stochastic volatility as the fluctuating rate of tradingvictor yakovenko 1 stochastic volatility as...

1Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko

Stochastic volatility as the fluctuating rate of trading: Comparison with the Heston model

Victor M. Yakovenko

Department of Physics, University of Maryland, College Park, USA

http://www2.physics.umd.edu/~yakovenk/econophysics.html

Publications

• A. A. Dragulescu and V. M. Yakovenko, Quantitative Finance 2, 443 (2002)APFA-3 London 2001 (Distribution of log-returns in the Heston model)

• A. C. Silva, R. E. Prange, and V. M. Yakovenko, Physica A 344, 227 (2004)APFA-4 Warsaw 2003 (Double-exponential Laplace distribution)

• A. C. Silva, Ph.D. Thesis (2005), Chapter V, physics/0507022

• A. C. Silva and V. M. Yakovenko, in preparation, APFA-5 Turin 2006(Heston model as subordination to the stochastic number of trades)


Double-exponential (Laplace) distribution of log-returns: An overlooked stylized fact

Detrended log-return is defined as xt = ln(S2/S1)-t, where S2 and S1 are the stock prices at times t2 and t1, t = t2t1 is the time lag (horizon), and is the mean growth rate.

We study the probability distribution Pt(x) of log-returns x after the time lag t.

A simple multiplicative Brownian motion gives the Gaussian distributionPt(x) exp(-x2/2vt), which does not agree with the data.

There are two aspects of discrepancy between the data and the Gaussian:

1. The tails of the distribution (about 1% of probability) follow a power law:Pt(x) 1/|x|a for large |x| - the Pareto law.

2. The central part of the distribution (about 99% of probability) follows a double-exponential (Laplace) law: Pt(x) exp(-|x|/ct).

The double-exponential law is a ubiquitous, but largely ignored stylized fact, because most studies focus on the tails and make plots in the log-log scale.The Laplace law becomes obvious in the log-linear scale.


Poland: D. Makowiec, Physica A 344, 36 (2004)

Germany: R. Remer and R. Mahnke, Physica A 344, 236 (2004)

India: K. Matia, M. Pol, H. Salunkay, and H. E. Stanley, Europhys. Lett. 66, 909 (2004)

Double-exponential distribution around the world

German DAXt = 1 hour

t = 1 day

Indian stocks

t = 1 dayPoland


Double-exponential distribution around the world

J. L. McCauley and G. H. Gunaratne, Physica A 329, 178 (2003)

Taisei Kaizoji, Physica A 343, 662 (2004)

US bonds

t = 4 hours

Japanese Yen

t = 1 hour

Deutsche Mark

t = 0.5 hour

Japanese Nikkei 225 index1990 – 2002

t = 1 day


Laplace distribution for short time lags

Pt(x)exp(-|x|/t):tent-shape, double-exponential, Laplacedistribution

Pt(x) rescales when plotted vs.the normalized log-return x/t, where t

2 = xt2

For x>0, we plot

( ) ( )CDFt t

x

x P x dx

and 1-CDF for x<0.


Mean-square variation of log-return

as a function of time lag

1863: Jules Regnault in “Calcul des Chances et Philosophie de la Bourse” observed t

2 = xt2 t for the

French stock market.

See Murad Taqqu http://math.bu.edu/people/murad/articles.html 134 “Bachelier and his times”.


Time evolution of Pt(x)

Dow Jones data for time lags from 1 day to 1 year.

Microsoft (MSFT) data for time lags from 5 min to 20 days.

The solid lines show a fit to the Heston model (to be discussed later).

The data points show evolution of Pt(x) from the double-exponential shape exp(-|x|) for short t to the Gaussian shape exp(-x2) for long t.


Short-time and long-time scaling

GaussianExponential


Science 284, 87 (1999)

J.E. Guilkey, A.R. Kerstein, P.A. McMurtry, J.C. Klewicki, Phys. Rev. E 56, 1753 (1997)

Turbulent pipe flow


R. Friedrich and J. Peinke, Phys. Rev. Lett. 78, 863 (1997)

Turbulent jet

Hydrodynamic turbulence

S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, Y. Dodge, Nature 381, 767 (1996)

There is amazing similarity between Pt(x) for financial markets and for hydrodynamic turbulence.


Stock price St is not a simple Markov processA Markov process satisfies the Chapman-Kolmogorov equation:

2 1 2 1 2 1 2 1, ( ) ( ) ( )t t t t t t t tP P P P x dx P x x P x

This is true for the Gaussian distribution: Normal * Normal = Normal

But not true for the Laplace distribution: Exp * Exp Exp

It is useful to introduce the characteristic function (the Fourier transform)

( ) ( ) x x i

i

ik x ik xt x t xP k dx P x e e

The Chapman-Kolmogorov equation demands

2 1 2 1

( ) log ( )( ) ( ) ( ) ( ) , ( )t t x t x t x t x x

t f kx P kxttP k P k P k P k e f k

We find that the stock-market data do not satisfy this condition.

It is important to describe the whole family of distributions Pt(x) for a range of t, not just for one t, such as 1 day.


Suppose the variance vt=t2 of a random walk x depends on time. Let

us introduce the integrated variance Vt, which acts as the effective time:

For stochastic variance, we use subordination (Feller’s book; P.K.Clark 1973):

2 / 2

0, t

t x Vt t tV dt P x e v

2 / 2

0( ) ( ) ,x Vt tP x e K V dV

where Kt(V) is the probability density of V for the time lag t.

Models with stochastic volatility

The Fourier and Laplace transforms of Pt(x) and Kt(V) are simply related:

2 / 2 2

0( ) ( ) , ( ) ( / 2)xVkt x t t x t xP k e K V dV P k K k

Markov processes xt can be written in the subordinated form, but stochastic volatility models can also describe non-Markovian Pt(x)


Steve Heston in Review of Financial Studies 6, 327 (1993) proposed a model where the stochastic variance vt follows the mean-reverting Feller or Cox-Ingersoll-Ross (CIR) process:

( )t t t td dt dW v v v

where Wt is a Wiener process. The model has 3 parameters: - the average variance: t

2 = xt2 = t.

- relaxation rate of variance, 1/ is relaxation time - volatility of variance, use dimensionless parameter = 2/2

Heston model with stochastic volatility

Solving the corresponding Fokker-Planck equation, we obtain

( | ) ( ) ( | ) ( ) ,andt V i t V t V i i iK k K k K k d v v v v

where kV is the Laplace variable conjugate to V, vi is the initial variance, and (vi) is the stationary probability distribution of vi.


/

( ) ,cosh s nh

,i

2

2

12 2 2

21 2

t

Vt Vt t

ekk t tK

For short times t « 1: Laplace distribution For =1, it also scales

( ) exp2

tP x x

t ( ) /t tP x f x

For long times t » 1: Gaussian distributionIt scales as

( ) exp2

2t

xP x

t

( ) /t tP x g x

Solution of the Heston model

Solution of the Heston model qualitatively agrees with the empirical data on time evolution of Pt(x).

Notice that is non-Markovian. We set .2( ) ( / 2)

t x t xP k K k


2 / 2

0 0( ) ( ) ( ) ( )x Nt N t tP x P x K N dN e K N dN

Number of trades as stochastic varianceMandelbrot & Taylor (1967) suggested that the integrated variance Vt may be

associated with the random number of trades Nt during the time lag t: Vt = Nt .

This scenario is related to the continuous-time random walk (CTRW) proposed by Montroll and Weiss (1965). Trades happen at random times ti, so we expect

1) After a fixed number of trades N (as opposed to a fixed time t), does the probability of returns x follow the Gaussian PN(x) exp(-x2/2N)?

2) What is the probability density Kt(N) to have N trades during the time t? These questions can be answered using tick-by-tick data.

There is some evidence in favor of (1) – Stanley et al., Phys. Rev. E 62, R3023 (2000), Ané & Geman, J. Finance 55, 2259 (2000).

It was disputed by Farmer et al., physics/0510007. However, they focused on the relatively small N and on the power-law tails of PN(x).

We study large N and focus on the central part of the distribution.


Three ranges for time horizon

1) Microscopic (atomic) range – up to ~30 minutes. It is dominated by discrete transitions.

2) Mesoscopic (diffusive) range – from ~30 minutes to days and weeks. Here continuous stochastic description makes sense.

3) Macroscopic (hydrodynamic) range – months-years-decades. It is dominated by macroeconomics: expansion vs. recession.

The Heston model is applicable only in the mesoscopic range and is compatible with CTRW and subordination formalisms.

Some other recent studies of CTRW in finance:

• Enrico Scalas et al. (2000-2006)• Peter Richmond et al. (2002-2004)• Jaume Masoliver et al. (2003-2006)• I.M. Dremin and A.V. Leonidov (2005)


Discrete price changes for short time horizonDiscrete price change mN = (SnSn-N)/h, where h=1$/64 is the quantum, vs. continuous return xN = lnSnlnSn-N (SnSn-N)/Sn = mNh/Sn smeared by Sn.

PN(m) linear scaleN=1 (1.5 sec)N=4000 (99 min)

PN(m) in log-linear scale

Derivative dPN(m) / dm


Discrete price changes for short time horizon

Blue line:price changemN=S/h

Black line:return xN=S/S


Discrete price changes for short time horizon

CDFt(x) for t = 5 min J.D. Farmer et al., Quant. Finance 4, 383 (2004)

For short time horizons (less than ~30 min), price changes and returns are dominated by discrete structure. Continuous (diffusive) models are not applicable at this microscopic scale.


Variance of returns and the number of trades2 / 2

0( ) ( )x Nt tP x e K N dN

Let us verify

for mesoscopic time horizons. We find:

, , ,2 2

tN tx N N t x t


Gaussian distribution for PN(x)

Verifying formula PN(x) exp(-x2/2N)

The Gaussian distribution works for, at least, 85% of probability (1.5 standard deviation). PN(x) is certainly more Gaussian than Pt(x).


PDF Kt(N) for the number of trades N during the time t is obtained for Heston model by inverse Laplace transform of

Comparing Kt(N) with the Heston model

/

( ) ,cosh si h

,n

2

2

12 2 2

21 2

t

t V Vt t

etkk tK


Comparing Pt(x) with the Heston model

2 / 2

0( ) ( )

x Nt tP x e K N dNVerifying for the Heston model.


Conclusions Probability distribution of log-returns x for mescoscopic time

lags t is subordinated to the number of trades N: Pt(x) = dN exp(x2/2N) Kt(N).

PN(x) is Gaussian, and Kt(N) is given by the Heston model with stochastic volatility. The stochastic process xt is a continuous-time random walk (CTRW).

The data and the Heston model exhibit the double-exponential (Laplace) distribution Pt(x)exp(|x|2/t) for short time lags and the Gaussian distribution Pt(x)exp(x2/2t

2) for long time lags. For all times, t

2 = xt2 = t.

The double-exponential distribution, found for many markets, should be treated as a stylized fact, besides the power laws, clustered volatility, etc.

Probability distributions for hydrodynamics turbulence look amazingly similar to those for financial markets – universality?


From short-time to long-time scaling


Brazilian stock market index IBOVESPA

R. Vicente, C. M. de Toledo, V. B. P. Leite, and N. Caticha, Physica A 361, 272 (2006)

Fits to the Heston model

stochastic volatility as the fluctuating rate of tradingvictor yakovenko 1 stochastic volatility as...

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