1. Practical model calibration
Michael Boguslavsky,
ABN-AMRO Global Equity Derivatives
Presented at RISK workshop,
New York, September 20-21, 2004

2. 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

4. 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

5. 1.1 Smile modelling approaches
models
parameterisations
“single maturity models”
fitting
estimation
Join estimation
& fitting

6. 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
...

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 parameterisations
Heston with parameters depending smoothly on time
SABR 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” 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

9. 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

10. 1.1 Example: low curvature spline fit to bid/offer/last
Hang Seng Index options
7 nodes
linear extrapolation
intraday fit

11. 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

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 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?

14. 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

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 observe
last (actual trade prices)
end-of-day mark
bid
ask

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 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

17. 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

18. 2.2 Standard approaches (cont)
Formally:

19. 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

20. 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:

21. 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

22. 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!

23. 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

24. Example: Heston model fit on level/skew/curvature
DAX Index options,
Heston model
global fit

25. 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

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 market
all models fit pretty well
compare then barrier, one-touch, lookback, and cliquet option prices
report huge discrepancies between prices

27. 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

28. 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

29. 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

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 T
Our three conditions are equivalent to
Non-negative integral of CDF
Non-negative CDF
Non-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 trade
if we can trade calls with all strikes then also
if we have options with strikes around 0

33. 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?

34. 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

35. 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

36. 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

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 of
long position in an option with strike K and maturity T
short 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 value
otherwise, we are left with
has 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 prices
again, make sure to use good interpolation

41. 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)

42. 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), ...

43. 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

44. 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

45. 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)

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 balanced
Or 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 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

48. 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

49. 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?

50. 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

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. 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).

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 , 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

56. 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.