Presentation on theme: "Practical model calibration"— Presentation transcript:
1 Practical model calibration Michael Boguslavsky,ABN-AMRO Global Equity DerivativesPresented at RISK workshop,New York, September 20-21, 2004
2 What is this talk about Pitfalls in fitting volatility surfaces Hints and tipsDisclaimer: 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 metric3: no-arbitrage and no-nonsense4: estimation techniques5: solving data quality issues6: using models for trading
4 1. Estimation vs fittingIn different situations one needs different models or smile representationsmarketmakingexotic pricingrisk managementbook markingTwo sources of data for the modelreal-world underlying past price seriescurrent derivatives prices and fundamental information
6 1.1 Smile modelling approaches (cont) Models:Bachelier & Black-ScholesDeterministic volatility and local volatilityStochastic VolatilityJump-diffusionsLevy processes and stochastic time changesUncertain volatility and Markov chain switching volatilityCombinations:Stochastic Volatility+JumpsLocal volatility+Stochastic volatility...
7 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 parameterisationsHeston with parameters depending smoothly on timeSABR in some forms
8 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” exerciseExample: a listed option marketmaker may be more interested in fit versatility than in consistency and exotic hedgesTypical parameterisations:splinestwo parabolas in strike or log-strikekernel smoothingetcMany fitting techniques are quite similar for models and parameterisations
9 1.1 Smile modelling approaches (cont) A nice parameterisation:cubic spline for each maturityfitting on tick datapenalties for deviations from last quotes (time decaying weights)penalties for approaching too close to bid and ask quotes, strong penalties for breaching thempenalty for curvature and for coefficient deviation from close maturities
10 1.1 Example: low curvature spline fit to bid/offer/last Hang Seng Index options7 nodeslinear extrapolationintraday fit
11 1.2 Models: estimation and fitting Will they give the same result?A tricky question, as one needsa lot of historic data to estimate reliablystationarity assumption to compare forward-looking data with past-lookingassumptions on risk preferences
12 1.2 Estimation vs fitting: example, Heston model with
13 1.2 Estimation vs fitting: example, Heston model (cont) However, very often estimated are very far from option impliedsome studies have shown that in practice skewness and kurtosis are much higher in option marketsBatesBakshi, Cao, ChenPossible causes:model misspecification e.g. extra risk factorspeso probleminsufficient data for estimation (>Javaheri)trade opportunity?
14 1.3: Similar problemsProblem 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
15 2: fitting: robustness and choice of metric 2.1 What do we fit to?There is no such thing as “market prices”We can observelast (actual trade prices)end-of-day markbidask
16 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 quotesnot synchronized in timeEnd-of day marksavailable once a dayindications, not real pricesBid/ask quotesmuch higher frequency than trade datasynchronized in timetradable immediatelyOften people use mid price or mid volatility quotesdiscarding extra information content of separate bid and ask quotes
17 2.2 Standard approachesGet somewhere “market” price for calls and puts (mid or cleaned last)Compose penalty functionleast squares fit in price (calls, puts, blend)least squares fit in volother point-wise metrics e.g. mean absolute error in price or volMinimize it using one’s favourite optimizer
19 2.2 Standard approaches (cont) Problem: why do we care about the least-squares?May be meaningful for interpolationuseless for extrapolationuseless for “global” or second order effectsalways creates unstable optimisation problem with multiple local minima
20 2.2 Standard approaches (cont) Some people suggest using global optimizers to solve the multiple local minima problemsimulated annealinggenetic algorithmsThey are slowAnd, actually, they do not solve the problem:
21 2.2 Standard approaches (cont) Suppose we have a perfect (and fast!) global optimizertrue local minima may change discontinuously with market prices!=> Large changes in process parameters on recalibration
22 2.3 Which metric to use?Ideally, we would want to have a low-dimensional linear optimization problemall process parameters are tradable/observable - not realisticIt is Ok if the problem is reasonably linearLuckily, in many markets we observe vanilla combination pricesFX: risk reversal and butterfly prices are availableequity: OTC quoted call and put spreads=>smile ATM skew and curvature are almost directly observable!
23 2.3 Which metric to use? (cont) Many models have reasonably linear dependence between process parameters and smile level/skew/curvature around the optimumActually, these are the models traders like most, because they think in terms of smile level/skew/curvature and can (kind of) trade themThus, one can e.g. minimize a weighted sum of vol level, skew, and curvature squared deviations from option/option combination quotes
24 Example: Heston model fit on level/skew/curvature DAX Index options,Heston modelglobal fit
25 2.4 Additional inputsSometimes, it is possible to use additional inputs in calibrationvariance 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
26 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 marketall models fit pretty wellcompare then barrier, one-touch, lookback, and cliquet option pricesreport huge discrepancies between prices
27 2.5 Fitting: a word of caution (cont) Examples:smile flattening in local volatility modelsLocal 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 probabilitythus, 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
28 2.5 Fitting: a word of caution (cont) Perfect fitting of the whole surface of Eurostoxx50 volatility with just 2-3 termsZero prices for variance butterflies that fall between volatility scenariosActually (almost) the same happens in Heston modelvolScenario 1, p=0.54Vol butterflyScenario 2, p=0.46T
29 3: no-arbitrage and no-nonsense Mostly important for parameterisations, not for modelsThis is one of the advantages of modelsHowever, some checks are useful, especially in the tails
30 3.1 No arbitrage: single maturity Fixed maturity European call prices:
31 3.1 No arbitrage: single maturity (cont) Breeden&Litzenberger’s formula:where f(X) is the risk-neutral PDF of underlying at time TOur three conditions are equivalent toNon-negative integral of CDFNon-negative CDFNon-negative PDF
32 3.1 No arbitrage: single maturity (cont) Are these conditions necessary and sufficient for a single maturity?Depends on which options we can tradeif we can trade calls with all strikes then alsoif we have options with strikes around 0
33 3.1 No arbitrage: single maturity (cont) ExampleNo dividends, zero interest rateC(80)=30, C(90)=21, C(100)=14is there an arbitrage here?
34 3.1 No arbitrage: single maturity (cont) All spreads are positive, butterfly is worth 30- 2*21+14=2>0...ButPayoff diagram:8090
35 3.2 No arbitrage: calendars Cross maturity no-arbitrage conditionsno dividends, zero interest ratelong call strike K, maturity T, short call strike K, maturity t<Tat time t,if S<K, then the short leg expires worthless, the long leg has non- negative valueotherwise, we are left with C(K,T)-S+K=P(K,T), again with non- negative value
36 3.2 No arbitrage: calendars Thus, with no dividends, zero interest rate,This is model independentWith dividends and non-zero interest rate, one has to adjust call strike for the carry on stock and cash positions
37 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
38 3.2 No arbitrage: calendars (cont) Consider a portfolio consisting oflong position in an option with strike K and maturity Tshort position in
39 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 valueotherwise, we are left withhas non-negative value
40 3.3 No nonsense Unimodal implied risk-neutral density can be interpolation-dependent if one is not careful!reasonable implied forward variance swap pricesagain, make sure to use good interpolation
41 3.3 No nonsense (cont) Model-specific constraints Example: Heston+Merton modelcorrelation should be negative (equities)mean reversion level should be not too far from the volatility of longest dated option at handvolatility goes to infinity for strike iff CDS price is positive, otherwise volatility can go 0 and stay around it (not a feasible constraint)
42 4: estimation techniques Most advances are for affine jump-diffusion modelsFirst one: Gaussian QMLE (Ruiz; Harvey, Ruiz, Shephard)does not work very well because of highly non-Gaussian dataGeneralized, Simulated, Efficient Methods of MomentsDuffie, Pan, and Singleton; Chernov, Gallant, Ghysels, and Tauchen (optimal choice of moment conditions), …FilteringHarvey (Kalman filter), Javaheri (Extended KF, Unscented KF), ...Bayesian (Markov chain Monte Carlo) (Kim, Shephard, and Chib), ...
43 4: Estimation techniques (cont) Can not estimate the model form underlying data alone without additional assumptionsEconometric criteria vs financial criteria: in-sample likelihood vs out-of- sample price predictionDifferent studies lead to different conclusions on volatility risk premia, stationarity of volatility, etcMuch to be done here
44 5: solving data quality issues Data aresparse,non-synchronised,noisy,limited in rangeNot everything is observeddividends and borrowing rates need to be estimatedNot all prices reported are propersome exchanges report combinations traded as separate trades
45 5: solving data quality issues (cont) If one has concurrent put and call prices, one can back-out implied forwardUsing high-frequency data when possibleUsing bid and ask quotes instead of trade prices (usually there are about times more bid/ask quote revisions than trades)
46 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 balancedOr we can put a lot of trust into our model and take a view based on it
47 6: using models for trading (cont) Example: realized skewness and kurtosis tradesCan be done parametrically, via calibration/estimation of a stochastic volatility model, or non-parametricallySkew:set up a risk-reversallong callshort putvega-neutral
48 6: using models for trading (cont) Kurtosis trade:long an ATM butterflyshort the wingsActually a vega-hedged short variance swap or some path- dependent exotics would do better
49 6: using models for trading (cont) Problems:what is a vega hedge - model dependentskew trade: huge dividend exposure on the forwardkurtosis trade: executionpeso problemwhen to open/close position?
50 6: using models for trading A simple example: historical vs implied distribution momentsBlaskowitz, Hardle, Schmidt:Compare option-implied distribution parameters with realizedDAX indexAssuming local volatility model
51 Historical vs implied distribution: StDev Image reproduced with authors’ permission from Blaskowitz, Hardle, Schmidt
52 Historical vs implied distribution: skewness Image reproduced with authors’ permission from Blaskowitz, Hardle, Schmidt
53 Historical vs implied distribution: kurtosis Image reproduced with authors’ permission from Blaskowitz, Hardle, Schmidt
54 ReferencesAt-Sahalia Yacine, Wang Y., Yared F. (2001) “Do Option Markets Correctly Price the Probabilities of Movement of the UnderlyingAsset?” Journal of Econometrics, 101Alizadeh Sassan, Brandt M.W., Diebold F.X. (2002) “Range- Based Estimation of Stochastic Volatility Models” Journal of Finance, Vol. 57, No. 3Avellaneda 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 5Bates David S. (2000) “Post-87 Crash Fears in the S&P500 Futures Option Market” Journal of Econometrics, 94Blaskowitz 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).
55 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 , 2003Coleman, 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 2Jacquier Eric, Polson N.G., Rossi P.E. (1994) “Bayesian Analysisof Stochastic Volatility Models” Journal of Business and Economic Statistics, Vol. 12, No, 4Javaheri Alireza, Lautier D., Galli A. (2003) “Filtering in Finance” WILMOTT, Issue 5
56 References (cont)Kim Sangjoon, Shephard N., Chib S. (1998) “Stochastic Volatility: Likelihood Inference and Comparison with ARCHModels” Review of Economic Studies, Volume 65Rookley, 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.