We study the exponential Ornstein-Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behavior in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these features make the model quite appealing since it appears to be more complete than other stochastic volatility models also based on a two-dimensional diffusion. We finally present an approximate solution for the return probability density designed to capture the kurtosis and skewness effects.
We adapt continuous time random walk (CTRW) formalism to describe asset price evolution and discuss some of the problems that can be treated using this approach. We basically focus on two aspects: (i) the derivation of the price distribution from high-frequency data, and (ii) the inverse problem, obtaining information on the market microstructure as reflected by high-frequency data knowing only the daily volatility. We apply the formalism to financial data to show that the CTRW offers alternative tools to deal with several complex issues of financial markets.
We present a stochastic volatility market model where volatility is correlated with return and is represented by an Ornstein-Uhlenbeck process. With this model we exactly measure the leverage effect and other stylized facts, such as mean reversion, leptokurtosis and negative skewness. We also obtain a close analytical expression for the characteristic function and study the heavy tails of the probability distribution.
Financial time series exhibit two different type of non-linear correlations: (i) volatility autocorrelations that have a very long-range memory, on the order of years, and (ii) asymmetric return-volatility (or 'leverage') correlations that are much shorter ranged. Different stochastic volatility models have been proposed in the past to account for both these correlations. However, in these models, the decay of the correlations is exponential, with a single time scale for both the volatility and the leverage correlations, at variance with observations. This paper extends the linear Ornstein-Uhlenbeck stochastic volatility model by assuming that the mean reverting level is itself random. It is found that the resulting three-dimensional diffusion process can account for different correlation time scales. It is shown that the results are in good agreement with a century of the Dow Jones index daily returns (1900-2000), with the exception of crash days.
We prove that Brownian market models with random diffusion coefficients provide an exact measure of the leverage effect ͓J-P. Bouchaud et al., Phys. Rev. Lett. 87, 228701 ͑2001͔͒. This empirical fact asserts that past returns are anticorrelated with future diffusion coefficient. Several models with random diffusion have been suggested but without a quantitative study of the leverage effect. Our analysis lets us to fully estimate all parameters involved and allows a deeper study of correlated random diffusion models that may have practical implications for many aspects of financial markets. The dynamics of particles in random media can be used for a large variety of phenomena in statistical physics and condensed matter ͓1͔. This class of models has been applied to polymer transport, electrospin dynamics of polarons, protein dynamics, and flux lines in high T c superconductors ͓1,2͔. Likewise, in modeling financial markets there also exist several approaches based on random diffusion although in mathematical finance these are known as stochastic volatility ͑SV͒ models ͓3,4͔. The aim of this paper is to stress the importance of the random diffusion approach in market dynamics by explaining the leverage effect, an old phenomenon only very recently quantified ͓5͔. In the past decade, there has been an increasing interest in applying the methods of statistical physics to the study of speculative markets ͓6͔. The present work adopts the same philosophy.The multiplicative diffusion process known as the geometric Brownian motion ͑GBM͒ has been widely accepted as a universal model for speculative markets. The model, suggested by Bachelier in 1900 as an ordinary random walk and redefined in its final version by Osborne in 1959 ͓7͔, presupposes a constant ''volatility'' , that is to say, a constant diffusion coefficient Dϭ 2 . However, and especially after the 1987 crash, there seems to be empirical evidence, embodied in the so-called ''stylized facts,'' that the assumption of constant volatility does not properly account for important features of markets ͓8͔. It is not a deterministic function of time either ͑as might be inferred by the evidence of nonstationarity in financial time series͒ but a random variable. In its more general form one therefore assumes that the volatility is a given function of a random process Y (t), i.e., (t)ϭ "Y (t)…. We may make an analogy from physics saying that speculative prices S(t) evolve in a random medium determined by a random diffusion coefficient. The randomness of the medium constrains the amplitude of price changes. It is commonly asserted that this amplitude is directly related to the market activity and the number of contracts negotiated. Hence, periods with high market activity indicate large variety of trading positions and this, in turn, implies a considerable dispersion of possible changes in prices at every time step. In the opposite case, low market activity indicates a small variety of market positions and finally, a small dispersion in possible future prices. In this sense,...
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