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Topics in Computational Financea view from the trenches
P. Hénaff
Télécom-Bretagne
20 March 2010
Stylized Facts about Financial Time SeriesFat TailsFitting a Density to Sample ReturnsDependenceVolatility Surface
Model Risk and Model CalibrationPrice and ValueCalibration Issues with Complex Models
Numerical Challenges in Monte-Carlo SimulationsDivision of LabourPractical Advantages of MC FrameworksOpen Topics for Research
Conclusion
Stylized Facts about Financial Time SeriesFat TailsFitting a Density to Sample ReturnsDependenceVolatility Surface
Model Risk and Model CalibrationPrice and ValueCalibration Issues with Complex Models
Numerical Challenges in Monte-Carlo SimulationsDivision of LabourPractical Advantages of MC FrameworksOpen Topics for Research
Conclusion
Stylized Facts about Financial Time SeriesFat TailsFitting a Density to Sample ReturnsDependenceVolatility Surface
Model Risk and Model CalibrationPrice and ValueCalibration Issues with Complex Models
Numerical Challenges in Monte-Carlo SimulationsDivision of LabourPractical Advantages of MC FrameworksOpen Topics for Research
Conclusion
Stylized Facts about Financial Time SeriesFat TailsFitting a Density to Sample ReturnsDependenceVolatility Surface
Model Risk and Model CalibrationPrice and ValueCalibration Issues with Complex Models
Numerical Challenges in Monte-Carlo SimulationsDivision of LabourPractical Advantages of MC FrameworksOpen Topics for Research
Conclusion
Stylized Facts about Financial Time Series
Fat Tails
Closing price of December 2009 WTI contract.
Stylized Facts about Financial Time Series
Fat Tails
Time Series of Daily Return
Time
TS
.1
2006−02−19 2006−09−08 2007−03−28 2007−10−15 2008−05−03 2008−11−20
−0.
050.
000.
050.
10
Stylized Facts about Financial Time Series
Fat Tails
Histogram of Daily Return
Daily return
r
Fre
quen
cy
−0.05 0.00 0.05 0.10
020
4060
8010
0
Stylized Facts about Financial Time Series
Fitting a Density to Sample Returns
Fitting a density to observed returns
The Johnson family of distributions. X : observed Z : N(0;1)
Z = + � ln(g(x � �
�)) (1)
where :
g(u) =
8>><>>:
u SLu +p
1 + u2 SUu
1�u SBeu SN
Stylized Facts about Financial Time Series
Fitting a Density to Sample Returns
Johnson SU Distribution - December 2009 WTIcontract.
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−0.04 −0.02 0.00 0.02 0.04
−0.0
50.
000.
050.
10Johnson SU distribution
q
y
Stylized Facts about Financial Time Series
Fitting a Density to Sample Returns
Johnson SU vs. Normal
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−0.0
50.
000.
050.
10
Johnson SU
q
y
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−3 −1 0 1 2 3
−0.0
50.
000.
050.
10
Normal
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Stylized Facts about Financial Time Series
Fitting a Density to Sample Returns
The Generalized Lambda Distribution
Tukey’s Lambda distribution :
Q(u) = �+u� � (1� u)�
�(2)
Generalized Lambda distribution :
Q(u) = �1 +u�3 � (1� u)�4
�2(3)
where :Pr(X < Q(u)) = u
Stylized Facts about Financial Time Series
Fitting a Density to Sample Returns
Density of daily return fitted with Generalized Lambdadensity
λ1: 1.10e−03 λ2: 1.31e+02 λ3: −1.76e−01 λ4: −5.32e−02
Daily return
Den
sity
−0.05 0.00 0.05 0.10
05
1015
2025
30
RPRSRMFMKLSTAR
Stylized Facts about Financial Time Series
Fitting a Density to Sample Returns
Gen. Lambda vs. Normal
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−0.0
50.
000.
050.
10
Gen Lambda
q
y
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−3 −1 0 1 2 3
−0.0
50.
000.
050.
10
Normal
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Stylized Facts about Financial Time Series
Dependence
Volatility ClusteringWTI Dec−09 return
Time
Daily
retu
rn
2006−02−13 2007−03−22 2008−04−27
−0.0
50.
000.
050.
10
FIGURE: Series of daily returns
Stylized Facts about Financial Time Series
Dependence
Autocorrelation of return
0 5 10 15 20 25
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Lag
ACF
autocorrelation of r(t)
0 5 10 15 20 25
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Lag
ACF
autocorrelation of |r(t)|
FIGURE: ACF of rt and jrt j
Stylized Facts about Financial Time Series
Dependence
Summary - Price Process
I No evidence of linear autocorrelation of returnI Large excess kurtosis, incompatible with normal densityI Distribution of return is well approximated by a Johnson
SU, to a lesser extend by a Generalized Lambdadistribution
I Observable autocorrelation of jrt j and r2t , suggesting
autocorrelation in the volatility of return.
Stylized Facts about Financial Time Series
Volatility Surface
Quoted option prices - March 2009 WTI contract
0
2
4
6
8
10
40 60 80 100 120 140
Call price vs. strike Mar09 WTI contract
0
20
40
60
80
100
20 40 60 80 100 120 140
Put price vs. strike Mar09 WTI contract
Stylized Facts about Financial Time Series
Volatility Surface
Volatility term structure - NYMEX WTI options
45
50
55
60
65
70
Feb09 Aug09 Feb10 Aug10 Dec11 Dec14
Forward Prices WTI futures
30
40
50
60
70
80
90
100
Feb09 Aug09 Feb10 Aug10 Dec11 Dec14
ATM vol WTI futures
Stylized Facts about Financial Time Series
Volatility Surface
Implied Volatility - March 2009 WTI contract
8990919293
40 50 60 70 80 90 100
Call vol vs. strike Mar09 WTI contract
889092949698
30 40 50 60 70 80 90 100
Put vol vs. strike Mar09 WTI contract
89909192939495969798
30 40 50 60 70 80 90 100
OTM vol vs. strike Mar09 WTI contract
Stylized Facts about Financial Time Series
Volatility Surface
Implied Volatility Cross-sections - NYMEX WTI options
30
40
50
60
70
80
90
30 40 50 60 70 80 90 100
OTM vol vs. strike WTI futures
Mar-09Apr-09Jun-09Jul-09
Dec-09Jun-10Dec-10Jun-11Dec-11
Stylized Facts about Financial Time Series
Volatility Surface
Implied Volatility Surface - NYMEX WTI options
Mar09Apr09Jun09Jul09Dec09
Jun10Mar09
Apr09Jun09 50
6070
8090
30
40
50
60
70
80
90
ATM vol surface WTI futures
Expiry
Strike
Stylized Facts about Financial Time Series
Volatility Surface
Summary - Volatility Surface
I Mean-reverting process for volatilityI Smile slope decreases as a function of 1p
T
I Smile convexity decreases as a function of 1T
I Assymetry between call and put smile
Stylized Facts about Financial Time Series
Volatility Surface
The Perfect Model
I Multi-factor to capture the dynamic of the term structureI Returns with fat tails : GL, VG, stochastic volatilityI Jumps (with up/down assymetry)I mean reverting stochastic volatility for volatiity clustering
Model Risk and Model Calibration
Stylized Facts about Financial Time SeriesFat TailsFitting a Density to Sample ReturnsDependenceVolatility Surface
Model Risk and Model CalibrationPrice and ValueCalibration Issues with Complex Models
Numerical Challenges in Monte-Carlo SimulationsDivision of LabourPractical Advantages of MC FrameworksOpen Topics for Research
Conclusion
Model Risk and Model Calibration
Price and Value
The One and Only Commandment of QuantitativeFinance
If you want to know the value of a security, use theprice of another security that’s as similar to it aspossible. All the rest is modeling.
Emanuel Derman, “The Boy’s Guide to Pricing and Hedging”
Model Risk and Model Calibration
Price and Value
Valuation by Replication
Replication can be :I static : useful even if only partialI dynamic : model are needed to describe possible outcome
Objective :I To minimize the impact of modeling assumptions.I Holy Grail : A model-free dynamic hedge
Model Risk and Model Calibration
Price and Value
Conclusion
I Pricing and hedging by replicationI Mesure of market risk :
I Not in terms of model parametersI but in terms of simple hedge instruments
I The reasons for the longevity of Black-ScholesI “The wrong volatility in the wrong model to obtain the right
price”I Black-Scholes as a formula to be solved for volatility : a
normalization of price.I Choose the model in function of the payoff pattern.
Model Risk and Model Calibration
Calibration Issues with Complex Models
Calibration Isues
I Market data is insufficient and of poor qualityI Model estimation is an ill-posed problem
Model Risk and Model Calibration
Calibration Issues with Complex Models
Option data : Settlement prices of options on theFeb09 futures contract
NEW YORK MERCANTILE EXCHANGENYMEX OPTIONS CONTRACT LISTING FOR 12/29/2008
TODAY’S PREVIOUS ESTIMATED DAILY DAILY--------CONTRACT-------- SETTLE SETTLE VOLUME HIGH LOW
LC 02 09 P 30.00 .53 .85 0 .00 .00LC 02 09 P 35.00 1.58 2.28 0 .00 .00LC 02 09 P 37.50 2.44 3.45 0 .00 .00LC 02 09 C 40.00 3.65 2.61 10 .00 .00LC 02 09 P 40.00 3.63 4.90 0 .00 .00LC 02 09 P 42.00 4.78 6.23 0 .00 .00LC 02 09 C 42.50 2.61 1.80 0 .00 .00LC 02 09 C 43.00 2.43 1.66 0 .00 .00LC 02 09 P 43.00 5.41 6.95 100 .00 .00
Model Risk and Model Calibration
Calibration Issues with Complex Models
Calibration of term structure model
Let F (t ;T ) be the value at time t of a futures contract expiringat T . Assume a two factor model for the dynamic of the futuresprices :
dF (t ;T )
F (t ;T )= B(t ;T )�SdWS + (1� B(t ;T ))�LdWL
with
B(t ;T ) = e��(T�t)
< dWS;dWL > = �
Model Risk and Model Calibration
Calibration Issues with Complex Models
First approach : non-linear least-square on impliedvolatility
Given the implied volatility per futures contract[�(Ti), find theparameters �L; �S; �; � that solve :
minPN
i=1[[�(Ti)�
pV (Ti ; �S; �L; �; �)]
2
such that�� � � � �+
��L � �L � �+L��S � �S � �+S�� � � � �+
Model Risk and Model Calibration
Calibration Issues with Complex Models
Calibration results
Parameters :
�S = 1:07; �L = :05; � = 1:0; � = 2:57
Model Risk and Model Calibration
Calibration Issues with Complex Models
Model Risk and Model Calibration
Calibration Issues with Complex Models
Calibration results, varying �
FIGURE: Mean error for various � fixed
Model Risk and Model Calibration
Calibration Issues with Complex Models
Model Risk and Model Calibration
Calibration Issues with Complex Models
Optimal parameters, � fixed
� mean error �S �L �
0.70 0.0026 1.07 0.0603 2.50.72 0.0026 1.07 0.0598 2.510.75 0.00259 1.07 0.0593 2.520.77 0.00259 1.07 0.0588 2.520.80 0.00259 1.07 0.0583 2.530.82 0.00259 1.07 0.0578 2.540.85 0.00259 1.07 0.0573 2.540.88 0.00259 1.07 0.0568 2.550.90 0.00259 1.07 0.0564 2.550.92 0.00259 1.07 0.0559 2.560.95 0.00259 1.07 0.0554 2.57
Model Risk and Model Calibration
Calibration Issues with Complex Models
Parameter relationships to historical data
dF (t ;T )
F (t ;T )= B(t ;T )�SdWS + (1� B(t ;T ))�LdWL
dF (t ;T )
F (t ;T )! �LdWL; T !1
dF (t ;T )
F (t ;T )! �SdWS; T ! 0
� �<dF (t ;T1)
F (t ;T1);
dF (t ;T0)
F (t ;T0)>
Model Risk and Model Calibration
Calibration Issues with Complex Models
Hybrid Calibration - Version 1
Estimate � and �L historically (� = :87, �L = :12), calibrate �Sand � to implied ATM volatility.
Model Risk and Model Calibration
Calibration Issues with Complex Models
Model Risk and Model Calibration
Calibration Issues with Complex Models
Hybrid Calibration - Version 2
Estimate � and �L historically, calibrate all parameters toimplied ATM volatility, with a penalty on � and �L for deviationfrom historical values.New objective function :
minNX
i=1
[[�(Ti)�p
V (Ti ; �S; �L; �; �)]2+w��(���)+w�L�(�L��L)
Penalty functions :
�(x) = x2
�(x) =
�0 if jx j < �
(jx j � �)2 otherwise
Numerical Challenges in Monte-Carlo Simulations
Stylized Facts about Financial Time SeriesFat TailsFitting a Density to Sample ReturnsDependenceVolatility Surface
Model Risk and Model CalibrationPrice and ValueCalibration Issues with Complex Models
Numerical Challenges in Monte-Carlo SimulationsDivision of LabourPractical Advantages of MC FrameworksOpen Topics for Research
Conclusion
Numerical Challenges in Monte-Carlo Simulations
Division of Labour
Division of Labour
Models used to price and hedge the portfolio of a typical exoticderivatives desk :
Nb Trades Model Reprice EODHedge 106 BS < 1 min < 10 minExoticAssets
103 Monte Carlo +StoVol, LocalVol, Jumpsetc.
< 30 min 2-6 hours
Numerical Challenges in Monte-Carlo Simulations
Practical Advantages of MC Frameworks
Practical Advantages of MC pricing
I Flexibility (with pay-off language)I Consistent pricing of Exotics and HedgeI Easy to switch model to assess model riskI Only feasible solution for large dimension risk models
Numerical Challenges in Monte-Carlo Simulations
Open Topics for Research
Open research topics :I Stability of Greeks in MC frameworkI Robust variance reduction methodsI Modeling the contract rather than the payoff
Conclusion
Conclusion
I With a model comes model risk : minimize this risk by :I looking first for replicating instruments (even partial)I using a model to price
I the residual payoffI the non-standard payoffs
I Express risk in terms of simple hedge instruments ratherthan model risk factors
I MC simulation is the workhorse of exotic pricing, but themethod suffers from many practical limitations.
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