Realized Volatility Analysis in A Spin Model of Financial MarketsTetsuya Takaishi

Hiroshima University of Economcis

H-PTh-5

Stylized properties of asset returns

Absence of autocorrelations

Slow decay of autocorrelation in absolute

returns

Fat-tailed (heavy tail) distributions

Volatility clustering

Leverage effect

Volume/volatility correlation

…..

Introduction

Gopikrishnan et. al(1999)

Fat-tailed return distribution

Gopikrishnan et al.,cond-mat/9905305

Fat-tailed

Oct. 19 1987

Gopikrishnan et al.,cond-mat/9905305

price return

volatility clustering

How can we explain the stylized facts?

Some of the stylized facts are explained by the mixture of distributions hypothesis (MDH)

tttR N(0,1) ～　t

Returns are described by Gaussian with time-varying volatility

Clark(1976)

Absence of autocorrelationsSlow decay of autocorrelation in absolute returnsFat-tailed (heavy tail) distributions

Consistent

t Volume, # of transactions

( # of information )

0 tttttttttt EEErrE

0))(( 2 cEEcrcrE tttttt trEc

Absence of autocorrelations

Non-vanishing autocorrelation in absolute returns

tttr 0trE

Not independent

ttE Slow decay(long autocorrelation time)

We assume

)2

exp()2()|(2

22/122

t

tt

rrP

22

0

2 )|()()( ttt drPPrP

)2/()(ln 221)(

th

t

t eh

hP

th

tt ehhP/1

)(

Unconditional return distribution

Lognormal distribution

Inverse gamma distribution

2

tth

Clark(1973)

Paretz(1972)

Volatility distribution

Student-t distribution

We do not know the form of volatility distributions

Fat-tailed distributions

tttR

t

t

tR

Standardized returns will be Gaussian variables with mean 0 and variance 1

N(0,1) ～　t

Volatility is unobserved in the markets.

Volatility is estimated by using high-frequency data.

t

t

t

RV

R

2/1

Normality of standardized returns has been examined

Variance =1 Kurtosis=3 ?

Consistency check of the MDH

Andersen et al. (2000) :Exchange Rate Returns

Fleming & Paye (2011)：Stock Returns

Andersen et al. (2001),(2007),(2010) :Stock Returns

Argents are assigned to spins

A dynamics is introduced and spins interact each other

Various models have been proposed

Cont and Bouchaud (2000)Stauffer and Penna (1998)Iori (1999)Sznajd-Weron and Weron (2002)Sanchez (2002)Bornholdt (2001)Takaishi (2005)etc

These models appear to show some of the stylized facts

Volatility clusteringFat-tailed distributionetc

Spin financial models

In this study we examine the MDH for the return dynamics in a spin financial markets(Bornholdt Model)

MDH?

)()()(ln tdWttpd

N

i

kiTtt rRV1

2

*

dsstt

TtT )()( 22

Integrated volatility (IV) for T period

IV

Realized Volatility

)(ln)(ln kipipri

Andersen, Bollerslev (1998)

Let us assume that the logarithmic price process follows

a stochastic diffusion as

Realized volatility is defined by a sum of

squared finely sampled returns.

return calculated using high-frequency data

t

TN

Δｔ: sampling period

)0( t

W(t): Standard Brownian motion

volatility spot:)(t

2/)3(2

1)2/)1((

)2/()(

k

k

y

kk

kyP

Peters and De Vilder (2006)

Standardized return distribution

kmkkkyE mm 42/112)( 22

Moments

)2

21(3

2

3)( 4

kk

kyE1)( 2 yE

Variance Kurtosis

Gaussian

k

k= # of samples

Finite-Sample Effect

5min (24)

30min (4)

Standardized return distribution（stock data）

Spin model

iSjS

iS

S.Bornholdt, Int.J.Mod.Phys.C12(2001) 667

takes +1 or -1

Buy Sell

We may assign +1 state to “Buy order” and

-1 state to “Sell order”

Agents live at sites on an n-dimensional lattice

Each site has a spin.

Two states spin model

(In this study we use 2-dimensional lattice.)

)(1

)( tSn

tMj

j

n

j

ijiji tMtStSJth1

|)(|)()()(

Local interaction

)))(2exp(1/(1 1)1( thptS ii

ptSi -1 1)1(

Spins are updated by the following probability

Global interaction

Local interaction: Majority effect

Global interaction: Minority effect

Difference between buy and sell orders

0

)(1

)( tSn

tMj

j

Simulation Study

2/)1()()( tMtMtr

Return distributions Cumulative return distributions

L=125 β=1.8 α=22

Realized volatility at Δt=1 update

One sweep (one day) = 125x125 updates

No finite-sample effect

Variance of standardized returns

Finite-sample effect is seen

kmkAk m 42/2

Fitting function

tNk /

Moments of standardized returns

variance kurtosis 6th 8th 10th

theory 1 3 15 105 945

Spin model 1.002(9) 2.96(3) 14.72(4) 102.8(4) 926(5)

Fitting results

Moments of the standardized returns are very close to the values expected for normal variables

Consistent with the MDH

Conclusion

Spin financial models can reproduce some stylized facts of financial markets.

Some stylized facts can be explained by the MDH. To examine the MDH in the spin financial market, we calculated the realized volatility and then calculated moments of the returns standardized by the realized volatility.

We confirmed that the moments of the standardized returns receive the finite-sample effect. However the values in the limit of Δt=1 are very close to the theoretical values for normal variables. Therefore it is concluded that the return dynamics of the spin financial market is consistent with the MDH

References

[1]R.Cont, Quantitative Finace 1 (2001) 223.[2]PK.Clark, Econometrica 41(1973) 135.[3]T.Takaishi, Evol. Inst. Econ. Rev. 7(1) (2010) 89.[4]T.Bollerslev, Journal of Econmetrics 31 (1986) 307[5]T.G.Andersen and T.Bollerslev, Int. Econ.Rev.39(1998)885.[6]T.Bornholdt, Int. J. Mod. Phys. C 12 (2002) 667[7]T.Takaishi, Int. J. Mod. Phys. C 16 (2005) 1311[8]R.T.Peters and R.G. De Vilder, J. of Bus. ¥& Econ. Stat. 24 (2006) 444.[9]J.Fleming and B.S.Paye, J. of Econ. 160 (2011) 119.[10]T.Takaishi, T.T.Chen and Z.Zhen, Prog. Theor. Phys. Suppl. 194 (2012) 43.[11]T.Takaishi, Procedia - Social and Behavioral Sciences 65 (2012) 968.