A dynamical model describing stock market price distributions

Abstract

High frequency data in finance have led to a deeper understanding on probability distributions of market prices. Several facts seem to be well stablished by empirical evidence. Specifically, probability distributions have the following properties: (i) They are not Gaussian and their center is well adjusted by Lévy distributions. (ii) They are long-tailed but have finite moments of any order. (iii) They are self-similar on many time scales. Finally, (iv) at small time scales, price volatility follows a nondiffusive behavior. We extend Merton's ideas on speculative price formation and present a dynamical model resulting in a characteristic function that explains in a natural way all of the above features. The knowledge of such distribution opens a new and useful way of quantifying financial risk. The results of the model agree -with high degree of accuracy-with empirical data taken from historical records of the Standard & Poor's 500 cash index.

Scaling properties and universality of first-passage-time probabilities in financial markets

“…3 we see that the SP collapse works properly, showing the diffusive hyperbolic decay t −1/2 , only if target returns are neither very small nor very large compared to their standard deviation. A little thought shows that this is the expected behavior since, as is well known, markets are approximately Gaussian away from the tails [10,30] and the center of the distribution [30,31].…”

Predator-Prey Model for Stock Market Fluctuations

“…5 we present the outcome of a statistical analysis performed with the stationary data set of fixed-time returns R(τ ; t) = R(t + τ ) − R(t), t > 100 minutes. We check that for τ ∼ τ 0 correlations are important, Gaussian limit is not attained and skewness is observed, like in actual markets [28,29,30,31]. This phenomenon is even more noticeable when the standard deviation of fixed-time returns, a measure of the volatility of the market, is analysed, Fig.…”

“…Let us consider the following paradigmatic example with N = 1000 investors. We have set τ 0 = 10 minutes, so it is of the same order of magnitude as a typical correlation length found in actual financial data [28]. Beyond this, the rest of values were not based on actual market observations.…”

“…Since X(t) and Y (t) are anti-correlated, and the return change is sensible to the difference of those magnitudes, we expect that volatility grows faster for small time scales, and reaches the diffusive regime for τ > τ 0 . Traits of abnormal (both sub-and super-) diffusion have been reported to be present in real markets as well [28,32,33].…”

Downside Risk Analysis Applied to the Hedge Funds Universe

“…As it has been observed this appears to be an unrealistic scenario in financial markets [7,8,9,10] with important implications in risk analysis and its mean-variance framework (see for instance [11]). The situation is much more dramatic in the Hedge Fund universe since these funds are clearly non-Gaussian having wild fluctuations and strong asymmetries in price changes.…”