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
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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
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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
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Historical volatility - MERVAL
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Historical volatility - IGPA
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Historical Volatility Estimation
Textbook Method: annualized SD of
Better Method: subtract RN drift instead of realized drift
Textbook method slightly underestimates volatility
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Commonly available information: open, close, high, low
Captures valuable volatility information
Parkinson estimate:
Garman-Klass estimate:
Estimates based on High/Low
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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..
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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
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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)
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GARCH Model
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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.
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GARCH results
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Options Warm-up
%30][%70][
=
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BlackRed
PP
Black if$0Red if100$
Roulette:
A lottery ticket gives:
You can buy it or sell it for $60Is it cheap or expensive?
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Risk Neutral Expectation
Buy6070 >
Sell6050 converges quickly to same volatility for all strike/maturity; breaks auto-correlation and vol/spot dependency.
?=>
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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.
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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
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Strike dependency Fair or Break-Even volatility is an average of returns,
weighted by the Gammas, which depend on the strike
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Strike dependency for multiple paths
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Exchange Rate: ARS Curncy BEVL
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Exchange Rate: CLP Curncy BEVL
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MERVAL Index
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IPSA (General Index)
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A Brief History of Volatility (1) : Bachelier 1900
: Black-Scholes 1973
: Merton 1973
: Merton 1976
: Hull&White 1987
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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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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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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
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European prices
StochasticVol
Models
StochasticVol
Models
Exotic prices
From simple to complex
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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)
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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
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Understanding Information embedded in Option prices: 4-way play
Underlying Probability density
Implied Vol
payoff distribution
Payoff
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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.
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OVIP overview
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OVIP Future likely paths
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OVV Draw a subjective view
Volatility & Correlation Map
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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
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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 =
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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
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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
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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
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5
10
15
x 103
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Three representations
3. Volatility Map
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USD
AUD
CAD
CHFEURGBP
JPY
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SEKSGD
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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
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An Example
1. Three currencies: USD, CLP, ARS (Two pairs USD-CLP, USD-ARS)
2. Coordinates:
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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
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A focused Latin American View
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Mix of stock indices and currencies
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MERVAL view
Banking
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IGPA index view
Banking
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Commodities view - Agriculture
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Commodities view Metals, Oil ..
Robust Asset Allocation
and
Portfolio Optimization
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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
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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
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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!!
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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
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Concentration Risk
Concentration risk can be defined as deviationfrom a market/prior portfolio
Conc. risk = New optimization problem
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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
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Concentration Risk Balance Scale
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Portfolio Optimizer Main Screen
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Output
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Efficient Frontier
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Black-Litterman Model
Use implied returns Consistent with APT A full version allows inputting your own
view of absolute or relative returns
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Semi-variance Semi-variance is
defined as one-sided quadratic risk below a benchmark return.
Is a tractable left tail risk measure.
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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
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Multi-Scenario Math Problem
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Handling Default risk
Add a jump factor to each asset
In Multi-scenario framework add defaultscenarios
Tail risk important in current market scenarios
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Multi-scenario optimization
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Bad out of sample performance (single scenario markowitz)
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Good Out of sample performance (Concentration risk in single scenario)
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Best out of sample performance (Multi-Scenario Optimization)
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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.