Victor M. Yakovenko A. Christian Silva Richard E. Prange

1. Time evolutionof the probability distribution of returns in the Heston model of stochastic volatility compared with the high-frequency stock-market data Victor M. Yakovenko A. Christian Silva Richard E. Prange Department of Physics University of Maryland College Park, MD, USA APFA-4 Conference, Warsaw, Poland, 15 November 2003

2. Mean-square variation of log-return as a function of time lag The log-return is xt = ln(S2/S1)-t, where S2 and S1are stock prices at times t2and t1, t = t2t1 is the time lag, and  is the average growth rate. 1863: Jules Regnault in “Calcul des Chances et Philosophie de la Bourse”observed t2 = xt2  t for the French stock market. See Murad Taqqu http://math.bu.edu/people/murad/articles.html 134 “Bachelier and his times”.

3. What is probability distribution Pt(x) of log-returns as a function of time lag t? 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 = vt = t. 1993: Steve Heston proposed a solvable model of multiplicative Brownian motion for xt with stochastic variancevt: Wt(1) & Wt(2) are Wiener processes. The model has 3 parameters:  - the average variance:t2 = xt2 = t.  - relaxation rate of variance, 1/ is relaxation time  - volatility of variance, use dimensionless parameter  = 2/2

4. characteristic function where is the dimensionless time. Short time: t « 1: exponential distribution For =1, it scales Long time: t » 1: Gaussian distribution It also scales Solution of the Heston model Dragulescu and Yakovenko obtained a closed-form analytical formula for Pt(x) in the Heston model: cond-mat/0203046, Quantitative Finance2, 443 (2002),APFA-3:

5. Comparison with the data Previous work: Comparison with stock-market indexes from 1 day to 1 year. Dragulescu and Yakovenko, Quantitative Finance2, 443 (2002), cond-mat/0203046; Silva and Yakovenko,Physica A324, 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

6. Cumulative probability distribution Solid lines – fits to the solution of the Heston model For very short time t ~ 5 min: Power-law(Student) For short time t ~ 30 min – several hours: exponential For long time t ~ few days: Gaussian

7. Short-time and long-time scaling Exponential Gaussian

8. From short-time to long-time scaling (t)

9. Characteristic function can be directly obtained from the data Direct comparison with the explicit formula for the Heston model:

10. Brazilian stock market index Fits 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

11. Comparison with the Student distribution The Student distribution works for shortt, but does not evolve into Gaussian for long t.

12. 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 scalingPt(x)exp(2|x|/t) and long-time scalingPt(x)exp(x2/2t2). For all times, t2 = xt2 = t. • For individual companies, the relaxation time1/ 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.