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. 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

Mean-square variation of log-return as a function of time lag The log-return is x t = ln(S 2 /S 1 )-  t, where S 2 and S 1 are stock prices at times t 2 and t 1, t = t 2  t 1 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 =  x t 2   t for the French stock market. See Murad Taqqu murad/articles.html 134 “Bachelier and his times”.

1900: Louis Bachelier wrote diffusion equation for the Brownian motion (1827) of stock price: P t (x)  exp(-x 2 /2vt) is Gaussian. However, experimentally P t (x) is not Gaussian, although  x t 2  = vt  Models with stochastic variance v:  x t 2  =  v  t =  t. 1993: Steve Heston proposed a solvable model of multiplicative Brownian motion for x t with stochastic variance v t : W t (1) & W t (2) are Wiener processes. The model has 3 parameters:  - the average variance:  t 2 =  x t 2  =  t.  - relaxation rate of variance, 1/  is relaxation time  - volatility of variance, use dimensionless parameter  = 2  /  2 What is probability distribution P t (x) of log-returns as a function of time lag t?

Solution of the Heston model Dragulescu and Yakovenko obtained a closed-form analytical formula for P t (x) in the Heston model: cond-mat/ , Quantitative Finance 2, 443 (2002), APFA-3: characteristic function whereis the dimensionless time. Short time:  t « 1: exponential distribution For  =1, it scales Long time:  t » 1: Gaussian distribution It also scales

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/ ; Silva and Yakovenko, Physica A 324, 303 (2003), cond-mat/ 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)  = 3.8x /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)(t)

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

Brazilian stock market index Fits to the Heston model by Renato Vicente and Charles Mann de Toledo, Universidade de Sao Paulo  = 1.4x /day = 35 %/year, 1/  = 10 days,  = 1.9

Comparison with the Student distribution 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 P t (x) for individual stocks from 15 min. to 20 days. The Heston model and the data exhibit short-time scaling P t (x)  exp(  2|x|/  t ) and long-time scaling P t (x)  exp(  x 2 /2  t 2 ). For all times,  t 2 =  x t 2  =  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 P t (x) for short t, but does not evolve into Gaussian for long t.