ars clp volatility portfolio opt

Volatility Estimation&

Portfolio Optimization

Dr Arun VermaQuantitative ResearchBloomberg, New York

09/10/2008 2Buenos Aires

Agenda

Volatility Estimation Historical Volatility Implied Volatility Stochastic Volatility Models

Portfolio Optimization Asset Allocation Correlation Map Tail risk measures Diversification


Straight from the LAB BEVL Break Even Volatility GRCH GARCH(1,1) volatility CORM Correlation Map CDFX CDS & FX options joint model FFIP Fed Fund Implied probability WIRP World Interest Rate probability OVV (upcoming) Option valuation with a

view OVIP (upcoming) Implied probability using

options CPIP (upcoming) Energy and Commodity

implied probability


Volatility : some definitions

Historical volatilityHistorical volatilityHistorical volatilityHistorical volatility :standard deviation of the returns; measure of

uncertainty/activity

Implied volatilityImplied volatilityImplied volatilityImplied volatility : measure of the option price given by the market. Expected

Future Volatility


Historical volatility - MERVAL


Historical volatility - IGPA


Historical Volatility Estimation

Textbook Method: annualized SD of

Better Method: subtract RN drift instead of realized drift

Textbook method slightly underestimates volatility

1

ln

i

i

it

tt S

Sx

2

1)(

1252

=

=

n

it xx

n i


Commonly available information: open, close, high, low

Captures valuable volatility information

Parkinson estimate:

Garman-Klass estimate:

Estimates based on High/Low

( )=

=

n

tttP du

n 1

22

2ln41

( ) ==

=

n

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22 39.05.0

Sln

open

upper

SS

u ln=

open

close

SS

c ln=

open

down

SSd ln=


Move based estimation

Leads to alternative historical vol estimation:

= number of crossings of log-price over [0,T]),( TL T

TLh

),(~

MoreRogers & Satchell , Yang-Zhang indicators..


Bloomberg Volatility estimators

HIVG Historical Implied volatility HVG Historical Vol with 5 estimation methods VCMP Volatility Comparison SKEW Implied Volatility Surface GRCH GARCH(1,1) model volatility BEVL Break Even Volatility


GRCH Standard GARCH(1,1) model Equivalent to a discrete term stochastic

volatility model Learns long term behavior of volatility

which standard historical volatility calculations ignore

Compare results with historical volatility Variance swap term structureCredit to Prof. Engle Nobel Prize (2003,

Economics)


GARCH Model

t

dWdtd

r

ttt

nnn

=

+=

++=+

)1( where

)(process OU lognormal a is equivalent timeContinuous

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2221


GARCH estimation

Maximum likelihood optimization

Find best value of parameters so that the discrete walk equation has the max likelihood as averaged over all discrete periods

We use Matlab optimization toolbox.


GARCH results


Options Warm-up

%30][%70][

=

=

BlackRed

PP

Black if$0Red if100$

Roulette:

A lottery ticket gives:

You can buy it or sell it for $60Is it cheap or expensive?


Risk Neutral Expectation

Buy6070 >

Sell6050 converges quickly to same volatility for all strike/maturity; breaks auto-correlation and vol/spot dependency.

?=>


Theoretical Skew from Prices (2)3) Discounted average of the Intrinsic Value from re-centered 3 month

histogram.4) -Hedging : compute the implied volatility which makes the -

hedging a fair game.


Theoretical Skewfrom historical prices (3)

How to get a theoretical Skew just from spot price history?Example: 3 month daily data1 strike a) price and delta hedge for a given within Black-Scholes

model b) compute the associated final Profit & Loss: c) solve for d) repeat a) b) c) for general time period and average e) repeat a) b) c) and d) to get the theoretical Skew

1TSkK =

( )PL( ) ( )( ) 0/ =kPLk

t

S

1T 2T

K1T

S


Strike dependency Fair or Break-Even volatility is an average of returns,

weighted by the Gammas, which depend on the strike


Strike dependency for multiple paths


Exchange Rate: ARS Curncy BEVL


Exchange Rate: CLP Curncy BEVL


MERVAL Index


IPSA (General Index)


A Brief History of Volatility (1) : Bachelier 1900

: Black-Scholes 1973

: Merton 1973

: Merton 1976

: Hull&White 1987

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t

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+=

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)(

2

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t

t dWdtrS

dS +=

dqdWdtkrSdS Q

tt

t ++= )(


A Brief History of Volatility (2)Dupire 1992, arbitrage modelwhich fits term structure of volatility given by log contracts.

Dupire 1993, minimal model to fit current volatility surface

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Tt

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t

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),( )(

KC

K

KC

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TK

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+=


A Brief History of Volatility (3)Heston 1993, semi-analytical formulae.

Dupire 1996 (UTV), Derman 1997, stochastic volatility model which fits current volatility surface HJM treatment.

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tolconditiona

varianceforward ousinstantane :


A Brief History of Volatility (4) Bates 1996, Heston + Jumps:

Local volatility + stochastic volatility: SABR: f is a power function

More..Levy Processes, Stochastic Clock..

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Volatility Model Requirements

Has to fit static/current data: Spot Price Interest Rate Structure Implied Volatility Surface

Should fit dynamics of: Spot Price (Realistic Dynamics) Volatility surface when prices move Interest Rates (possibly)

Has to be Understandable In line with the actual hedge Easy to implement


European prices

StochasticVol

Models

StochasticVol

Models

Exotic prices

From simple to complex


Heston Model

- Calibrate Heston model to all available strikes and maturities (relatively robust to missing strikes)

- Interpolate differences between the best-fit Heston Implied Vols (squared) and Market Implied vols (squared).

- Flat extrapolation on errors (asymptotes are thus Shifted-Heston curves)

dtdWdBdBvdtvdv

dWvdtqrS

dS

tt

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t

=

+=

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)(

)(


Role of parameters Correlation gives the short term skew Mean reversion level determines the long term

value of volatility Mean reversion strength

Determine the term structure of volatility Dampens the skew for longer maturities

Volvol gives convexity to implied vol Functional dependency on S has a similar effect

to correlation


Understanding Information embedded in Option prices: 4-way play

Underlying Probability density

Implied Vol

payoff distribution

Payoff


Upcoming functions OVV & OVIP

OVIP Implied probability from Options Prices Show Implied probabilities

Compare Historical densities to Implied densities

Illustrate Future implied paths OVV Option valuation with a view

Draw a density of possible underlying values at a given horizon

Draw your own payoff or choose from selected European or structured profiles

On the fly scenario analysis and risk analysis

Use your subjective view to find an optimal portfolio for you.


OVIP overview


OVIP Future likely paths


OVV Draw a subjective view

Volatility & Correlation Map


Correlation Map

Correlation is a key data in risk management. Alternative way of representing covariance and

moreover effective for computation : the Vol Map.Basic idea : construct a visual map such that: where A, B are two assets and

is the volatility of A/B.

where is the correlation of A returns.

ABBA 2 AB

),cos( ACAB


Correlation Map

It is possible since if we consider a vectorized Black-Scholes model.

Two similar assets will be placed close together on the map (useful for hedging, proxy for substitutions)

Clusters of assets can be identified (risk aggregation & management, hedging)

Flat Triangles : could indicate potential arbitrage

BAAB =


Geometry of random variables Representation in risk space n random variables represented as n points in dimension

n from the covariance matrix (of the log returns) Standard deviation: distance Covariance: scalar product Correlation: cosine of the angle

X1

X2


Visualization: Three representations

1. Raw returns (relative to USD, period 1/1/2000-1/1/2005)

0 200 400 600 800 1000 1200 14000.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

days

Representation I : small dataset. Normalized currency pairs timeseries

AUDEURGBPJPYMXP


Three representations

2. Covariance matrix

=

0.0075 0.0008- 0.0009- 0.0017- 0.0003 0.0008- 0.0111 0.0018 0.0027 0.0016 0.0009- 0.0018 0.0057 0.0049 0.0022 0.0017- 0.0027 0.0049 0.0100 0.0037 0.0003 0.0016 0.0022 0.0037 0.0106

C

USDAUD

EURGBP

JPY

USDAUD

EURGBP

JPY

5

0

5

10

15

x 103


Three representations

3. Volatility Map

0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08

0.08

0.06

0.04

0.02

0

0.02

0.04

0.06

0.08

USD

AUD

CAD

CHFEURGBP

JPY

KRWMXP

NZD

SEKSGD


Correlation Map Computation Given N assets, choose one as a base, and build the NxN

covariance matrix C of the currencies expressed in this base Find X (for instance by Cholesky decomposition) s.t. C = X.XT X's rows (X)i are our mapping of currencies since :

Defines a unique simplex Shift the origin to the barycenter of the simplex Using implied vol data potentially builds a different map. the

implied map may not exist (Cimp


An Example

1. Three currencies: USD, CLP, ARS (Two pairs USD-CLP, USD-ARS)

2. Coordinates:

=

=

xx

xyyy

xx

yxxx

xx

xyyy

xx

yx

xx

yyyx

xyxx

VV

V

VV

V

VV

VVV

V

VVVV

C2

2

0

0

USD EUR

JPY( )

, )

( 31

,

0 , )0,0(

2

2

+=

+

=

+=

+=

xx

xyyyxx

xx

yx

xx

xyyy

xx

yx

xx

VV

VVVV

VV

VVV

BRL

VCOP

USD

CLP

ARSUSD


Dimension Reduction To represent the map, project it on a 2-3D space

Projection must minimize the loss of "information Linear projection

PCA algorithm : keep the 2 or 3 largest eigenvalues Non linear projection (more optimal)

minimize with respect to the (dij) where d/o are the new/original distances


A focused Latin American View


Mix of stock indices and currencies


MERVAL view

Banking


IGPA index view

Banking


Commodities view - Agriculture


Commodities view Metals, Oil ..

Robust Asset Allocation

and

Portfolio Optimization


BPORT PREP Portfolio Reporting RVP Equity Relative Value EQS Equity Screening PRT Equity Real-Time Analysis NPH Monitor Portfolio News ALRT Portfolio Alerts OSA Monitor Equity Options BLP Monitor Portfolios in Launchpad BERR Portfolios on Blackberry RSE Analyst Research VAR Value-at-Risk

TRK Tracking Error WRSTStress Tests KRR Key Rate Risk LRSK Liquidity Risk BBAT Equity Return Attribution HFA Historical Fund Analysis PFST Excel Drag & Drop BBU Automatic Portfolio

Uploader PRTU Create/Edit Portfolios PLST List all Portfolios PDIS Share PortfoliosCOMING UP : 1) ACA Asset Allocation2) Equity Factor Models


Agenda A problem at the core - Estimation risk Risk measure choice Variance/semi-

variance/ CVaR? Multi-scenario Robust optimization Default risk Concentration risk Black-Litterman and beyondMarkowitz invented Mean-Variance Portfolio

Optimization in 1950s, Nobel Prize in Economics in 1990


Review pitfalls of Markowitz approach

Corner Solutions Estimation risk Out-of-sample performance is usually bad

(Over-optimized!) Not consistent with APT/CAPM Market

portfolio is not efficient!!


Improving Markowitz

Approaches that work (in practice) Assign higher variance to non-principal

factors Adding Concentration/Liquidity risk

measures

Use Black-Litterman Multi-scenario robust optimization Tail risk measures


Concentration Risk

Concentration risk can be defined as deviationfrom a market/prior portfolio

Conc. risk = New optimization problem

)()( wwww T

11 :

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=

+

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T

TT

w

wtosubjectwwCdiagwwCwwMin


Data Test Bed

23 Asset classes (from Fixed income, Equities, Commodities, International Equities and Alternative investments)

Historical periods 1997-2007

GARCH volatilities used in place of historical volatilities


Concentration Risk Balance Scale


Portfolio Optimizer Main Screen


Output


Efficient Frontier


Black-Litterman Model

Use implied returns Consistent with APT A full version allows inputting your own

view of absolute or relative returns


Semi-variance Semi-variance is

defined as one-sided quadratic risk below a benchmark return.

Is a tractable left tail risk measure.

var

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][]})[{(var

])[(2

2

SrEioSortinoRat

VarrE

oSharpeRati

TrESTrEVar

=

=

=

=


Multi-Scenario Optimization Why put all faith in one-scenario (standard

Markowitz)? Use historical periods as stress test

scenario Black-Litterman view can be just another

stress test scenario Re-sample and generate additional

scenarios Robust Optimization framework


Multi-Scenario Math Problem

e.performanc sample-of-outgreat a Has

sense.robust in theonly but period singleany in efficient benot may Portfolio Optimal

11

,..,1,:

:Risk Case- WorstMinimize

=

==T

Tii

Ti

Ti

w

NiwtosubjectwCwMaxMin


Handling Default risk

Add a jump factor to each asset

In Multi-scenario framework add defaultscenarios

Tail risk important in current market scenarios


Multi-scenario optimization


Bad out of sample performance (single scenario markowitz)


Good Out of sample performance (Concentration risk in single scenario)


Best out of sample performance (Multi-Scenario Optimization)


Conclusions

Out of sample performance is a key measure

Scenario optimization is important for a robust framework

Portfolio and risk managers view should be incorporated into the asset allocation process.

ars clp volatility portfolio opt

Documents

volatility grch garch1

buenos airesstraight

buenos airesvolatility

buenos airesmove

model volatility bevl

buenos airestheoretical

buenos airesoptions

model equivalent