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**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

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**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**

models parameterisations “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 Volatility+Jumps Local volatility+Stochastic 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 (>Javaheri) 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 Breeden&Litzenberger’s 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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**discarding extra information content of separate bid and ask quotes**

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 one’s 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! => 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 =>smile 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 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, p=0.54 Vol butterfly Scenario 2, p=0.46 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)**

Breeden&Litzenberger’s 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, butterfly is worth 30- 2*21+14=2>0... But Payoff diagram: 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 t<T at time t, if S<K, then the short leg expires worthless, the long leg has non- negative value otherwise, we are left with C(K,T)-S+K=P(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 S<K, 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: Heston+Merton 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 times more bid/ask quote revisions than trades)

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**6:using 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 S&P500 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), , Duffie, Darrell, Pan J., Singleton, K., ``Transform Analysis and Asset Pricing for Affine Jump-Diffusions,’’ Econometrica 68, (2000), 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.

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