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A. Meucci - Fully Flexible Views (Entropy Pooling)

Attilio Meucci

Fully Flexible Views:Fully Flexible Views: Entropy Pooling in Theory and Practice

Swissquote Conference on Asset Management

Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerlandy q ( ),

October 21, 2011


ESTIMATION RISK

SCENARIO ANALYSIS

THE BLACK-LITTERMAN APPROACH

ENTROPY POOLINGENTROPY POOLING

CASE STUDIES

REFERENCES AND CONCLUSIONS

A. Meucci - Fully Flexible Views (Entropy Pooling) Estimation risk

E T m R

CovS R

: expected returns

: covariance

argmax ' mw w

Cov T S R : covariance

argmaxvw

mw wC

subject to ' vw Sw

w : portfolio weights

v : grid of target variances

' 1 01C : investment constraints, e.g. ' 1, 0w 1 w

A. Meucci - Fully Flexible Views (Entropy Pooling) Estimation risk

E T m R

CovS R

: expected returns

: covariance

argmax 'w w m

Cov T S R : covariance

argmaxvw

w w mC

subject to ' vw Sw

0 7

0.8

0.9

1

wei

ghts

efficient frontier

S10

0.4

0.5

0.6

0.7

port

folio

Dell S4S5

S7S8

S9S10

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6

A. Meucci - Fully Flexible Views (Entropy Pooling) N , 1, ,t t T r m S~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?

argmax 'w w m

Cov T S R : covariance?

argmaxvw

w w mC

subject to ' vw Sw

0 7

0.8

0.9

1

wei

ghts

efficient frontier

S10

0.4

0.5

0.6

0.7

port

folio

Dell S4S5

S7S8

S9S10

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6


1 T

N , 1, ,t t T r m S~

~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

1t

tT

m r

1 'T

S r m r m

~

~ ~ ~

argmax 'w w m

Cov T S R : covariance? 1

t ttT

S r m r m

argmaxvw

w w mC

subject to ' vw Sw

0 7

0.8

0.9

1

wei

ghts

efficient frontier

S10

0.4

0.5

0.6

0.7

port

folio

Dell S4S5

S7S8

S9S10

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6


1 T

N , 1, ,t t T r m S~

~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

1t

tT

m r

1 'T

S r m r m

~

~ ~ ~

argmax 'w w m


argmax ' mw w

1

t ttT

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

w

mw wC

subject to ' vw Sw~

0 7

0.8

0.9

1

wei

ghts

efficient frontier

S10

0.4

0.5

0.6

0.7

port

folio

Dell S4S5

S7S8

S9S10

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6


1 T

N , 1, ,t t T r m S~

~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

1t

tT

m r

1 'T

S r m r m

~

~ ~ ~

argmax 'w w m


argmax ' mw w

1

t ttT

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

w

mw wC

subject to ' vw Sw~

0 7

0.8

0.9

1

wei

ghts

efficient frontier

LIVES10

0.4

0.5

0.6

0.7

port

folio

LIVEDell S4S5

S7S8

S9S10

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6


1 T

N , 1, ,t t T r m S 1~

~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

1t

tT

m r

1 'T

S r m r m

~

~ ~ ~

argmax 'w w m


argmax ' mw w

1

t ttT

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

w

mw wC

subject to ' vw Sw~

0 7

0.8

0.9

1

wei

ghts

0 7

0.8

0.9

1

wei

ghts

efficient frontier efficient frontier

S10

0.4

0.5

0.6

0.7po

rtfo

lio

0.4

0.5

0.6

0.7

port

folio

Dell S4S5

S7S8

S9S10

0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

v0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6

A. Meucci - Fully Flexible Views (Entropy Pooling) N , 1, ,t t T r m S

1 T

2~

~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

1t

tT

m r

1 'T

S r m r m

~

~ ~ ~

argmax 'w w m


argmax ' mw w

1

t ttT

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

w

mw wC

subject to ' vw Sw~

0 7

0.8

0.9

1

0 7

0.8

0.9

1

wei

ghts

wei

ghts


S10

0.4

0.5

0.6

0.7

0.4

0.5

0.6

0.7

port

folio

port

folio

Dell S4S5

S7S8

S9S10

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

v0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6

A. Meucci - Fully Flexible Views (Entropy Pooling) N , 1, ,t t T r m S

1 T

3~

~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

1t

tT

m r

1 'T

S r m r m

~

~ ~ ~

argmax 'w w m


argmax ' mw w

1

t ttT

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

w

mw wC

subject to ' vw Sw~

0 7

0.8

0.9

1

wei

ghts

0 7

0.8

0.9

1

wei

ghts


S10

0.4

0.5

0.6

0.7

port

folio

0.4

0.5

0.6

0.7po

rtfo

lio

Dell S4S5

S7S8

S9S10

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

v 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

vIBM

S3 S6


ESTIMATION RISK

SCENARIO ANALYSIS



CASE STUDIES

CONCLUSIONS AND REFERENCES


Market distribution

Scenario analysis / stress-testing

returns on asset classes/funds

Market distribution

DEL

L

IBM


Scenario analysisMarket distribution




FULL specification:

vQ = identity

DEL

L

IBM






FOCUS: Q = portfolios

DEL

L

IBM

v






Conditional formula






Conditional formula

Optimization


ESTIMATION RISK

SCENARIO ANALYSISSCENARIO ANALYSIS

THE BLACK-LITTERMAN APPROACH - Estimation risk

- Views

- Discussion

ENTROPY POOLING

CASE STUDIESCASE STUDIES



Market distribution

Black-Litterman – estimation risk


Market distribution

DEL

L

IBM


Market distribution



Market distribution

estimation risk?


Market distribution



Market distribution

estimated by exponential smoothing


Market distribution



Market distribution


?


Market distribution



Market distribution



Market distribution



Market distribution


?


Market distribution



Market distribution


mean-variance


Market distribution



Market distribution


112

w Σ πλ λ

mean-variance


Market distribution



Market distribution


equilibrium portfolio

equilibrium returns


Market distribution



Market distribution


DEL

L

D

IBM

equilibrium returns


Market distribution



Market distribution

0.9

1

hts

efficient frontier

0.5

0.6

0.7

0.8

ortfo

lio w

eig

DEL

L

S8

S9S10

0.1

0.2

0.3

0.4po

D

IBM

Dell

S3

S4S5

S6

S7S8

0.05 0.1 0.15 0.2 0.25 0.3 0.350 v IBM

IBM

Optimization


ESTIMATION RISK



- Views

- Discussion

ENTROPY POOLING




Market distribution Views

Black-Litterman – views



scenario analysis with uncertainty

DEL

LD

IBM






FOCUS: Q = portfolio

DEL

LD

IBM

vv


ViewsMarket distribution




Bayes’ formulaBayes formula






small uncertainty

Ω 0

Bayes’ formula conditional formulaBayes formula

v

conditional formula

v






Ω large

uncertainty

Bayes’ formulaBayes formula

vv






Bayes’ formula






Bayes’ formula

Optimization Optimization

-






LIVELIVEOptimization Optimization

-






0.8

0.9

1

0.8

0.9

1

wei

ghts

wei

ghts


0 4

0.5

0.6

0.7

0.5

0.6

0.7

port

folio

w

port

folio

w

S8

S9S10

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

IBM

Dell

S3

S4S5

S6

S7

Optimization0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05 0.1 0.15 0.2 0.25 0.3 0.350v v

Optimization

-


ESTIMATION RISK



- Views

- Discussion

ENTROPY POOLING




Views/ScenariosMarket distribution

Black-Litterman – pro’s


Views/Scenarios

+ uncertainty

Market distribution

Optimization



Black-Litterman – con’s


Views/Scenarios

+ uncertainty

Market distribution

Market is not only returns: implied volatilities (derivatives)rates paths (mortgages)implied correlations (CDO’s)….

Optimization





Views/Scenarios

+ uncertainty

Market distribution

Market is not only returns

Market is not only normal: fat tails skewness tail-risk codependenceMarket is not only normal: fat tails, skewness, tail risk codependence,…

Optimization





Views/Scenarios

+ uncertainty

Market distribution


Market is not only normalMarket is not only normal

Market is not only equilibrium: historical estimates, implied values, …

Optimization





Views/Scenarios

+ uncertainty

Market distribution



Market is not only equilibrium

Views are not only portfolios: generic non-linear functions

Optimization





Views/Scenarios

+ uncertainty

Market distribution




Views are not only portfolios

Views are not only on expectations: correlations, volatilities,tail behavior, copulas, …

Optimization





Views/Scenarios

+ uncertainty

Market distribution





Views are not only on expectations

Views are not only equalities: stock ranking, qualitative views

Optimization





Views/Scenarios

+ uncertainty

Market distribution





Views are not only on expectations

Views are not only equalities

Optimization is not only mean variance: mean-CVaR, mean-VaR, …

Optimization


ESTIMATION RISKESTIMATION RISK

SCENARIO ANALYSIS


ENTROPY POOLING - Theory

- Implementation- Implementation

CASE STUDIES



Market distr. not returns, not normal, not equilibrium 1Xe.g.

2X2-yr swap rate

5-yr swap rate2X 5-yr swap rate5y

r sw

ap

22yr swap


not returns, not normal, not equilibriumMarket distr. 1Xe.g.

2X2-yr swap rate


r sw

ap

22yr swap

P i i d l / / f ll i ie.g.

d ti itPricing delta/gamma/vega, full pricing, … duration + convexity



2X2-yr swap rate


r sw

ap

22yr swap

P i i d l / / f ll i ie.g.

d ti itPricing

Optimization

delta/gamma/vega, full pricing, …

utility, mean-CVaR, … mean-variance

duration + convexity



2X2-yr swap rate

5-yr swap rate

Focus non-linear functions and external factors2X 5-yr swap rate

2 21 2V X X

convexity factor



2X2-yr swap rate

5-yr swap rate

Focus non-linear functions and external factors2X 5-yr swap rate

1V X

5yr s

wap

22yr swap



2X2-yr swap rate

5-yr swap rate

Focus

Views

non-linear functions and external factors

full distribution specification

2X 5-yr swap rate

1V X

22yr swap



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X

view on expectations (BL), medians

22yr swap



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X


ranking



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X


ranking

views on volatilities

22yr swap



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X


ranking


correlation stress-testing



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X


ranking


correlation stress-testing

i t il b h iview on tail behavior



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X

partial distribution specification



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X


2yr swap



2X2-yr swap rate

5-yr swap rate

Focus

Views

non-linear functions and external factors2X 5-yr swap rate

1V X

r sw

ap ?

5yr

?

2yr swap



2X2-yr swap rate

5-yr swap rate

Focus

Views

non-linear functions and external factors2X 5-yr swap rate

1V X

r sw

ap ?

5yr

?

Posterior ?2yr swap



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X


Posterior ?

“distance” btw. distributionsrelative entropy



2X2-yr swap rate

5-yr swap rate

Focus

Views



2X 5-yr swap rate

1V X


Posterior least distance from prior



not returns, not normal, not equilibriumMarket distr.

Focus

Views




Posterior least distance from prior, views satisfied




Focus

Views







Focus

Views





LIVELIVE


Market distr. 1X2X

2-yr swap rate

5-yr swap rate2X 5-yr swap rate

5yr s

wap

prior USD SWAP 2Y rate prior USD SWAP 5Y rate

2yr swap

prior USD SWAP 2Y rate p

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap


Market distr. 1X2X

2-yr swap rate


prior USD SWAP 2Y rate prior USD SWAP 5Y rateprior USD SWAP 2Y rate p

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap


Market distr. 1X2X

2-yr swap rate


Focus1V X

prior USD SWAP 2Y rate prior USD SWAP 5Y rateprior USD SWAP 2Y rate p

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap


Market distr. 1X2X

2-yr swap rate


Focus

3 26V

m V m V bp

Views

1V X


3.26 2

m V m V bp


normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap


Market distr. 1X2X

2-yr swap rate


1V XFocus

Views 3 26

Vm V m V bp


3.26 2

m V m V bp


normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Posterior

5

0posterior USD SWAP 2Y rate

correctnormal fit

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

2yr swap


Market distr. 1X2X

2-yr swap rate


1V XFocus

Views 3 26

Vm V m V bp


3.26 2

m V m V bp


normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Posterior

5

0posterior USD SWAP 2Y rate posterior USD SWAP 5Y rate

correctnormal fit

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap



Focus

Views







Focus

Views





Confidence multi-user, multi-confidence



Focus

Views






100(1-c) % of times: PRIOR



Focus

Views






100c % of times: POSTERIOR



Focus

Views





Confidence

P i i

multi-user, multi-confidence

d l / / f ll i iPricing delta/gamma/vega, full pricing, …



Focus

Views





Confidence

P i i

multi-user, multi-confidence

d l / / f ll i iPricing

Optimization mean-variance, mean-CVaR, …

delta/gamma/vega, full pricing, …


ESTIMATION RISKESTIMATION RISK

SCENARIO ANALYSIS


ENTROPY POOLING - Theory

- Implementation- Implementation

CASE STUDIES



Entropy PoolingParametric Fully Flexible

Implementation

DistributionsFully Flexible Probabilities


Entropy PoolingParametric

Implementation

Parametric

Distributions

Parametric Entropy Pooling



Implementation

Parametric

Distributions


Normal (+dimension reduction)



Implementation

Parametric

Distributions



NormalLinear-Quadratic


Market distr.

Analytical implementation

Focus

Views

Posterior

Confidence

P i iPricing

Optimization


Market distr.

Analytical implementation

Focus

Views

Posterior

Confidence

P i i …

…

Pricing

Optimization


Fully FlexibleEntropy Pooling

Parametric

Non-parametric implementation

Fully Flexible ProbabilitiesDistributions

joint scenario of N risk drivers probability of joint scenario



Parametric


Fully Flexible Probabilities

Parametric N P i

Distributions


Non-Parametric Entropy Pooling





Parametric


Fully Flexible Probabilities

Parametric N P i

Distributions



Fuzzy Membership

Bayesian networks


State Conditioning

KernelsCrisp

ConditioningNormalLinear-Quadratic

g

Time Conditioning

A. Meucci - Fully Flexible Views (Entropy Pooling) Non-parametric implementation

probability = 1 / num scenarios

“prior”


probability < 1 / num scenarios



probability > 1 / num scenarios

“prior”

bearish bullish



regular market


probability < 1 / num scenarios



probability > 1 / num scenarios

regular market

low volatility high volatility



market distribution


probability = 0

probability = 1

scenario analysis

market distribution

scenario analysis


Market distr.



Market distr.


joint scenario of N risk driversj

Probability of joint scenariojoint scenario


Market distr.


joint scenario of securities prices

P i i Pricing Probability of joint scenario


Market distr.


P i i Pricing

Optimization …


Market distr.


Focus

Views

Posterior

Confidence

P i iPricing

Optimization


Market distr.


Focus

Views scenario index

Posterior

Confidence

P i iPricing

Optimization


Market distr.


Focus

Views

Xe.g.

2-yr swap rate

1X2X

2 yr swap rate

5-yr swap rate

1V X

m V

J

j jp V

Posterior

1j j

jp

V

Confidence

P i iPricing

Optimization


Market distr.


Focus

Views

Posterior


Posterior

Dual formulation: li l i dlinearly constrained

convex optimization in # variables = # views


Market distr.


Focus

Views

Posterior Confidence

P i iPricing

Optimization


Market distr.


Focus

Views

Posterior Confidence P i iPricing

Optimization


Market distr.


Focus

Views


P i i

Pricing

Optimization


Market distr.


Focus

Views


P i i

…

Pricing

Optimization


ESTIMATION RISK

SCENARIO ANALYSIS


ENTROPY POOLING

CASE STUDIES - Portfolio from sorts

- Implied expected returns

Distributional stress testing- Distributional stress-testing

- Option trading



expected returns no views1

frontier no viewsexpected returns efficient frontier

Case study: portfolios from sorts

6789

10

0.6

0.8

wei

ghts S7

S10

S8S9S10

S7S6

12345

0.2

0.4

port

folio

w

IBM

DellS3

S4

S5

S6S8

S9

IBMDell

S5

S3S4

0 2 4 6 8 10 12

1

8 9 10 11 12 13 14 150

expected return volatility

IBMIBM





6789

10

0.6

0.8

wei

ghts S7

S10

S8S9S10

S7S6

12345

0.2

0.4

port

folio

w

IBM

DellS3

S4

S5

S6S8

S9

IBMDell

S5

S3S4

0 2 4 6 8 10 12

1

8 9 10 11 12 13 14 150

view: 3 4E R E Rexpected return volatility

IBMIBM





6789

10

0.6

0.8

wei

ghts S7

S10

S8S9S10

S7S6

12345

0.2

0.4

port

folio

w

IBM

DellS3

S4

S5

S6S8

S9

IBMDell

S5

S3S4

0 2 4 6 8 10 12

1

8 9 10 11 12 13 14 150

view: 3 4E R E Rexpected return volatility

IBMIBM

89

10

expected returns after views

0.8

1frontier after viewsexpected returns efficient frontier

ts S7S10

S8S9S10

345

678

0.4

0.6

rtfo

lio w

eigh

t

Dell S3S6

S7

S8

S9S8S7S6S5

S3S4

0 2 4 6 8 10

123

8 9 10 11 12 13 14 150

0.2por

volatilityexpected return

IBM

DellS5

IBMDellS3

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: portfolios from sorts

Views:

AA

Sample means Standard ranking Centroid Entropy Pooling

UTXPGHDDISKOBA

CAT

DDGE

MMMKFTMCD

VZUTX

XOMCSCOWMT

JNJAXPIBM

T

PFEHPQCVXMRK

MSFTINTCBACXOM

-20 0 20 40

JPMTRVPFE

Prior [% p.a.]-20 0 20 40

Common [% p.a.]-20 0 20 40

AC [% p.a.]-20 0 20 40

EP [% p.a.]


ESTIMATION RISK

SCENARIO ANALYSIS


ENTROPY POOLING




- Option trading


A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: implied expected returns

XOM

market capital. weights

sample means Black-Litterman implied exp. returns

Entropy Pooling implied exp. return

PGT

TRVUTXVZ

WMTXOM

KFTKO

MCDMMMMRK

MSFTPFE

GEHD

HPQIBM

INTCJNJJPM

BABACCAT

CSCOCVX

DDDISGE

0 5 10

AAAXPBA

Weights [%]0 2 4P i [% ]

0 2 4BL [% ]

0 2 4EP [% ]

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: implied expected returns

10 10

Expectation -covariance ellipsoids

0

5

BA

0

5

DD

Sample

-10 -5 0 5 10

-5

AXP-10 -5 0 5 10

-5

AA

Sample

Black-Litterman

AXP AA

105

10Entropy-Pooling

0

5

CA

T

-5

0

CS

CO

-10 -5 0 5 10

-5

BAC-10 0 10

-10

CAT


ESTIMATION RISK

SCENARIO ANALYSIS


ENTROPY POOLING




- Option trading


A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: distributional stress-testing

Prior (exponential time decay)

State indicator (2yr swap rate - current level)State indicator (2yr swap rate current level)

State indicator (1->5yr ATM implied swaption vol - current level)

Membership (pseudo Gaussian kernel)

Posterior (Entropy Pooling mixture)

time

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: time decay + membership kernel

P&L scenarios

time

P&L distributionP&L distribution

P&L ex ante performance statisticsP&L ex-ante performance statistics


ESTIMATION RISK

SCENARIO ANALYSIS


ENTROPY POOLING




- Option trading



Black-Scholes formula: deterministic function of risk into price

Case study: option trading

deterministic function of risk into price

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: option trading

empirical smirk and smile


call option price at horizon





Portfolio: Microsoft 1 month Microsoft 2 months Microsoft 6 months

curve change (growth/inflation) not directly in pricing

Yahoo 1 month Yahoo 2 monthsYahoo 6 months Google 1 month G l 2 thGoogle 2 months Google 6 months




profit and loss is highly non-linear, highly non-normal?




?Mean-CVaR optimization ?

- long-short delta-neutral - no cash upfront - limit on leverage


Market distr.


P i i ?Pricing

Optimization

??




simulations




simulations

linear programming


Market distr.


P i i Pricing

Optimization linear programming


Market distr.


Focus

Views


view: G 6m G 2m


Market distr.


Focus

Views


P i i

…

Pricing

Optimization


view: G 6m G 2m


ESTIMATION RISK

SCENARIO ANALYSIS



CASE STUDIES


A. Meucci - Fully Flexible Views (Entropy Pooling) References


Entropy Pooling

Meucci (2008) http://symmys.com/node/158

References


Entropy Pooling


References

Parametric N P iParametric Entropy Pooling


Meucci (2008)NormalLinear-Quadratic



Entropy Pooling


References



Bayesian networks


Meucci (2008)NormalLinear-Quadratic

Meucci (2008) http://symmys.com/node/158 Kernels



Entropy Pooling

ReferencesMeucci (2008) http://symmys.com/node/158



Meucci, Ardia, Keel (2011) http://symmys.com/node/160

Bayesian networks


Meucci (2011)Normal (+dimension reduction)

Meucci (2008)

Fuzzy Membership State Conditioning

Crisp



Meucci (2008) http://symmys.com/node/158 Kernels

Crisp Conditioning

Time ConditioningMeucci (2010)

http://symmys.com/node/150

A. Meucci - Fully Flexible Views (Entropy Pooling) conclusions

Entropy Pooling:

Market represented by generic non-linear risk factors, not only returns

Market distribution fully general, not only normal

Views/stress-testing on any function of the market, not only linear portfolios

Views on any feature, not only on expectations: median, volatility, correlations, tails

Views are equalities and inequalities: ranking is possible

Applications span several areas of portfolio management and risk management

meucci fully flexible views - theory and practice.ppt · a. meucci - fully flexible views (entropy...

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