on trades, volume, and the martingale estimating function approach for stochastic volatility models...

8/9/2019 On Trades, Volume, And the Martingale Estimating Function Approach for Stochastic Volatility Models With Jumps

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On Trades, Volume, and the MartingaleEstimating Function Approach for Stochastic

Volatility Models with Jumps

Friedrich Hubalek (Joint work with Petra Posedel)

PRisMa 2008 One-Day Workshop on Portfolio Risk

Management, Vienna University of Technology,September 29,2008.


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Our papers

Friedrich Hubalek and Petra Posedel, Joint analysis andestimation of stock prices and trading volume inBarndorff-Nielsen and Shephard stochastic volatility models,arXiv:0807.3464 (July 2008)

Friedrich Hubalek and Petra Posedel, Asymptotic analysis fora simple explicit estimator in Barndorff-Nielsen and Shephardstochastic volatility models, arXiv:0807.3479 (July 2008)

Friedrich Hubalek and Petra Posedel, Asymptotic analysis for

an optimal estimating function for Barndorff-Nielsen-Shephardstochastic volatility models, Work in progress.


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The Barndorff-Nielsen and Shephard stochastic volatilitymodels with jumps

Logarithmic returns (discounted)

dX(t) = ( + V(t))dt +

V(t)dW(t) + dZ(t)

Instantaneous variance

dV(t) = V(t)dt + dZ(t)

W. . . Brownian motion, Z. . . subordinator,

Z(t) = Z(t) [. . . ] Parameters: R...linear drift, R. . . I t o drift,

R...leverage, > 0. . . acf parameter.


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Analytical tractability

(X(t), V(t), t 0). . . Markov, affine model (in continuoustime)

simple Riccati-type equations for characteristic resp. momentgenerating function

general solution (up to one integral)

-OU and IG-OU completely explicitly in terms of elementaryfunctions

Exploited in

Option pricing (Nicolato and Venardos)

Portfolio optimization (Benth et al.) Minimum entropy martingale measure (Benth et al.,

Rheinlander and Steiger)

Semimartingal Esscher transform (Hubalek and Sgarra)

. . .


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But statistical inference seems difficult! Bayesian, MCMC computer intensive approaches!

Barndorff-Nielsen O.E., Shephard N. (2001), Non-GaussianOrnstein-Uhlenbeck-based models and some of their uses infinancial economics.

Roberts G.O., Papaspiliopoulos O., Dellaportas P. (2004),Bayesian inference for non-Gaussian Ornstein-Uhlenbeck

stochastic volatility processes, J.E. Griffin, M.F.J. Steel (2006), Inference with non-Gaussian

Ornstein-Uhlenbeck processes for stochastic volatility

Matthew P.S. Gandera and David A. Stephens (2007),

Stochastic volatility modelling in continuous time with generalmarginal distributions: Inference, prediction and modelselection

Sylvia Fruhwirth-Schnatter and Leopold Sogner (2007?),Bayesian estimation of stochastic volatility models based on

OU processes with marginal Gamma laws.


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Discrete observations

Grid ti = i, i 0, fixed width > 0, discrete time observations

Xi = X(ti) X(ti1), Vi = V(ti)Discrete dynamics

Xi = + Yi +

YiWi + Zi, Vi = eVi1 + Ui

Auxiliary quantities (no discretization error!)

Zi = Z(ti) Z(ti1), Ui =titi1

e(tis)dZ(s)

and

Yi =

titi1

V(s)ds, Wi = 1Yi

titi1

V(s)dW(s).

(Xi, Vi, i N). . . Markov affine model (in discrete time)


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Construction and moments

Two starting points

L . . . infinitely divisible distribution on R+ subordinator Zwith Z(1)

d= L (OU-L)

D . . . self-decomposable distribution on R+ stationaryOrnstein-Uhlenbeck process V with V(t)

d= D

(D-OU)

Moments of D resp. L all (mixed, conditional, unconditional)integer moments by simple algebra (multivariate Faa di Brunoformula resp. Bell polynomials, practical calculations best byrecursions!)

E[Xni ], E[Vni ], E[X

mi V

ni ], E[X

i Vmi V

ni1],

E[Xni |Vi1], E[Vni |Vi1], E[Xmi Vni |Vi1], . . .

method of moments estimation


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Various methods of moments

Method of moments MM (Pearson 1893)

Generalized method of moments GMM (Hansen 1982)

Simulated method of moments SMM (. . . )

Efficient method of moments EMM (Gallant and Tauchen1996),

. . .

[Methods of moments for weak convergence]


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Estimation: Setting and problems

Grid, fixed width, horizon (number of observations) going toInfinity for asymptotics! (Several other possibilities. . . )

Rich, well-informed financial institutions and traders observeand trade in continous-time

Poor, academic statisticians and econometers do inferencewith daily (or less frequent!) observations

[But: High-frequence analyses . . . ]

Discrete time observations

Vi not observed, BNS becomes

non-Markovian, (a hidden Markov model)!


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Remedies

Substitute unobserved Vi model-implied Vi from optiondata, i.e., joint analysis of P and Q. Cf. Jun Pan, The Jump-Risk Premia Implicit in Options: Evidence

from an Integrated Time-Series Study (2002).

(GMM, realistic, complicated, many assumptions.)Also our long term goal!

Ignore the problem. Purely theoretical study, exhibitsmethodology, provides an upper bound for the accuracy forthis type of methods. See our first paper!

NOW: Substitute unobserved Vi by an observable proxy,volume or number of trades.


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Prices, volatility, trading intensity

Our incentive

Carl Lindberg, The estimation of the Barndorff-Nielsen andShephard model from daily data based on measures of tradingintensity. Applied Stochastic Models in Business and Industry24 (4), 2008.

Some earlier/classical references

J. M. Karpoff, The relation between price changes and tradingvolume: a survey. JFQA 22, 1987.

R.P.E. Gallant, A.R. and G. Tauchen, Stock prices andvolume, Rev.Fin.Stud. 5:199242, 1992.

K.G. Jones, C. and M.L. Lipson, Transactions, volume andvolatility. Rev.Fin.Stud. 7:631651, 1994.

G.E. Tauchen and M.Pitts, The Price Variability-VolumeRelationship on Speculative Markets Econometrica 51,(1983).


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The new variant/interpretation of the BNS models

Bold simplification/assumption: Instantaneous variance IS a(constant) multiple of the trading volume resp. number of trades.Introduce a proportionality parameter > 0. [. . . ]

Logarithmic returns

dX(t) = ( + V(t))dt +

V(t)dW(t) + dZ(t) Trading volume (or number of trades)

dV(t) = V(t)dt + dZ(t)

W. . . Brownian motion, Z. . . subordinator,Z(t) = Z(t) [. . . ]

Parameters: R...linear drift, R. . . I t o drift, > 0. . . proportionality, R...leverage, > 0...acfparameter.


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What about maximum likelihood ?

Practical issue: Bivariate Markov, known transition probability(in terms of characteristic resp. cumulant function) invertfor each observation in each iterations [Possible remedies,approximate inversions, LeCams trick,. . . ]

Theoretical issue: For infinite activity BDLP (e.g., IG-OU)

fine, for finite activity (e.g., -OU with exponential compoundPoisson BDLP)

P[V1 = ve|V0 = v] = e (no jump)

No dominating sigma-finite measure! Usual MLframework does not apply! Generalized ML (Kiefer and Wolfowitz 1956) [. . . ] Much better than

n by ad hoc (?) methods! [. . . ]


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Martingale estimating functions

E.g., (, )-OU: Parameter vector (3 + 4 = 7)

= (,,,,,,)

Moments

i = (Vi, ViVi1, V2i , Xi, XiVi1, XiVi, X

2i ), i = (Vi1, V

2i1)

Martingale estimating function

Gn() =1

n

ni=1

[i f(Vi1, )] , f(v, ) = E[1|V0 = v]

Estimator: Solve Gn() = 0 ! Sample moments

n =1

n

ni=1

i, n =1

n

ni=1

i,


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Consistency

The basic (and only!) assumption: V0 self-decomposable rv on R+with

E[Vn0 ] < n N.The basic convergence result

1

n

n

i=1

Xpi V

qi V

ri1

a.s.

E[Xp

1

Vq

1

Vr0 ]

p, q, rN.

Remark: Ergodicity vs. simple proof. Martingale differences uncorreclated elementary convergence result.

TheoremWe have P(Cn) 1 and the estimator n is consistent on Cn,namely

nICna.s. 0

as n

.


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Asymptotic normality delta method

Explicit estimator Delta-Method applies: Sample moments

(n, n)D

N(0, )estimator

n = h(n, n)

result n(n 0) D N(0, T) T = JJ

Jacobian J = h. Messy. Better: General framework (implicit function theorem)

Michael Srensen, Statistical inference for discretely observeddiffusions, Lecture Notes, Berlin, 1997. Michael Srensen, On asymptotics of estimating functions,

Brazil. J. Prob. Stat. (1999).

Also when estimating functions Gn() explicit, but estimator

n is not [. . . optimal estimating functions].

f


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Asymptotic normality general framework

Basic result: asymptotic normality of estimating functions

1n

Gn(0)D N(0, ), = E[Var[1]|V0]

Proof by multivariate martingale central limit theorem.

TheoremThe estimator nICn is asymptotically normal, namely

n

n 0 D N(0, T), T = A1(A1)

as n , with Jacobian A = E[f(V0, 0)]. Recall f(v, ) = E[1|V0 = v] and E = E0 . Matrices A and simple, explicit, (slightly lengthy).

Fi i l f h ll d i l i


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Finite sample performance the controlled simulationexperiment

-OU: Volume V(t)

(, ) stationary, BDLP Z compoundPoisson, intensity exponential jumps with mean 1/.

Parameter values (annual, 250 trading days)

= 6.17, = 1.42, = 177.95,

= 0.015, = 0.00056, = 0.435, = 0.087. BDLP: 4.4 jumps per day (interesting pieces of news

arriving?), each jump with mean and stddev 0.704.

Volume (in Mio): Stationary mean 4.35, variance 0.033Volatility 18%. ACF half-life 1 day.

Log returns: Mean -6.5%, volatility 18%.

Experiments: n=2500 (10 years), n = 8000 (32 years,theoretical check).

Si l d h 1


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Simulated paths 1

Volume

0 500 1000 1500 2000 25000

5

10

15

t

Volatility

0 500 1000 1500 2000 25000.05

0.1

0.15

0.2

0.25

0.3

0.35

Volatility

t

Si l t d th 2


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Simulated paths 2

Returns

0 500 1000 1500 2000 25000.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

Xt

A t ti f


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Asymptotic performance

True values = (,,,,,,)

= (6.17, 1.42, 177.95, 0.015, 0.00056, 0.435, 0.087) Asymptotic stddev s/

n

s = (12.0, 2.8, 440, 9.0, 2.6, 0.066, 0.007)

Asymptotic correlation r

r =

1 0.9 0.6 0.007 0.05 0.006 0.0030.9 1 0.6 0.007 0.05 0.01 0.0040.6 0.6 1 0.01 0.09 0.0006 0.00

0.007 0.008 0.01 1 0.8 0.01 0.030.05 0.05 0.09 0.8 1 0.01 0.5

0.006 0.01 0.0006 0.01 0.01 1 0.0050.003 0.004 0.00 0.03 0.5 0.005 1

Bigr

in AR(1)-part! Optimal estimating function.

Histog a s 1000 e licatio s each 2500


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Histograms: m = 1000 replications, each n = 2500observations, volume parameters

5.5 6 6.5 70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

nu

1.2 1.3 1.4 1.5 1.6 1.70

1

2

3

4

5

6

7

8

alpha

150 160 170 180 190 200 2100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

lambda

Histograms : m 1000 replications each n 2500


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Histograms : m = 1000 replications, each n = 2500observations, return parameters

0.25 0.2 0.15 0.1 0.05 0 0. 05 0.1 0. 15 0.20

1

2

3

4

5

6

7

8

beta

10 8 6 4 2 0

x 104

0

500

1000

1500

2000

2500

3000

3500

rho

0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

mu

0.082 0.084 0.086 0.088 0.09 0.092 0.0940

50

100

150

200

250

300

sigma

A first empirical analysis data


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A first empirical analysis data

Closing price and volume

IBM: March 23, 2003 March 23, 2008 [NYSE], 1259

observations MSFT: April 11, 2003 Feb 4, 2008 [Nasdaq], 1212

observations

IBM data


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IBM dataPrice

2004 2005 2006 2007 200870

80

90

100

110

120

130

Volume

2

4

6

8

10

12

14

16

18

20

MSFT data


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MSFT dataPrice

2004 2005 2006 2007 200820

22

24

26

28

30

32

34

36

38

Volume

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Estimation results


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Estimation results

IBM stddev 6.17 0.339 1.42 0.079

177.95 12.509

0.435 0.254

-0.015 0.072 0.087 0.002 -0.00056 0.0002

MSFT stddev 4.496 0.247 67.895 3.773

201.99 14.420

0.4162 0.265

-0.464 5.059 0.81 0.018 -0.025 0.013

Interpretation: [. . . ]

Unconditional return distributions


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Unconditional return distributionsTheoretical BNS (dashed) versus kernel estimates (solid)

0.1 0.08 0.06 0.04 0.02 0 0.02 0.04 0.060

5

10

15

20

25

30

35

40

45

0.15 0.1 0.05 0 0.05 0.10

5

10

15

20

25

30

35

40

45

Log densities

8 6 4 2 0 2 4 625

20

15

10

5

0

8 6 4 2 0 2 4 635

30

25

20

15

10

5

0

Autocorrelation function (volume)


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Autocorrelation function (volume)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Autocorrelation for variance

ACF IBM

estimated theoretical ACF

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

1.2

Autocorrelation for variance

ACF MSFT

estimated theoretical ACF

BNS with Superposition of OU-processes [ . . . ]

Model fit residual analysis


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Model fit residual analysis

Volume: Usual (and exact) AR(1) analysis, though with funnyinnovations (Ui) iid,

Vi e

= Ui, Ui =titi1 e

(tis)

dZ(s)

Returns: Not exact (?), Euler approximation

. . .

Further developments and directions 1


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Superposition

V(t) = w1V1(t)+ +wmVm(t), dVi(t) = iVi(t)dt+dZi(it)

(X, V1, . . . , Vm) Markov affine Observations? V1. . . common factor (market volume,. . . )

V2. . . idiosyncratic factor (asset volume,. . . )

V3... ? (similar asset? ...?)



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p

Number of trades (Lindberg!) Optimal martingale estimating functions

Gn() =

ni=1

B(, Vi1) [i f(, Vi1)) f(, v) = E[i|Vi1 = v]

Comparison with ML and related methods (for infinite activity)

Comparison with GMM

Hybrid approaches

Other moments (trigonometric, c.f., Singleton, . . . )

Other time-scales (!!!)

Integrated analysis for asset and derivatives

on trades, volume, and the martingale estimating function approach for stochastic volatility models...

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