swiss quant

8/12/2019 Swiss Quant

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t it ii t r ,,)(

I D D t t t ~~~ 21

)()(

)),(()(

,

,

t x P

S Dt y

t x f t y

l

e

l el

l eel

t t

t m

t

t

,

,1

m

ik ikik t t r t x

1

)()()(


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66

Risk Modeling MethodologiesDelta NormalApproximation

Pure HistoricalSimulation

Filtered HistoricalSimulation

Monte-CarloNormal Simulation

Modeling assumptions

Simplified but highperformance approach

Accounting for covariance

dynamicsImproved forward looking

Handling of instrument withInsufficient historical data

Linearization

Normality assumption

Cross-product dependencyby Covariance

Risk factor compression

No distribution assumption

Easy to understand / oftenused

Exact treatment ofderivatives

Modeling assumptions

Modeling assumptionsModeling assumptions Future = past

Constant volatility

Constant correlations


Dynamic correlation/vola

Non parametricassumption of residuals


Normality assumption

Cross-product dependency

by VCV No inclusion of fat-tail


Accounting for covariancedynamics

Inclusion of fat-tails

Improved forward looking


Handling of instrument withInsufficient historical data

Accounting for covariancedynamics

Improved forward looking


Features Features Features Features

1

3

Ensured by:Volatility filter &Quantilecorrection 1

Ensured by:Inclusion ofspecific risk

2

3

Large-scaleMedium-term

Medium-scaleShort-term

Ensured by:Analyticalmapping ofinstruments

3

2

1

2

2

Improvements


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T

t

m

ikii

m

iikik T t r t x

ki 1 1

2

1,

)()(min

see Tibshirani, R. J. (1996)


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, 1

, 2

, 3

2


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21

0,

2,

1

0

,

)(11

,)(

1

1

J

j jt ii

j J t i

J

j

i j

J t i

jt r

jt r

t it ii t r ,,)(

.

)

)1((

,

,

1, t i

t ii

t i

t r

t iT iT i t r

,,,

)(~

1

see Barone-Adesi et al.(1998)


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1


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1

see Franconi and Herzog (2012)


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1


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0 500 1000 1500 2000 2500-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time points

F r e q u e n c y o f V i o l a t i o n s

0 500 1000 1500 2000 250-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time points

F r e q u e n c y o

f V i o l a t i o n s

1


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Portfolio Aggregation using the Delta Normal Approximation

Risk Factor Data it r ,

r : Returnt : Time

mt

t

t

r

r

r

,

1,

...)( t r mean

Client PositionWeights

Instrument Data

Derivatives andStructured Products

, z k v

,k t x

,k e

Betas on RiskFactors k,i , k

v : Weights z : Client portfolio id x : Instrument returnsk : Instrument no.

: Delta (first order derivative)e : Derivative no.

k : Idiosyncratic risk (specific risk) : Intrinsic covariance matrix: Beta value of k-th instrument

to i-th benchmark/risk factor: covariance matrix risk factors

k,i = OLS(x t,k , r t,i , selection rules) = [ k,i ] matrix nxm

ndiag ...1

Risk LT

, z k v

ik ,

z corrected

T T T z

VaR

Dv Dvv D Dv

)(

1

k e

k

D

,

,1

.00

.....0

0.....

00.

D

3

3


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Portfolio Aggregation using scenarios

Risk Factor Data it r ,

r : Returnt : Time

mt

t

t

r

r

r

,

1,

...)( t r mean

Client PositionWeights

Instrument Data

Derivatives andStructured Products

, z k v

,k t x

,k e f

Betas on RiskFactors k,i , k

v : Weights z : Client portfolio id x : Instrument returnsk : Instrument no.

f : derivative fomulae : Derivative no.

k : Idiosyncratic risk (specific risk): scenarios of the risk factors: Beta value of k-th instrument

to i-th benchmark/risk factor: scenarios of the idiosyncratic risk

k,i = OLS(x t,k , r t,i , selection rules) = [ k,i ] matrix nxm

Risk)(~ , sr T k

, z k v

ik , ~ ))(~())(~(

)(~)(~)(~)(~)(~

2

1

2

111 1

sr quantileVaR sr

s sr f v s sr v sr

port Z port

S

s z

l i

m

ilie

E

ee

K

k k i

m

ikik port

corrected

e f

3

3

)(~ , sr T k

)(~ sk


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-0.6

-0.4

-0.2

0

0.2

0.4

-0.6 -0.4 -0.2 0 0.2

A n a

l y t i c a

l B e t a s

( P C 1

)

Regressed Betas (PC1)-0.4

-0.2

0

0.2

-0.4 -0.2 0 0.2 0.4

A n a

l y t i c a

l B e t a s

( P C 2

)

Regressed Betas (PC2)

-1

-0.5

0

0.5

1

-0.6 -0.4 -0.2 0 0.2 0.4

A n a

l y t i c a

l B e t a s

( P C 3

)

Regressed Betas (PC3)

2


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1

)()()( t t t r j j

23

In order to improve the forward-looking properties of themodel

A volatility filtering procedure is applied A weighing factor is used in the computation of historical

volatilities (EWMA) or a GARCH type model

23

96.0

95.0

94.0 9.0

99.5% 99% 95% 90%

99.87% 99.62% 96.55% 91.86%

99.78% 99.43% 96.02% 91.24%

99.73% 99.35% 95.81% 90.93%

99.68% 99.27% 95.61% 90.74%

Problem: While forward-looking properties are ensured, thereis still uncertainty in tail estimation

Solution: Use weighing factor and quantile correction

Original quantile (before correction)

However: leads to the usage of a smaller effectivenumber of historical data used in the computation of thevolatilities

To mitigate this effect a quantile correction is calculatedbased on a look-up table

Depending on the value of a higher quantile is computedthan was originally aimed at.

The difference to the original quantile (before correction) isthe quantile correction add-on

C o r r e c t e

d q

u a n t i l e

R e

t u r n s

Scenario Scenario

Volatilityfiltering

Instrument returnsbefore volatility filteri ng

Instrument residualsafter volatility filtering

Positions showinstabilities in volatilities

Positions showstationary behavior (noinstabilities, thus goodforecastability)


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