Levy Innovation, Dynamic Volatility, Leverage effect and ...€¦ · About Modeling Stock Prices 1,...
Transcript of Levy Innovation, Dynamic Volatility, Leverage effect and ...€¦ · About Modeling Stock Prices 1,...
Levy Innovation, Dynamic Volatility,
Leverage effect and Option Pricing
------ Option Pricing for Tempered Stable distributed ARMA-NGARCH models
Fumin ZHU
Tel.631-885-3585
For Option Pricing
We should solve two basic and important problems: (1)To specify an appropriate process for underlying assets.
Stochastic factors: innovations Drift :The risk-premium value/market price of the risk Volatility: of the stock price-risk measurement
(2) To obtain the equivalent martingale measurement. Non-arbitrage valuation for the derivatives(efficient markets) Neutral –risk distribution (incomplete market)
A fine option model should capture: (1)Volatility smirk on Maturity and strike price. IV(O,S,K,T)
Different Volatility factor between Short term and long term.
(2)Jump activity in the price and volatility.
Non-Arbitrage between call and put option Have a robust test result (parity law)
Literature: Fama, B.S., Merton, Hull, Duffie, Cochrane, Duan, Heston, Carr, Wu, Chiristoffersen, Rosinski, Kim, Rachev,……
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0
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About Modeling Stock Prices
1, Risk-premium from Return rate/holding yield. Linear and exponential expectation or drift rate.
2, Dynamic volatility for Risk measurements Clustering , persistence/reversion and feedback
3, Innovation of (Semi)Martingale processes Levy Innovation: stochastic part- continue and jump
4, Leverage effect --- big drop cause big volatility Negative relationship between return and volatility
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A figure evidence: H&S300 index
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0 1000 2000 3000 4000
1
1.5
2
2.5
3
x 104 Stock price
0 1000 2000 3000 4000
-6
-4
-2
0
2
Noise series
0 1000 2000 3000 4000
-0.1
0
0.1
Return innovation
0 1000 2000 3000 40000
0.02
0.04
0.06
Volatility
Measurements for Option Pricing
Latent states: unobservable variables 1, Market price of Risk/risk premium 2, Jump activity in the market(jump sides) 3, Dynamic volatility of the stock
Measurements: Observable variables Stock price/Return rates (risk-free bonds) Realized Volatility from high-frequency data Implied Volatility from option prices
Sequential Bayesian Filtering methods To estimate the models, capture the states To sequential option pricing
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Motivation
What will we do?
Specify stochastic process with Leverage effect for underlying assets and option pricing based on the time series analysis and particle filtering, on the condition of that, the Dynamic Levy innovation or the noise is tempered stable distribution, which is one kind of infinite activity pure jump processes.
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Literature Research
Levy Processes Brownian Motion: L. Bachelier(1900), Samuelthon(1965) Geometric B.M.: Black-Scholes Model(1973) Finite jumps: Diffusion + Jump Models.(1976-1999) Infinite activity processes: VG, NIG, GH, CGMY, a-Stable, Tempered Stable(2000-
2012) Volatility Model
Stochastic Volatility processes.(Merton, Carr, Heston, Wu, Huang,2001-2005) Term Rate Structure, such as CIR, SQR
Conditional Volatility model.(Engle,Bollerslev,Duan,Christoffersen,Kim,2011) ARCH, GARCH
Leverage Effect Negative relationship coefficient (Carr,2004,2008,2011) Asymmetrical GARCH models(Heston,Christoffersen,2000,2010)
Bayesian analysis MCMC: strict, accurate requirements for prior density of the variables and
parameters(Polson.2003,Li. 2008,) Particle filtering: Sequential Monte Carlo simulation, Bayesian filtering based
on simulation technology.(Pit,2002,Li,2012)
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Theoretical Model (1)
ARMA-N GARCH: Obtain stationary i.i.d noise Conditional expectation Conditional volatility Leverage effect
Levy Innovation: jump intensity Jump intensity of all size: Jump intensity of per size Jump scale of each size: Levy innovation:
Gaussian noise Diffusion Jump noise Infinite Jump noise: V.G., Tempered Stable
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0
0
( ) ( )
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R
R v dx
x d v dx v dx
X x x v dx
00
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X v dx X X s iid X s
Theoretical Model (2)
For Time series analysis(ARMA-GARCH) Autocorrelation and Heteroscedasticity.(Engle,Ng,1991)
Levy innovation and Levy noise
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2 2 2
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leverage
Theoretical Model (3)
Exponential Dynamic Levy Process
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1
2 2 2
0
1 1
log ( / ) ( )
( )
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volatility
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Theoretical Model (4)
3-d Dynamic State Space Model (combined model(2) with (3))
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1 1
2
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(Mean=0,std=1)
Theoretical Model (4)
A simplest example: Gaussian distribution without jump
Asymmetric N GARCH model
Duan’s GARCH option pricing model
Jump-Diffusion model with finite jumps
Historical filtering simulation GARCH Non-parametric distribution model
Infinite Pure jump/infinite activity model Pure jump without diffusion part
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(0, )t tN h(0, ), 0t t iN h
Theoretical Model (5)
Risk-Neutral model (Local Non-arbitrage martingale measurement)
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( )
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Noise structure
Theoretical Model (6)
Radan-Nickodym derivative
Escher Transformation(if and only if)
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( )
11
| /[ | ]
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dynamic
Solutions may be not only one
2 3 4
2 3 4
( ) (0) (0) (0) / 2 (0) / 6 (0) / 24
(0) (0) / 2 / 6 / 24
u u u u u
u u skew u kurt u
A simplest example:
Complete Market
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2 /2. (0,1), ( ) ,u
XIf X N u e
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u u u u u
u u skew u kurt u
Theoretical Model (7)
Alternative model in stochastic volatility(SV) Heston’s SQR(1995) Carr’s CIR(2004)
For simulation:
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/
ln ( / 2)
ln ln ( / 2) , (0,1)
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( )
t t t t t
t t t t t
t t t t t t t
t t t t t t
t t t t t t
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d S V dt V dW
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V V k V t V t z
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t t t t t t t
V V k V t V t z
y V t V t z zz z N i i d
Tempered Stable
Theoretical Model (7)
Option pricing of S.V. models: risk-neutral measurements:
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*
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d S r V dt V dW
dV k V dt V dt V dZ dW dZ dt
dV k k V dt V dZ
dV k k k V dt V dZ
dV k V dt V dZ k k k k
Volatility premium
Markov Chain Monte Carlo Simulation
for Option Pricing
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1
1
( , ) ( ( ) ) ( )
1( , ) max( ( ) ,0)
1( , ) max( ( ),0)
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P t K e K S in
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t t t tt t tr
TS i S e
Discussion for Numerical Solution
Fast Fourier Transformation by the Characteristic function:
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( ) ( )ivk k
TF v e e C k dk
( ) (1 )
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Discrete F. T.
★ ★ ★ Find the C.F of S(T)!!!
Significance of the model
Could use Time series analysis method directly
Easy to estimate and test models(one factor)
Useful to obtain conditional predications
Well defined to MCMC for option pricing
Simplest Local equivalent martingale measurement
Particle filtering and parameter learning(further research)
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Econometric Methodology (1)
Time series analysis : Two-step estimation
Quasi-Maximum Likelihood for ARMA-NGARCH model with historical filtering distribution.
Moment method or Maximum Likelihood based on Fourier Transformation method.
Sequential Bayesian Analysis
Markov Chain Monte Carlo Simulation
Particle filtering and learning
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Econometric Methodology (2)
Time series analysis
Find the lag orders of expectation and volatility for the time series (return and noise sequences).
Estimate the Levy noise by Characteristics Function(Moment Generating or Fourier Trans)
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( ) 2
1 1( )
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Return sequence(ACF,PACF)
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0 2 4 6 8 10 12 14 16 18 20-0.5
0
0.5
1
Lag
Auto
corr
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tion
Return Autocorrelation Function (ACF)
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Part
ial A
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corr
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tions
Return Partial Autocorrelation Function
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Auto
corr
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tion
Return2 Autocorrelation Function (ACF)
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0
0.5
1
Lag
Part
ial A
uto
corr
ela
tions
Return2 Partial Autocorrelation Function
Fourier Transform method Rachev(2011)
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Econometric Methodology (3)
Bayesian analysis Method(Particle Filtering)
Three dimension state variables and one observation
Maximum Likelihood
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{ ; ; }
( )t
t t t t
t t dL t t t
x dL h
y h h dL
1 1 0|0 | 1
1 12
1 1({ } | ) ( | ; ) ( | ; )
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t t t t t
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p y p y x p y xN N
Bayesian Analysis
Filtering Finding the current state
Condition contribution
Prediction
Evidence
Sequential Update
1:1:
1:
1: 1
1: 1
1: 11: 1
1: 1 1: 1
1: 1 1: 1 1: 1
1: 1 1: 1
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1: 1( | )t tp y y
Bayesian Analysis
General Bayesian Filter
Initialization on 0 and t-1
Find the predictive density(prior):
Find the update density(posterior)
0 0 0 1 1: 1( | ) ( ), ( | )t tp x y p x p x y
1: 1 1 1: 1 1 1: 1 1( | ) ( | , ) ( | )t t t t t t t tp x y p x x y p x y dx
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1: 1
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Sequential Importance Sampling
In order to prevent the particles from degeneration: resample and reweight by a easy sampling density(proposal PDF)
Importance sampling density:
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1: 11:
1: 1
( | ) ( | )( | )
( | ) ( | )
t t t tt t
t t t t t
p y x p x yp x y
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p x yf x q x y dx
q x y
p x yf x
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Kalman filter
Smooth-joint likelihood function
1: 0: 0:0: 1:
1:
1: 0: 11
1: 1 1: 1
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1: 1 1: 1
1
1 1
1: 1
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Econometric Methodology (4)
Kalman Filter:
For linear and Gaussian Model
Particle Filtering(simulation for non-Gaussian and non-linear)
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1( )
[ ] ( )
[ ]
xy y
T
x y
K P P
x x K y y
x P KP K
( ) ( ) ( )( ) ( ) 1
1 ( ) ( )
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Particle Filtering
Monte Carlo P.F.
Dirac-delta function
Bootstrap P.F.
Auxiliary P.F.
Kalman SIR P.F.
Extended K.F.
Unscented K.F.
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Econometric Methodology (5)
Why do we need Particle filtering?
For capturing the latent state variables, especially formore than one stochastic factors.
For parameter learning, in order to capture the dynamic parameter phenomenon.
To make the observation variables integrated with option pricing and realized volatility.
To analyze which kind of jump or activity is.
Particle filtering method is important!
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Empirical Research (1)
ARMA-N GARCH Model Conditional dynamic with Leverage effect
Levy innovation No jump: GBM Finite jumps: Merton’s Jump-Diffusion model Infinite jump with thin tail: Variance Gamma, Normal
Inverse Gaussian and Mixner Infinite jump with heavy tail: a-stable, CGMY, Tempered
stable, rapid decreasing TS.
Data: Hang seng Index and Index options(Hong Kong market) H&S300(SH,SZ combined index of Chinese market)
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Empirical Research (2)
Tempered Stable(Rachev (2011)):
Rapid Decreasing Tempered Stable:
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1 1
0 0( ) ( / 1 / | | 1 )x x
CTS x xv dx C e x C e x dx
2 2/2 /21 1
0 0( ) ( / 1 / | | 1 )x x
RDTS x xv dx C e x C e x dx
( ) exp{ ( ) [( ) ] ( ) [( ) ]}CTS u C iu C iu
2 12 212 2
2 2
1 1 1 3( ; , ) 2 ( )( ( , ; ) 1) 2 ( )( ( , ; ) 1)
2 2 2 2 2 2 2 2
x xG x M x M
( ) exp{ ( ; , ) ( ; , )}RDTS u C G iu G uC i
Tempered Stable simulation
Classical Tempered Stable(j=500)
Rapidly decreasing Tempered Stable(j=2000)
2013/11/25Quantitative finance, AMS, Stony Brook
University, Fumin Zhu35
1 1
1
0
1
| |[( ) | | ] ( )
|| ||
exp(1), , (0,1), ( , , ,1 )
j
dj j
j j
j j
j
j j i j j
i
vX e u v i
T v
e e u U v F P P
1 11
12
0
| |[( ) 2 | | ] ( )
|| ||j
j
dj j
j
j j
vX e u v i
T v
Empirical Research (3)
Other Levy processes(MJ: finite jump and VG: infinite jump but thin-tailed)
2013/11/25Quantitative finance, AMS, Stony Brook
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2
2
( )
21( , , ( )) ( , , )
2
t J
J
dL
t JD J
J
v dL e
0 0( , , ( )) ( , , / | | 1 / 1 )Gx Mx
t VG x xv dL Ce x Ce x
2 22 2
2( ) ( 1)( )2( ) [ ]
uJiu J
JdLt t
t
uiu t e tt uiudL
MJ u e e e
2ln( ) ln( )( ) [ ]tiudL tC GM tC GM iMu iGu u
VG u e e
Empirical Research (4)
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Coeff\Market SZZZ SZCZ H&S300 S&P500
a -0.7827(-5.6885) -0.7365(-5.7398) 0.9854 (55.3270) 0.0368(5.1909)
b 0.8114(6.2580) 0.7766(6.4720) -0.9782 (-44.2149) -0.0331(4.6684)
c -0.0001(-0.1980) -0.0003(-0.7804) 4.5e-6(0.6265) 0.0003(3.1192)
Alpha0 2.04e-6(7.7245) 3.5e-6 (6.3219) 2.8e-6(2.9645) 2.4e-6 (18.927)
Alpha1 0.0694(9.0572) 0.0644(10.128) 0.0456 (5.6215) 0.0095(2.006)
Beta 0.8885(126.56) 0.9030(140.40) 0.9452 (137.59) 0.9062(254.45)
Delta 0.0529(5.7239) 0.0408(4.7591) 0.0056(0.6023) 0.1278(22.967)
Noise mean 0.0041 0.0096 -0.0004 -0.0046
Noise std. 1.0015 1.0003 0.9993 0.9998
Noise skew -0.1718 -0.1439 -0.3941 -0.3975
Noise kurtosis 5.3300 4.9332 4.7595 8.6327
JBtest(H,P) (1,0.001) (1,0.001) (1,0.001) (1,0.000)
) Table 3 Parameters Estimation Results of the ARMA-HNGARCH Models (t-value)
Empirical Research (4)
Parameter estimation for ARMA-HN GARCH(H&S300)
2013/11/25Quantitative finance, AMS, Stony Brook
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ARMA p,q a b c Mu
Coeff 1,1 0.0095 -0.0205 0.0006
Std 0.0091 0.0133 0.0004
GARCH m,n Alpha0 Alpha1 Beta1 delta
Coeff 1,1 2.5e-006 0.0501 0.9438 0.0451
Std 8.6e-007 0.0067 0.0074 0.0058
White Noise mean variance skewness kurtosis
KS(P=0.000) -0.0001 0.9992 -0.3883 4.5800
Table 1 Parameter estimation results for ARMA-GARCH models by Generalized least-square method
Empirical Research (4)
Estimation For Levy Processes(H&S300)
2013/11/25Quantitative finance, AMS, Stony Brook
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Mean Variance Skewness Kurtosis C G M Y
BM -0.0001 0.9992 -0.3883 4.5800 0 0 0 0
VG 1.9117 1.9537 1.9539 0
TS 1.9332 1.5544 2.1719 0.0006
JD Mu_w Sigma_w Lambda
_J
Mu_J Sigma_J
Coeff 0.0001 0.7047 0.4740 -0.3101 0.9805
Table 2 Parameter estimation results for Levy processes
Good of fit
2013/11/25Quantitative finance, AMS, Stony Brook
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-4 -2 0 2 4-5
0
5
10
Standard Normal Quantiles
Quantile
s o
f In
put
Sam
ple
Normal
-5 0 5 10-10
-5
0
5
10
X Quantiles
Y Q
uantile
s
Merton Jump
-5 0 5 10-5
0
5
10
X Quantiles
Y Q
uantile
s
Variance Gamma
-5 0 5 10-5
0
5
X Quantiles
Y Q
uantile
s
Tempered Stable
-8 -6 -4 -2 0 2 4 60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5PDF of Noise
Empirical
MJ
Normal
-6 -4 -2 0 2 4 6 80
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5PDF curve
Empirical
VG
Normal
-8 -6 -4 -2 0 2 4 60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5PDF of Noise
Empirical
TS
Normal
Particle filtering for the jump and volatility
2013/11/25Quantitative finance, AMS, Stony Brook
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0 200 400 600 800 1000 1200 1400 1600 18000
1
2
3x 10
-3
Time steps
Con
ditio
nnal
Var
ianc
e
Conditional Variance Particle estimator
markets
Particle Filter
Kalman Filter
0 200 400 600 800 1000 1200 1400 1600 1800-5
0
5
10
Time steps
Levy
Jum
ps
Innovation Particle estimator
markets
Particle Filter
Kalman Filter
0 200 400 600 800 1000 1200 1400 1600 18000
0.5
1
1.5x 10
-3
Time steps
Con
ditio
nnal
Var
ianc
e
Conditional Variance Particle estimator
markets
Particle Filter
Kalman Filter
0 200 400 600 800 1000 1200 1400 1600 1800-5
0
5
10
Time steps
Levy
Jum
ps
Innovation Particle estimator
markets
Particle Filter
Kalman Filter
0 200 400 600 800 1000 1200 1400 1600 18000
0.5
1
1.5
2x 10
-3
Time steps
Conditio
nnal V
ariance
Conditional Variance Particle estimator
markets
Particle Filter
Kalman Filter
0 200 400 600 800 1000 1200 1400 1600 1800-5
0
5
10
Time steps
Levy J
um
ps
Innovation Particle estimator
markets
Particle Filter
Kalman Filter
0 200 400 600 800 1000 1200 1400 1600 1800-5
0
5
10
Time steps
Jum
ps
market
Particle Filter
Kalman Filter
0 200 400 600 800 1000 1200 1400 1600 18000
0.01
0.02
0.03
0.04
Time steps
Vola
tilit
y
t
market
Particle Filter
Kalman Filter
Estimation errors under Particle filtering
Jump Linear Drift Volatility
AAE APE MSE RMSE AAE APE MSE RMSE AAE APE MSE RMSE
BSKF 0.7333 0.9804 0.9800 0.9806 9.0E-5 0.4659 1.2E-4 5.2488 6.8E-4 2.3148 7.3E-4 2.6556
BSPF 0.5383 1.6052 0.7192 21.272 9.5E-5 0.4810 1.3E-4 5.6059 6.7E-4 2.3103 7.2E-4 2.6514
JDKF 0.7333 0.9803 0.9800 0.9806 1.1E-4 0.3746 1.3E-4 2.9083 8.3E-5 0.2030 1.3E-4 0.2499
JDPF 0.4477 2.3084 0.5491 19.551 1.1E-4 0.3721 1.3E-4 2.8077 8.3E-5 0.2026 1.3E-4 0.2495
VGKF 0.7333 0.9807 0.9800 0.9808 9.1E-5 0.4320 1.2E-4 4.2360 4.3E-4 1.5760 4.7E-4 1.8547
VGPF 0.5358 0.9551 0.7128 4.3832 9.2E-5 0.4307 1.2E-4 4.1778 4.3E-4 1.5725 4.6E-4 1.8509
TSKF 0.7333 0.9804 0.9800 0.9806 9.4E-5 0.3285 1.2E-4 3.5196 1.4E-4 0.2790 2.0E-4 0.3202
TSPF 0.3383 1.8765 0.4376 19.561 9.5E-5 0.3076 1.2E-4 3.2424 1.3E-4 0.2785 2.0E-4 0.3197
2013/11/25Quantitative finance, AMS, Stony Brook
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Table 3 Loss Function of state variables
Empirical Research (5)
For option pricing:
Measurement correction and transformation
Market price of Risk:
Measure correction:
Neutral-risk volatility:
Estimate the initial neutral-risk states
Initial implied volatility:
Initial risk premium:
2013/11/25Quantitative finance, AMS, Stony Brook
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( ) /t t t tr h
( ), +Q
t t t th k z 2 2 2 2
0
1 1
( )p q
t i t i i i
Q
t t i j t j
i j
h h k h
[ ( ) ( )]/Q
t t t t t tk h h h h
0 0h0h
Empirical Research (5)
Option Pricing(Loss function)
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University, Fumin Zhu44
1
1| _ ( ) _ ( ) |
N
i
AAE C mark i C model iN
1 1
| _ ( ) _ ( ) | / _ ( )N N
i i
APE C mark i C model i C mark i
1
1[| _ ( ) _ ( ) | / _ ( )]
N
i
ARPE C mark i C model i C mark iN
2
1
1( _ ( ) _ ( ))
N
i
C mark i C model iN
RMSE
0
1
1| _ ( ) _ ( ) | /
N
i
RMRE C mark i C model i SN
Empirical Research (5)
Maturity Type AAE APE ARPE RMSE RMRE
Short Term(One month)
Call-BS147.4873
(292.7745)0.2757
(0.3898)0.5369
(2.4444)149.8889
(302.0304)0.0069
(0.0137)
Call-TS145.5624
(278.1271)0.2721
(0.3703)0.5391
(2.3274)147.6076
(286.8308)0.0068
(0.0131)
Call-RDTS142.4067
(280.1710)0.2662
(0.3731)0.4951
(2.2744)145.4930
(288.8251)0.0067
(0.0131)
Put-BS33.5364
(179.2151)0.0683
(0.3186)0.1418
(1.2398)36.7853
(192.4949)0.0016
(0.0084)
Put-TS33.5335
(182.7449)0.0683
(0.3249)0.1499
(1.3088)37.6403
(194.3589)0.0016
(0.0086)
Put-RDTS30.4343
(178.2130)0.0620
(0.3168)0.1073
(1.3147)35.1640
(190.2302)0.0014
(0.0084)
Medium Term(Three Months)
Call-BS409.5443
(591.2013)0.3800
(0.5485)0.8220
(1.3760)414.0673
(593.5411)0.0150
(0.0277)
Call-TS387.3018
(514.5302)0.3593
(0.4774)0.7664
(1.1929)392.1542
(517.1219)0.0150
(0.0241)
Call-RDTS310.9689
(505.0065)0.2885
(0.4685)0.2518
(1.1033)373.0926
(507.1175)0.0147
(0.0237)
Put-BS90.5858
(227.5224)0.1031
(0.2589)0.2388
(0.4658)98.2503
(233.5263)0.0102
(0.0107)
Put-TS89.5572
(237.5857)0.1019
(0.2703)0.2464
(0.4911)97.4499
(243.1389)0.0100
(0.0111)
Put-RDTS62.6149
(97.3005)0.0712
(0.1107)0.1280
(0.2308)74.1962
(105.8420)0.0069
(0.0046)
Long Term(Six Months)
Call-BS584.3153
(689.9162)0.4128
(0.4873)0.6669
(0.8654)586.4369
(693.5971)0.0274
(0.0324)
Call-TS509.1911
(633.7717)0.3597
(0.4477)0.5753
(0.7817)510.9088
(635.6138)0.0239
(0.0297)
Call-RDTS567.7610
(483.6827)0.4011
(0.3417)0.6147
(0.5911)572.3610
(485.1091)0.0266
(0.0297)
Put-BS231.3504
(292.3183)0.2246
(0.2838)0.3567(0.3732)
236.5914 (300.9854)
0.0109(0.0137)
Put-TS74.1031
(219.9883)0.0720
(0.2136)0.1272(0.2883)
80.7329 (225.3138)
0.0035(0.0103)
Put-RDTS65.0967
(123.5411)0.0632
(0.1200)0.0928(0.1559)
72.4104 (129.0487)
0.0031 (0.0058)
2013/11/25Quantitative finance, AMS, Stony Brook
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Table 4 Loss Function of short, medium and long terms options( parentheses is non-leverage effect)
Empirical Research (6)
Short term—non leverage effect
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0.85 0.9 0.95 1 1.05 1.10
0.02
0.04
0.06
0.08
0.1
0.12
Moneyness(K/S)
Option p
rice(p
/S)
Short term Options(1month)
Call
Put
CBS
PBS
CTS
PTS
CRDTS
PRDTS
0.85 0.9 0.95 1 1.05 1.10.1
0.15
0.2
0.25
0.3
0.35
Moneyness(K/S)
I.V
.
Implied Volatility of 1 month maturity
IVCall
Cmark
CBS
CTS
CRDTS
IVput
Pmark
PBS
PTS
PRDTS
dispersion
Empirical Research (6)
Medium term—non leverage effect
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0.85 0.9 0.95 1 1.05 1.10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Moneyness(K/S)
Option p
rice(p
/S)
Short-term Options
Call
Put
CBS
PBS
CTS
PTS
CRDTS
PRDTS
0.85 0.9 0.95 1 1.05 1.10.1
0.15
0.2
0.25
0.3
0.35
Moneyness(K/S)
Implied Volatility of 3-month maturity
IVCall
Cmark
CBS
CTS
CRDTS
IVput
Pmark
PBS
PTS
PRDTS
Empirical Research (6)
Short term—with leverage effect
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0.8 0.85 0.9 0.95 1 1.05 1.1 1.15-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Moneyness(K/S)
Option p
rice(p
/S)
Short term Option
Call
Put
CBS
PBS
CTS
PTS
CRDTS
PRDTS
0.8 0.85 0.9 0.95 1 1.05 1.1 1.150.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Moneyness(K/S)
Implied Volatility of short maturity
IVCall
Cmark
CBS
CTS
CRDTS
IVput
Pmark
PBS
PTS
PRDTS
tightness
Empirical Research (6)
Medium term—with leverage effect
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0.85 0.9 0.95 1 1.05 1.10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Moneyness(K/S)
Option p
rice(p
/S)
Medium-term Options
Call
Put
CBS
PBS
CTS
PTS
CRDTS
PRDTS
0.85 0.9 0.95 1 1.05 1.10.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Moneyness(K/S)
Implied Volatility of Medium-term options
IVCall
Cmark
CBS
CTS
CRDTS
IVput
Pmark
PBS
PTS
PRDTS
Analysis of results
Infinite activity pure jump model captures the sequential volatility and noise of ARMA-N GARCH model accurately.
Tempered Stable process performs better than other Levy processes for the return good-of-fit.
Tempered Stable GARCH model for option pricing is superior to Gaussian-GARCH model
NGARCH models with Leverage effect improves the option pricing ability significantly. Tempered Stable for option remains better than Gaussian models.
Tempered Stable for Long term option is not too significant.
2013/11/25Quantitative finance, AMS, Stony Brook
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Conclusion
Theoretical or practical Researcher chose Infinite Pure jump/infinite activity stochastic processes among the Levy innovation model for index stock market would be better.
Time series for analysis Dynamic Levy processes is easy to manipulate.
Leverage effect must be considered for option pricing model.
2013/11/25Quantitative finance, AMS, Stony Brook
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THANK YOU
FOR YOUR ATTENTION!
Speaker: Fumin ZHU;
Place: Stony Brook University, New York.
Date:11/08/2012
2013/11/25Quantitative finance, AMS, Stony Brook
University, Fumin Zhu52