1. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 1 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)

2. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 2 Double-exponential (Laplace) distribution of log-returns: An overlooked stylized fact Detrended log-return is defined as x t = ln(S 2 /S 1 )-  t, where S 2 and S 1 are the stock prices at times t 2 and t 1, t = t 2  t 1 is the time lag (horizon), and  is the mean growth rate. We study the probability distribution P t (x) of log-returns x after the time lag t. A simple multiplicative Brownian motion gives the Gaussian distribution P t (x)  exp(-x 2 /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: P t (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: P t (x)  exp(-|x|/c  t). 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.

3. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 3 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 DAX t = 1 hour t = 1 day Indian stocks t = 1 day Poland

4. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 4 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 index 1990 – 2002 t = 1 day

5. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 5 Laplace distribution for short time lags P t (x)  exp(-|x|/  t ): tent-shape, double- exponential, Laplace distribution P t (x) rescales when plotted vs. the normalized log-return x/  t, where  t 2 =  x t 2  For x>0, we plot and 1-CDF for x<0.

6. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 6 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 =  x t 2   t for the French stock market. See Murad Taqqu http://math.bu.edu/people/ murad/articles.html 134 “Bachelier and his times”.

7. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 7 Time evolution of P t (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 P t (x) from the double-exponential shape exp(-|x|) for short t to the Gaussian shape exp(-x 2 ) for long t.

8. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 8 Short-time and long-time scaling GaussianExponential

9. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 9 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

10. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 10 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 P t (x) for financial markets and for hydrodynamic turbulence.

11. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 11 Stock price S t is not a simple Markov process A Markov process satisfies the Chapman-Kolmogorov equation: 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) The Chapman-Kolmogorov equation demands We find that the stock-market data do not satisfy this condition. It is important to describe the whole family of distributions P t (x) for a range of t, not just for one t, such as 1 day.

12. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 12 Suppose the variance v t =  t 2 of a random walk x depends on time. Let us introduce the integrated variance V t, which acts as the effective time: For stochastic variance, we use subordination (Feller’s book; P.K.Clark 1973): where K t (V) is the probability density of V for the time lag t. Models with stochastic volatility The Fourier and Laplace transforms of P t (x) and K t (V) are simply related: Markov processes x t can be written in the subordinated form, but stochastic volatility models can also describe non-Markovian P t (x)

13. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 13 Steve Heston in Review of Financial Studies 6, 327 (1993) proposed a model where the stochastic variance v t follows the mean-reverting Feller or Cox-Ingersoll-Ross (CIR) process: where W t is a Wiener process. 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 Heston model with stochastic volatility Solving the corresponding Fokker-Planck equation, we obtain where k V is the Laplace variable conjugate to V, v i is the initial variance, and  (v i ) is the stationary probability distribution of v i.

14. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 14 For short times  t « 1: Laplace distribution For  =1, it also scales For long times  t » 1: Gaussian distribution It scales as Solution of the Heston model Solution of the Heston model qualitatively agrees with the empirical data on time evolution of P t (x). Notice that is non-Markovian. We set .

15. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 15 Number of trades as stochastic variance Mandelbrot & Taylor (1967) suggested that the integrated variance V t may be associated with the random number of trades N t during the time lag t: V t =  N t. This scenario is related to the continuous-time random walk (CTRW) proposed by Montroll and Weiss (1965). Trades happen at random times t i, 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 P N (x)  exp(-x 2 /2  N)? 2)What is the probability density K t (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 P N (x). We study large N and focus on the central part of the distribution.

16. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 16 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)

17. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 17 Discrete price changes for short time horizon Discrete price change m N = (S n  S n-N )/h, where h=1$/64 is the quantum, vs. continuous return x N = lnS n  lnS n-N  (S n  S n-N )/S n = m N h/S n smeared by S n. P N (m) linear scale N=1 (1.5 sec) N=4000 (99 min) P N (m) in log-linear scale Derivative dP N (m) / dm

18. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 18 Discrete price changes for short time horizon Blue line: price change m N =  S/h Black line: return x N =  S/S

19. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 19 Discrete price changes for short time horizon CDF t (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.

20. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 20 Variance of returns and the number of trades Let us verify for mesoscopic time horizons. We find:

21. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 21 Gaussian distribution for P N (x) Verifying formula P N (x)  exp(-x 2 /2  N) The Gaussian distribution works for, at least, 85% of probability (1.5 standard deviation). P N (x) is certainly more Gaussian than P t (x).

22. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 22 PDF K t (  N) for the number of trades N during the time t is obtained for Heston model by inverse Laplace transform of Comparing K t (N) with the Heston model

23. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 23 Comparing P t (x) with the Heston model Verifying for the Heston model.

24. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 24 Conclusions  Probability distribution of log-returns x for mescoscopic time lags t is subordinated to the number of trades N: P t (x) =  dN exp(  x 2 /2  N) K t (N). P N (x) is Gaussian, and K t (N) is given by the Heston model with stochastic volatility. The stochastic process x t is a continuous-time random walk (CTRW).  The data and the Heston model exhibit the double-exponential (Laplace) distribution P t (x)  exp(  |x|  2/  t ) for short time lags and the Gaussian distribution P t (x)  exp(  x 2 /2  t 2 ) for long time lags. For all times,  t 2 =  x t 2  =  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?

25. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 25 From short-time to long-time scaling

26. Stochastic volatility as the fluctuating rate of tradingVictor Yakovenko 26 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