Transcript Practical model calibration

Practical model calibration
Michael Boguslavsky,
ABN-AMRO Global Equity Derivatives
Presented at RISK workshop,
New York, September 20-21, 2004
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
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
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
1.1 Smile modelling approaches
models
parameterisations
“single maturity models”
fitting
estimation
Join estimation
& fitting
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
–...
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
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
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
1.1 Example: low curvature spline fit to bid/offer/last

Hang Seng Index options

7 nodes

linear extrapolation

intraday fit
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
1.2 Estimation vs fitting: example, Heston model
dSt  t St dt  vt St dBt
with
dvt   (  vt )dt   vt dZt
 db, dZ  
 and  are thesame under physicaland risk neutralmeasures(Girsanov)
1.2 Estimation vs fitting: example, Heston model
(cont)

However, very often estimated  and  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?
1.3: Similar problems
 Problem of fitting/estimating smile is similar to
fitting/estimating (implied) risk-neutral density (via
Breeden&Litzenberger’s formula)
 2C (T , K )
 exp(rT ) f ( X ),
2
X
 But smile is two integrations more robust
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
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
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
2.2 Standard approaches (cont)
 Formally:
i - index number for option with given strikeand maturity
Pi - measuresof marketprices,e.g. cash pricesof OT M
optionsor implied volatilities
 - vectorof modelparameters
 i ( ) - modelprice measurefor parameters and strike/maturity i
wi - weights
 - metricexponentused; usually 1,2,or 
penaltyfunction
M ( )   wi ( Pi   i ( ))  min
i
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
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:
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
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!
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
Example: Heston model fit on
level/skew/curvature
•DAX Index options,
•Heston model
• global fit
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
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
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):
dSt  (r (t )  y(t ))St dt   (t , St )St dwt

at time 0+
volatility starts following of of the few prescribed trajectories
 i (S , t )
with probability i
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  i (t ), not i (t , St )
), so the density is a mixture of lognormals
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
vol
Scenario 1, p=0.54
Vol butterfly
Scenario 2, p=0.46
T
 Actually (almost) the same happens in Heston model
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
3.1 No arbitrage: single maturity
 Fixed maturity European call prices:
C ( K )  0 (non- negativeprices)
C ( K1 )  C ( K 2 )  0 (monotonicprices)
aC( K1 )  bC( K 2 )  (b  a )C ( K 3 )  0,
a
b

(non- negativeratiobutterflies)
K 2  K1 K 3  K 2
3.1 No arbitrage: single maturity (cont)
Breeden&Litzenberger’s formula:
 2C (T , K )
 exp(rT ) f ( X ),
2
X
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
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
lim C ( K )  0
K 
if we have options with strikes around 0
(S  K )  S  K  C' (0)  1
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?
3.1 No arbitrage: single maturity (cont)
 All spreads are positive, 80-90-100 butterfly is worth 302*21+14=2>0...
 But
1
9
1
S  C (80)  C (90)    0
8
8
4
 Payoff diagram:
0
80 90
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 nonnegative value
otherwise, we are left with C(K,T)-S+K=P(K,T), again with nonnegative value
3.2 No arbitrage: calendars
 Thus, with no dividends, zero interest rate,
C(K , T )  C(K , t )
 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
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)
Let us haveinterestrater (t ) and dividend yield y(t )
possibly with discrete components
t
(only time integrals
Local volatility model
yi (ti )
Y (t )   y (t )dt of y(t) will matter)
0
dSt  (r (t )  y(t ))St dt   (t , St )St dwt
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
eY (T ) Y (t ) units of call with maturityt and strike
k  eY (T ) Y (t ) ( R (T )  R (t )) K ,
T
T
Y (T )   y (t )dt,
R(T )   r (t )dt,
0
0
3.2 No arbitrage: calendars (cont)
 As before, at time t,
if S<K, then the short leg expires worthless, the long leg has nonnegative value
otherwise, we are left with
Y (t ) Y (T )
C( K , T )  e
has non-negative value
(S  K )  P( K , T )
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
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  0 iff CDS price is positive
1
2
   2 , otherwise volatility can go 0 and stay around it (not a
feasible constraint)
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), ...
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-ofsample price prediction

Different studies lead to different conclusions on volatility risk premia,
stationarity of volatility, etc

Much to be done here
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
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)
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
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
6: using models for trading (cont)
 Kurtosis trade:
long an ATM butterfly
short the wings
Actually a vega-hedged short variance swap or some pathdependent exotics would do better
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?
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
Historical vs implied distribution: StDev
Image reproduced with authors’ permission from Blaskowitz, Hardle, Schmidt
Historical vs implied distribution: skewness
Image reproduced with authors’ permission from Blaskowitz, Hardle, Schmidt
Historical vs implied distribution: kurtosis
Image reproduced with authors’ permission from Blaskowitz, Hardle, Schmidt
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).
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
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.