Practical%20model%20calibration

1
Practical model calibration

• Michael Boguslavsky,
• ABN-AMRO Global Equity Derivatives
• Presented at RISK workshop,
• New York, September 20-21, 2004

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What is this talk about

• Pitfalls in fitting volatility surfaces
• Hints and tips
• Disclaimer The models and trading opinions
  presented do not represent models and trading
  opinions of ABN-AMRO

3
Overview

• 1 estimation vs fitting
• 2 fitting robustness and choice of metric
• 3 no-arbitrage and no-nonsense
• 4 estimation techniques
• 5 solving data quality issues
• 6 using models for trading

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1. Estimation vs fitting

• In different situations one needs different
  models or smile representations
• marketmaking
• exotic pricing
• risk management
• book marking
• Two sources of data for the model
• real-world underlying past price series
• current derivatives prices and fundamental
  information

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1.1 Smile modelling approaches
parameterisations
models
single maturity models
fitting
estimation
Join estimation fitting
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1.1 Smile modelling approaches (cont)

• Models
• Bachelier Black-Scholes
• Deterministic volatility and local volatility
• Stochastic Volatility
• Jump-diffusions
• Levy processes and stochastic time changes
• Uncertain volatility and Markov chain switching
  volatility
• Combinations
• Stochastic VolatilityJumps
• Local volatilityStochastic volatility
• ...

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1.1 Smile modelling approaches (cont)

• Some models do not fit the market very well, less
  parsimonious ones fit better (does not mean they
  are better!)
• Multifactor models can not be estimated from
  underlying asset series alone (one needs either
  to assume something about the preference
  structure or to use option prices)
• Some houses are using different model parameters
  for different maturities - a hybrid between
  models and smile parameterisations
• Heston with parameters depending smoothly on
  time
• SABR in some forms

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1.1 Smile modelling approaches (cont)

• Parameterisations
• In some cases, one is not interested in the model
  for stock price movement, but just in a joining
  the dots exercise
• Example a listed option marketmaker may be more
  interested in fit versatility than in consistency
  and exotic hedges
• Typical parameterisations
• splines
• two parabolas in strike or log-strike
• kernel smoothing
• etc
• Many fitting techniques are quite similar for
  models and parameterisations

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1.1 Smile modelling approaches (cont)

• A nice parameterisation
• cubic spline for each maturity
• fitting on tick data
• penalties for deviations from last quotes (time
  decaying weights)
• penalties for approaching too close to bid and
  ask quotes, strong penalties for breaching them
• penalty for curvature and for coefficient
  deviation from close maturities

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1.1 Example low curvature spline fit to
bid/offer/last

• Hang Seng Index options
• 7 nodes
• linear extrapolation
• intraday fit

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1.2 Models estimation and fitting

• Will they give the same result?
• A tricky question, as one needs
• a lot of historic data to estimate reliably
• stationarity assumption to compare
  forward-looking data with past-looking
• assumptions on risk preferences

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1.2 Estimation vs fitting example, Heston model

• with

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1.2 Estimation vs fitting example, Heston model
(cont)

• However, very often estimated are
  very far from option implied
• some studies have shown that in practice skewness
  and kurtosis are much higher in option markets
• Bates
• Bakshi, Cao, Chen
• Possible causes
• model misspecification e.g. extra risk factors
• peso problem
• insufficient data for estimation (gtJavaheri)
• trade opportunity?

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1.3 Similar problems

• Problem of fitting/estimating smile is similar to
  fitting/estimating (implied) risk-neutral
  density (via BreedenLitzenbergers formula)
• But smile is two integrations more robust

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2 fitting robustness and choice of metric

• 2.1 What do we fit to?
• There is no such thing as market prices
• We can observe
• last (actual trade prices)
• end-of-day mark
• bid
• ask

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2.1 What do we fit to?

• Last prices
• much more sparse than bid/ask quotes
• not synchronized in time
• End-of day marks
• available once a day
• indications, not real prices
• Bid/ask quotes
• much higher frequency than trade data
• synchronized in time
• tradable immediately
• Often people use mid price or mid volatility
  quotes
• discarding extra information content of separate
  bid and ask quotes

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2.2 Standard approaches

• Get somewhere market price for calls and puts
  (mid or cleaned last)
• Compose penalty function
• least squares fit in price (calls, puts, blend)
• least squares fit in vol
• other point-wise metrics e.g. mean absolute error
  in price or vol
• Minimize it using ones favourite optimizer

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2.2 Standard approaches (cont)

• Formally

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2.2 Standard approaches (cont)

• Problem why do we care about the least-squares?
• May be meaningful for interpolation
• useless for extrapolation
• useless for global or second order effects
• always creates unstable optimisation problem with
  multiple local minima

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2.2 Standard approaches (cont)

• Some people suggest using global optimizers to
  solve the multiple local minima problem
• simulated annealing
• genetic algorithms
• They are slow
• And, actually, they do not solve the problem

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2.2 Standard approaches (cont)

• Suppose we have a perfect (and fast!) global
  optimizer
• true local minima may change discontinuously with
  market prices!
• gt Large changes in process parameters on
  recalibration

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2.3 Which metric to use?

• Ideally, we would want to have a low-dimensional
  linear optimization problem
• all process parameters are tradable/observable -
  not realistic
• It is Ok if the problem is reasonably linear
• Luckily, in many markets we observe vanilla
  combination prices
• FX risk reversal and butterfly prices are
  available
• equity OTC quoted call and put spreads
• gtsmile ATM skew and curvature are almost
  directly observable!

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2.3 Which metric to use? (cont)

• Many models have reasonably linear dependence
  between process parameters and smile
  level/skew/curvature around the optimum
• Actually, these are the models traders like most,
  because they think in terms of smile
  level/skew/curvature and can (kind of) trade them
• Thus, one can e.g. minimize a weighted sum of vol
  level, skew, and curvature squared deviations
  from option/option combination quotes

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Example Heston model fit on level/skew/curvature

• DAX Index options,
• Heston model
• global fit

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2.4 Additional inputs

• Sometimes, it is possible to use additional
  inputs in calibration
• variance swap price dictates the downside skew
  (warning dependent on the cut-off level!)
• Equity Default Swap price far downside skew
  (warning very model-dependent!)
• view on skew dynamics from cliquet prices

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2.5 Fitting a word of caution

• Even if your model perfectly fits vanilla option
  prices, it does not mean that it will give
  reasonable prices for exotics!
• Schoutens, Simons, Tistaert
• fit Heston, Heston with exponential jump process,
  variance Gamma, CGMY, and several other
  stochastic volatility models to Eurostoxx50
  option market
• all models fit pretty well
• compare then barrier, one-touch, lookback, and
  cliquet option prices
• report huge discrepancies between prices

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2.5 Fitting a word of caution (cont)

• Examples
• smile flattening in local volatility models
• Local Volatility Mixture of Densities/Uncertain
  volatility model of Brigo, Mercurio, Rapisarda
  (Risk, May 2004)
• at time 0 volatility starts following of of
  the few prescribed trajectories
  with probability
• thus, the marginal density of S at time t is a
  linear combination of marginal densities of
  several different local volatility models
  (actually, the authors use
  ), so the density is a mixture of lognormals

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2.5 Fitting a word of caution (cont)

• Perfect fitting of the whole surface of
  Eurostoxx50 volatility with just 2-3 terms
• Zero prices for variance butterflies that fall
  between volatility scenarios
• Actually (almost) the same happens in Heston model

vol
Scenario 1, p0.54
Vol butterfly
Scenario 2, p0.46
T
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3 no-arbitrage and no-nonsense

• Mostly important for parameterisations, not for
  models
• This is one of the advantages of models
• However, some checks are useful, especially in
  the tails

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3.1 No arbitrage single maturity

• Fixed maturity European call prices

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3.1 No arbitrage single maturity (cont)

• BreedenLitzenbergers formula
• where f(X) is the risk-neutral PDF of underlying
  at time T
• Our three conditions are equivalent to
• Non-negative integral of CDF
• Non-negative CDF
• Non-negative PDF

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3.1 No arbitrage single maturity (cont)

• Are these conditions necessary and sufficient for
  a single maturity?
• Depends on which options we can trade
• if we can trade calls with all strikes then also
• if we have options with strikes around 0

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3.1 No arbitrage single maturity (cont)

• Example
• No dividends, zero interest rate
• C(80)30, C(90)21, C(100)14
• is there an arbitrage here?

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3.1 No arbitrage single maturity (cont)

• All spreads are positive, 80-90-100 butterfly is
  worth 30-221142gt0...
• But
• Payoff diagram

0
80
90
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3.2 No arbitrage calendars

• Cross maturity no-arbitrage conditions
• no dividends, zero interest rate
• long call strike K, maturity T, short call strike
  K, maturity tltT
• at time t,
• if SltK, then the short leg expires worthless,
  the long leg has non-negative value
• otherwise, we are left with C(K,T)-SKP(K,T),
  again with non-negative value

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3.2 No arbitrage calendars

• Thus, with no dividends, zero interest rate,
• This is model independent
• With dividends and non-zero interest rate, one
  has to adjust call strike for the carry on stock
  and cash positions

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3.2 No arbitrage calendars

• The easiest way to get calendar no-arbitrage
  conditions is via a local vol model (Reiner) (the
  condition will be model-independent)
• possibly with discrete components
• (only time integrals
  of y(t) will matter)
• Local volatility model

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3.2 No arbitrage calendars (cont)

• Consider a portfolio consisting of
• long position in an option with strike K and
  maturity T
• short position in

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3.2 No arbitrage calendars (cont)

• As before, at time t,
• if SltK, then the short leg expires worthless, the
  long leg has non-negative value
• otherwise, we are left with
• has non-negative value

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3.3 No nonsense

• Unimodal implied risk-neutral density
• can be interpolation-dependent if one is not
  careful!
• reasonable implied forward variance swap prices
• again, make sure to use good interpolation

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3.3 No nonsense (cont)

• Model-specific constraints
• Example HestonMerton model
• correlation should be negative (equities)
• mean reversion level should be not too far
  from the volatility of longest dated option at
  hand
• volatility goes to infinity for strike
  iff CDS price is positive
• , otherwise volatility can
  go 0 and stay around it (not a feasible
  constraint)

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4 estimation techniques

• Most advances are for affine jump-diffusion
  models
• First one Gaussian QMLE (Ruiz Harvey, Ruiz,
  Shephard)
• does not work very well because of highly
  non-Gaussian data
• Generalized, Simulated, Efficient Methods of
  Moments
• Duffie, Pan, and Singleton Chernov, Gallant,
  Ghysels, and Tauchen (optimal choice of moment
  conditions),
• Filtering
• Harvey (Kalman filter), Javaheri (Extended KF,
  Unscented KF), ...
• Bayesian (Markov chain Monte Carlo) (Kim,
  Shephard, and Chib), ...

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4 Estimation techniques (cont)

• Can not estimate the model form underlying data
  alone without additional assumptions
• Econometric criteria vs financial criteria
  in-sample likelihood vs out-of-sample price
  prediction
• Different studies lead to different conclusions
  on volatility risk premia, stationarity of
  volatility, etc
• Much to be done here

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5 solving data quality issues

• Data are
• sparse,
• non-synchronised,
• noisy,
• limited in range
• Not everything is observed
• dividends and borrowing rates need to be
  estimated
• Not all prices reported are proper
• some exchanges report combinations traded as
  separate trades

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5 solving data quality issues (cont)

• If one has concurrent put and call prices, one
  can back-out implied forward
• Using high-frequency data when possible
• Using bid and ask quotes instead of trade prices
  (usually there are about 10-50 times more bid/ask
  quote revisions than trades)

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6using models for trading

• What to do once the model is fit?
• We can either make the market around our model
  price and hope our position will be reasonably
  balanced
• Or we can put a lot of trust into our model and
  take a view based on it

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6 using models for trading (cont)

• Example realized skewness and kurtosis trades
• Can be done parametrically, via
  calibration/estimation of a stochastic volatility
  model, or non-parametrically
• Skew
• set up a risk-reversal
• long call
• short put
• vega-neutral

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6 using models for trading (cont)

• Kurtosis trade
• long an ATM butterfly
• short the wings
• Actually a vega-hedged short variance swap or
  some path-dependent exotics would do better

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6 using models for trading (cont)

• Problems
• what is a vega hedge - model dependent
• skew trade huge dividend exposure on the forward
• kurtosis trade execution
• peso problem
• when to open/close position?

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6 using models for trading

• A simple example historical vs implied
  distribution moments
• Blaskowitz, Hardle, Schmidt
• Compare option-implied distribution parameters
  with realized
• DAX index
• Assuming local volatility model

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Historical vs implied distribution StDev
Image reproduced with authors permission from
Blaskowitz, Hardle, Schmidt
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Historical vs implied distribution skewness
Image reproduced with authors permission from
Blaskowitz, Hardle, Schmidt
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Historical vs implied distribution kurtosis
Image reproduced with authors permission from
Blaskowitz, Hardle, Schmidt
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References

• At-Sahalia Yacine, Wang Y., Yared F. (2001) Do
  Option Markets Correctly Price the Probabilities
  of Movement of the UnderlyingAsset? Journal of
  Econometrics, 101
• Alizadeh Sassan, Brandt M.W., Diebold F.X. (2002)
  Range- Based Estimation of Stochastic Volatility
  Models Journal of Finance, Vol. 57, No. 3
• Avellaneda Marco, Friedman, C., Holmes, R., and
  Sampieri, D., Calibrating Volatility Surfaces
  via Relative-Entropy Minimization, in Collected
  Papers of the New York University Mathematical
  Finance Seminar, (1999)
• Bakshi Gurdip, Cao C., Chen Z. (1997) Empirical
  Performance of Alternative Option Pricing Models
  Journal of Finance,Vol. 52, Issue 5
• Bates David S. (2000) Post-87 Crash Fears in the
  SP500 Futures Option Market Journal of
  Econometrics, 94
• Blaskowitz Oliver J., Härdle W., Schmidt P
  Skewness and Kurtosis Trades, Humboldt
  University preprint, 2004.
• Bondarenko, Oleg, Recovering risk-neutral
  densities a new nonparametric approach, UIC
  preprint, (2000).

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References (cont)

• Brigo, Damiano, Mercurio, F., Rapisarda, F.,
  Smile at Uncertainty, Risk, (2004), May
  issue.
• Chernov, Mikhail, Gallant A.R., Ghysels, E.,
  Tauchen, G "Alternative Models for Stock Price
  Dynamics," Journal of Econometrics , 2003
• Coleman, T. F., Li, Y., and Verma, A
  Reconstructing the unknown local volatility
  function, The Journal of Computational Finance,
  Vol. 2, Number 3, (1999), 77-102,
• Duffie, Darrell, Pan J., Singleton, K.,
  Transform Analysis and Asset Pricing for Affine
  Jump-Diffusions, Econometrica 68, (2000),
  1343-1376.
• Harvey Andrew C., Ruiz E., Shephard Neil (1994)
  Multivariate Stochastic Variance Models Review
  of Economic Studies,Volume 61, Issue 2
• Jacquier Eric, Polson N.G., Rossi P.E. (1994)
  Bayesian Analysis
• of Stochastic Volatility Models Journal of
  Business and Economic Statistics, Vol. 12, No, 4
• Javaheri Alireza, Lautier D., Galli A. (2003)
  Filtering in Finance WILMOTT, Issue 5

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References (cont)

• Kim Sangjoon, Shephard N., Chib S. (1998)
  Stochastic Volatility Likelihood Inference and
  Comparison with ARCH
• Models Review of Economic Studies, Volume 65
• Rookley, C., Fully exploiting the information
  content of intra day option quotes applications
  in option pricing and risk management,
  University of Arizona working paper, November
  1997.
• Riedel, K., Piecewise Convex Function
  Estimation Pilot Estimators, in Collected
  Papers of the New York University Mathematical
  Finance Seminar, (1999)
• Schonbucher, P., A market model for stochastic
  implied volatility, University of Bonn
  discussion paper, June 1998.