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,……

2013/11/25Quantitative Finance, AMS, Stony Brook

University, Fumin Zhu2

0

rtC Ke S P

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

2013/11/25Quantitative Finance, AMS, Stony Brook


A figure evidence: H&S300 index

2013/11/25Quantitative finance, AMS, Stony Brook


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



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.



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)



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



0

0

( ) ( )

( ) ( ) ( )

( ) ( )

R

R

R v dx

x d v dx v dx

X x x v dx

00

( , , ( )) : 0; ( ), ;lim ( ) 0.s

X v dx X X s iid X s


For Time series analysis(ARMA-GARCH) Autocorrelation and Heteroscedasticity.(Engle,Ng,1991)

Levy innovation and Levy noise



2 2 2

0

1

1

1 1

2

-1 -1

1

2

log( / )

|

( )

[ ] 0, [ ] ,| / .

q p

t i t i i t i j t j

i

m n

t t t i t i j t j t

i j

t

j

t t ttt t t

y S S c a y b

y

h h

y h

h

h z

Log-return(lag)>0i Asymmetric

leverage


Exponential Dynamic Levy Process



1

2 2 2

0

1 1

log ( / ) ( )

( )

( ) log [ ] [ ]t t t

t

t

t t t t t t t

q p

t i t i i t i j t j

h z y

z

j

t

i

h E e E

y S S h h z

h h h

e e

drift

volatility

Mean correction term

1 1( | ( )( | )) ( )t t tt t tt t tVar y F Var F Vah z r zh h


3-d Dynamic State Space Model (combined model(2) with (3))



1 1

2

0

1 1

2 2 2

( ; , , ( ));

[ ( ) ( )] ;

( )

t t

p q

t i t i t i j t j

i j

m

t

n

i i t i j i

t

t i t

j

t

i

t i t

h h

h h

h h h

dL Levy x v x

a b c

z

2 2| | 1{ /2 ( 1 ) ( )}( )

( ) 1 1( ) | [ | ]iux

x tt iu u e iux v dxiuX t

X t t tu e e

(Mean=0,std=1)


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



(0, )t tN h(0, ), 0t t iN h


Risk-Neutral model (Local Non-arbitrage martingale measurement)



( )

( ( ) )

( )+

( )+[ ( ) ( )]

[ - ( ) ( )]( )+{ + }

[ ( ) ( )]+

[ ( ) ( )]/

t t t t t

Q Q Q

t t t t t

t t t t t t

Q Q

t t t t t t t t

QQ t t t t

t t t t

t

Q

t t t tt t

t

Q

t t t t t t

y h h z

y r h h

r h h h z

r h h h h h z

h h hr h z h

h

h h hz

h

k h h h h

( ) /t t t tr h

. ( )= ( )Q

t tif h h

.

( )

t tt t t t

t

t t t tt

t t

rso z z

h

r c h r

h h

Noise structure


Radan-Nickodym derivative

Escher Transformation(if and only if)



( )

11

| /[ | ]

tt x t

t

xx

xtt

dQ e e eedP

( ) ( ) ( )[ / ] [ ]

( ) ( ) ( )

( ) ( ) ( )

t z t t t t t x t t x t t th h z z r h h x

t z t t z t t x t

t t z t x t z t t

e e e e

h h r

r h h

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



2 /2. (0,1), ( ) ,u

XIf X N u e

. ( )= ( )Q

t tSet h h

, ( ) /t t t tand r h

2( ) / 2x

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


Alternative model in stochastic volatility(SV) Heston’s SQR(1995) Carr’s CIR(2004)

For simulation:



/

ln ( / 2)

ln ln ( / 2) , (0,1)

( ) ,[ , ]

( )

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

dS S dt V dW

d S V dt V dW

S S V t V t N

dV k V dt V dZ dW dZ dt

V V k V t V t z

1,

2

1, 2,

1 2

( )

( / 2) ( 1 ), (0,1), . .

t t t t t t

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


Option pricing of S.V. models: risk-neutral measurements:



*

*

* * *

*

*

* * * * *

/

ln ( / 2)

( ) ,[ , ]

[ ( ) ]

( )[ / ( ) ]

( ) . , / ( )

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 t

t t t t t t

dS S rdt V dW

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



1

1

( , ) ( ( ) ) ( )

1( , ) max( ( ) ,0)

1( , ) max( ( ),0)

rt

TK

nr t

T T

in

r t

T T

i

C t K e S T K f S dS

C t K e S i Kn

P t K e K S in

1 1 1( )

0( )T T TQ Q

t t t tt t tr

TS i S e

Discussion for Numerical Solution

Fast Fourier Transformation by the Characteristic function:



( ) ( )ivk k

TF v e e C k dk

( ) (1 )

( ) (1 )

(1 ) (1 )

( ) ( )

( ) ( )

( ) ( )

( )[ ]

( )[

ivk k rt s k

k

srt s k ivk k

srt s iv k k iv

s srt s iv k k iv

iv s iv srt

e e e e e f s dsdk

e f s e e e dkds

e f s e e dkds

e f s e dk e dk ds

e ee f s

iv

(1 )

2 2

( ( 1) )

2 2

2 2

]1

( )(2 1)

( )(2 1)

( ( 1) )(2 1)

iv srt

rti v i s

rt

dsiv

ee f s ds

v iv

ef s e ds

v iv

ev i

v iv

0( ) ( )

kivk

T

eC k e F v dv

1

0

( ) ( )j

Niv kivk

j

j

e F v dv e F v v

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)



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




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)



( ) 2

1 1( )

2

( ) 1( ; ) log , 2,...,

2 2

i

i i i j ji

i

i i

y c a y bl i N

1

0

1 1 1( ) ( ) ( ) ( )

2 2 2

j j j n

Naiuz iuz iz u

j na

n

f z e u du e u du e u u

( )( ) [log( ( ))] / |n n

n u oC X u i

Return sequence(ACF,PACF)



0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Auto

corr

ela

tion

Return Autocorrelation Function (ACF)

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Part

ial A

uto

corr

ela

tions

Return Partial Autocorrelation Function

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Auto

corr

ela

tion

Return2 Autocorrelation Function (ACF)

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

Lag

Part

ial A

uto

corr

ela

tions

Return2 Partial Autocorrelation Function

Fourier Transform method Rachev(2011)



1

1 1( ) ( ) ( )

2 21

( ) ,|| || 2 , 2 / .2( ) [ ( ) 2 ( )] / 3.

n

aiuz iuz

a

Nizu

n

n

j left j mid j

f z e u du e u du

e u u u a u a N

f x f x f x

* *

*

1

1 1* *

0 021 ( )

*

02 2

0

2. : ( ) / 2 , 2 .

1 1 2( ) ( ) ( )

2 21 2

( )2

( ,|| || )

, 0,2

j n j n

j

q

n n n

N Nix u ix u

j n n

n na aN ix a nN N

n

na

i x n i njN N

j

j

a aMid p u u u a n N

N Na

f x e g u u e g uN

ae g u

N

e e x x N x c ja a a

Nx j j

a a

*

1

21 ( )( )*2

01 21 ( )( 1)

*2

01 21 ( )

*2 2

021

2 2

1,2, , 1.

2( ) / 2

( ) ( )

( )

( )

n n n

N a aN i j a na a N N

j n

nN nN i j

N Nn

nN j jnN i n j

N Nn

njN

i i nji ii n N N

N

a au u u a n

N Na

f x e g uN

ae g u

Na

e g uNa

e e e e eN

1

*

021

*2 2

021

*2

021

*2

021

0

( )

( )

1 ( 1) ( 1) ( )

( 1) ( 1) ( )

.

( 1) (

Ni j

n

nji N Ni i nji

i j i nN Nn

nji Ni i nj

j nN Nn

nji Ni i nj

j nN Nn

nN i nj

Nj n

n

n

n n

e g u

ae e e e e e g u

Na

e e e g uN

ae e g u

N

C e g

g g u

* 2), ( 1) .ji

ij N

j

aC e

N

1 1

0 021 ( )

0

21 ( )( )2

0

2. . : , 2 , 0,1,2,3, , 1.

1 1 2( ) ( ) ( )

2 21 2

( )2

, 0,1,2, , 1.2

( ) ( )

j n j n

j

q

n

N Nix u ix u

j n n

n naN ix a n

Nn

n

j

N aN i j a na a N

j n

n

aLeft p u a n N n N

Na

f x e g u u e g uN

ae g u

NN

x j j Na aa

f x e g uN

21 ( )( 1)

2

021 ( )

2

021

2

021

021

02

0

( )

( )

( )

( 1) 1 ( 1) ( )

( 1) ( 1) ( )

N nN i jN

n

nN jnN i j n

Nn

nNN i nji

i j i n Nn

nN i nj

j n Nn

nN i nj

j nNn

n

i njN

j n

n

ae g u

Na

e g uNa

e e e e g uNa

e g uN

ae g u

N

C e g

1

.

( 1) ( ), ( 1) .

N

n j

n n j

ag g u C

N


Bayesian analysis Method(Particle Filtering)

Three dimension state variables and one observation

Maximum Likelihood



{ ; ; }

( )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({ } | ) ( | ; ) ( | ; )

TN NT i i

t t t t t

i it

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

1: 1

( | ) ( )( | )

( )

( , | ) ( )

( , )

( | , ) ( )( | )

( | ) ( )

( | , ) ( ) ( | ) ( )

( | ) ( ) ( )

( |

t t tt t

t

t t t t

t t

t t t tt t

t t t

t t t t t t t

t t t t

t t

p y x p xp x y

p y

p y y x p x

p y y

p y y x p xp y x

p y y p y

p y y x p x p x y p y

p y y p y p x

p y y

1: 1

1: 1

, ) ( | )

( | )

t t t

t t

x p x y

p y y

1: 1( | )t tp x y

1: 1( | , )t t tp y y x

1: 1 1: 1 1: 1( | ) ( | , ) ( | )t t t t t t t tp y y p y y x p x y dx

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

1: 11:

1: 1

( | ) ( | )( | )

( | ) ( | )

t t t tt t

t t t t t

p y x p x yp x y

p y x p x y dx

Sequential Importance Sampling

In order to prevent the particles from degeneration: resample and reweight by a easy sampling density(proposal PDF)

Importance sampling density:



1: 11:

1: 1

( | ) ( | )( | )

( | ) ( | )

t t t tt t

t t t t t

p y x p x yp x y

p y x p x y dx

1:

1:1:

1:

1:1:

1:

( )( ) 1:

( )1 1:

( ( )) ( ) ( | )

( | ; )( ) ( | )

( | ; )

( | )( ) ( | ; )

( | ; )

( | )( )

( | ; )

t t t t t

t tt t t t

t t

t tt t t t

t t

iNi t t

t ii t t

E f x f x p x y dx

q x yf x p x y dx

q x y

p x yf x q x y dx

q x y

p x yf x

q x y

1:( | ; )t tq x y

( )( ) 1:

( )

1:

( )( )

( )

1

( | )

( | ; )

ˆ

ii t t

t i

t t

ii t

t N i

ti

p x yw

q x y

ww

w

Kalman filter

Smooth-joint likelihood function

1: 0: 0:0: 1:

1:

1: 0: 11

1: 1 1: 1

11 1

1: 1 1: 1

1

1 1

1: 1

( | ) ( )( | )

( )

( | ) ( ) ( | )

( | ) ( )

( | ) ( ) ( | )

( | ) ( )

( | ) (( | ) ( | )

( | )

t t tt t

t

t

t t o i ii

t t t

t t

i i o i ii i

t t t

t

i i ot t t t i

t t

p y x p xp x y

p y

p y x p x p x x

p y y p y

p y x p x p x x

p y y p y

p y x p xp y x p x x

p y y

1

11

1: 1

11

1: 1

) ( | )

( )

( | ) ( | )

( | )

t

i ii

t

t t t tt t

t t

p x x

p y

p y x p x xw w

p y y


Kalman Filter:

For linear and Gaussian Model

Particle Filtering(simulation for non-Gaussian and non-linear)



1( )

[ ] ( )

[ ]

xy y

T

x y

K P P

x x K y y

x P KP K

( ) ( ) ( )( ) ( ) 1

1 ( ) ( )

1

( | ) ( | )

( | , )

i i ii i t t t t

t t i i

t t t

p y x p x xw w

q x x y

( ) ( ) ( ) ( )

1 1( | , ) ( | )i i i i

t t t t tq x x y p x x Bootstrap PF

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.




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!



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)




Tempered Stable(Rachev (2011)):

Rapid Decreasing Tempered Stable:



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)



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


Other Levy processes(MJ: finite jump and VG: infinite jump but thin-tailed)



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




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)


Parameter estimation for ARMA-HN GARCH(H&S300)



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


Estimation For Levy Processes(H&S300)



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



-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



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


markets

Particle Filter

Kalman Filter

0 200 400 600 800 1000 1200 1400 1600 1800-5

0

5

10

Time steps

Levy

Jum

ps


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


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


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



Table 3 Loss Function of state variables


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:



( ) /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


Option Pricing(Loss function)



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


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)



Table 4 Loss Function of short, medium and long terms options( parentheses is non-leverage effect)


Short term—non leverage effect



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


Medium term—non leverage effect



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


Short term—with leverage effect



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


Medium term—with leverage effect



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.



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.



THANK YOU

FOR YOUR ATTENTION!

Speaker: Fumin ZHU;

Place: Stony Brook University, New York.

Date:11/08/2012



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