5 1.0 1.5 2.0

0.5

1.0

1.5


0.5 1.0 1.5 2.0

0.5

1.0

1.5


The quantiles at probabilities : 0.05, 0.10, 0.25, 0.50,0.75, 0.90, 0.95, 0.99


Empirical Quantiles of Inclan-Tiao and Monte Carloresults (22)

T 100 200 300 400 500p (2) (3) (4) (5) (6)

0.05 0.44400 0.46629 0.47822 0.48363 0.486110.10 0.49569 0.51878 0.52776 0.53491 0.539910.25 0.59764 0.62448 0.63078 0.64160 0.642690.50 0.74846 0.77064 0.78345 0.79195 0.790560.75 0.94364 0.96711 0.97387 0.98700 0.983700.90 1.1516 1.1664 1.1738 1.1938 1.18680.95 1.2846 1.2951 1.3076 1.3319 1.31500.99 1.5454 1.5531 1.5749 1.6146 1.6341


The masking effect (23)

√T2 E(D58) =

√50(E( C58

C100)− 58

100) =√

50( 251 · 58 + 3 · 22 + 6 · 20

− 58100) =

√50( 58

244 −58100) =

√50− 0.342295) = −2.4203.

(17)


The ICSS algorithm of Inclan-Tiao (1994) (24)

a[t1; t2] represents the series at1 , at2+1, ....., at2 , t1 < t2

Dk(a[t1 : t2]) indicates the range over which thecumulative sum of squares is calculated.

D∗ is a critical value (5%-quantile)two successive condition-control loops :

1 1. loop : compute D([kfirst : klast ]) until kfirst = klast1 a : search the smallest change point k∗ = kfirst ∈ [1,T ]2 b : search for the greatest change point k∗ = klast ∈ [1,T ]

2 Let NT be the number of found change points (�) in cp3 2. loop : Keep only the change points :

1 Set : cp0 = 0, cpNT+1 = T2 Keep change point cpj if,

max(Dk([cpj−1 + 1 : cpj+1])) > D∗, j = 0, ..,T


α-stable driven Brownian bridges (25)

What happens, when the Brownian process is replacedby an α-stable process ?

Replace the normal by a stable distribution !

Monte Carlo simulation of α-stable bridges

0 100 200 300 400 500 600 700 800 900 1000−5.0

−2.5

0.0

2.5

5.0 alpha=2, sigmasqr=2, N(0,2)Alpha−stable Distribution

0 100 200 300 400 500 600 700 800 900 1000−2

−1

0

1

2Distribution of Dk

0 100 200 300 400 500 600 700 800 900 1000−15

−10

−5

0

5

10

15alpha=1.9, sigmasqr=2, beta=0Alpha−stable Distribution

0 100 200 300 400 500 600 700 800 900 1000−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5Distribution of Dk


MC Simulation : Quantiles of α-stable distrib. (26)

0 1 2 3 4 5 6 70.0

0.5

1.0

1.5

2.0 DensityDistribution of the max|Dk|, formulae (1), alpha−stable

0.0 2.5 5.0 7.5 10.0

0.5

1.0

1.5


0.0 2.5 5.0 7.5 10.0 12.50.0

0.5

1.0

1.5


0.0 2.5 5.0 7.5 10.0 12.5 15.0

0.5

1.0

1.5


0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

0.5

1.0

1.5


T 100 200 300 400 500p (2) (3) (4) (5) (6)

0.05 0.8893 1.2225 1.4684 1.6677 1.83120.10 1.0610 1.4486 1.7453 1.9971 2.21790.25 1.4365 1.9985 2.4402 2.8112 3.14890.50 2.1294 2.9942 3.6303 4.2364 4.71890.75 3.1813 4.5032 5.4509 6.3732 7.13810.90 4.3189 6.1712 7.5005 8.7465 9.74170.95 4.9964 7.2552 8.7029 10.212 11.3240.99 6.0915 8.7263 10.710 12.215 13.895


Conclusions (27)

Monte Carlo simulations confirm the results of Inclanand Tiao !

Change points exist, when the driving distribution isnormal.

Experiment : α-stable driven cumulative sums ofsquares Dk show that the quantiles become larger andchange points no longer appear !

Thesis : The notion of change points is naturallyinherent to the chosen distribution !

Future work : The application of the ICSS-criterionmust be justified through a some Limit Theoremrelated to self similar processes (Kesten and Spitzer)


The Cumulative Sums of Squares for Retrospective Detection of Changes in Variance

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