Time evolution of the probability distribution of returns in the Heston

model of stochastic volatility compared with the high-frequency

stock-market data

Victor M. YakovenkoA. Christian SilvaRichard E. Prange

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

APFA-4 Conference, Warsaw, Poland, 15 November 2003

Mean-square variation of log-return as a function of time lag

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

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”.

1900: Louis Bachelier wrote diffusion equation for the Brownian motion (1827) of stock price: Pt(x) exp(-x2/2vt) is Gaussian.

However, experimentally Pt(x) is not Gaussian, although xt2 = vt

Models with stochastic variance v: xt2 = vt = t.

1993: Steve Heston proposed a solvable model of multiplicative Brownian motion for xt with stochastic variance vt:

( );12tv

t t tdx dt v dW ( )( ) 2t t t tdv v dt v dW

Wt(1) & Wt

(2) are Wiener processes. 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

What is probability distribution Pt(x) of log-returns as a function of time lag t?

Solution of the Heston modelDragulescu and Yakovenko obtained a closed-form analytical formula for Pt(x) in the Heston model: cond-mat/0203046,Quantitative Finance 2, 443 (2002), APFA-3:

( )( ) ,2tF kikxdk

tP x e e

characteristic function ( )( ) tF k

tP k e

( ) ln cosh sinh ,2 1

2 2 2 2t t t

tF k ( ) ,2

01 kx 0x

where t t is the dimensionless time.

Short time: t « 1: exponential distribution For =1, it scales

( ) exp2

tP x xt

( ) /t tP x f x

Long time: t » 1: Gaussian distributionIt also scales

( ) exp2

2t

xP x

t

( ) /t tP x g x

Comparison with the data

Previous work: Comparison with stock-market indexes from 1 day to 1 year. Dragulescu and Yakovenko, Quantitative Finance 2, 443 (2002),cond-mat/0203046; Silva and Yakovenko, Physica A 324, 303 (2003), cond-mat/0211050.

New work: Comparison with high-frequency data for several individual companies from 5 min to 20 days. The plots are for Microsoft (MSFT).Silva, Prange, and Yakovenko (2003) = 3.8x10-4 1/day = 9.6 %/year, 1/=1:31 hour, =1

Cumulative probability distribution

For short time t ~ 30 min – several hours:exponential

Solid lines – fits to the solution of the Heston model

For very short time t ~ 5 min:Power-law (Student)

For long time t ~ few days: Gaussian

Short-time and long-time scaling

GaussianExponential

From short-time to long-time scaling

(t)

Characteristic function can be directly obtained from the data

( )

Re t

t

t

ikx

x

P k

e

Direct comparisonwith the explicitformula for theHeston model:

( )( ) tF ktP k e

Brazilian stock market indexFits to the Heston model by Renato Vicente and

Charles Mann de Toledo, Universidade de Sao Paulo

= 1.4x10-3 1/day = 35 %/year, 1/ = 10 days, = 1.9

Comparison with the Student distribution

1

2 2 2

1( )

1 /t d

t

P x

x

The Student distribution works for short t, but does not evolve into Gaussian for long t.

Conclusions• The Heston model with stochastic variance well describes

probability distribution of log-returns Pt(x) for individual stocks from 15 min. to 20 days.

• The Heston model and the data exhibit short-time scaling Pt(x)exp(2|x|/t) and long-time scaling Pt(x)exp(x2/2t

2). For all times, t

2 = xt2 = t.

• For individual companies, the relaxation time 1/ is of the order of hours, but, for market indexes, 1/ is of the order of ten days.

• The Heston model describes Brazilian stock market index from 1 min. to 150 days.

• The Student distribution describes Pt(x) for short t, but does not evolve into Gaussian for long t.