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We consider the structure functions S^(q)(T), i.e. the moments of order q of the increments X(t+T)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S^(q)(T)~T^z(q)). We demonstrate that the nonlinearity of the observed scaling exponent z(q) is incompatible with monofractal additive stochastic models usually introduced in finance: Brownian motion, Levy processes and their truncat
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We discuss price variations distributions in foreign exchange markets, characterizing them both in calendar and business time frameworks. The price dynamics is found to be the result of two distinct processes, a multi-variance diffusion and an error process. The presence of the latter, which dominates at short time scales, leads to indeterminacy principle in finance. Furthermore, dynamics does not allow for a scheme based on independent probability distributions, since volatility exhibits a strong correlation even at the shortest time scales.
A new approach to the understanding of complex behavior of financial markets index using tools from thermodynamics and statistical physics is developed. Physical complexity, a magnitude rooted in Kolmogorov-Chaitin theory is applied to binary sequences built up from real time series of financial markets indexes. The study is based on NASDAQ and Mexican IPC data. Different behaviors of this magnitude are shown when applied to the intervals of series placed before crashes and to intervals when no financial turbulence is observed. The connection between our results and The Efficient Market Hypothesis is discussed.
We perform a scaling analysis on NYSE daily returns. We show that volatility correlations are power-laws on a time range from one day to one year and, more important, that they exhibit a multiscale behaviour.
The dynamics of prices in financial markets has been studied intensively both experimentally (data analysis) and theoretically (models). Nevertheless, a complete stochastic characterization of volatility is still lacking. What it is well known is that absolute returns have memory on a long time range, this phenomenon is known as clustering of volatility. In this paper we show that volatility correlations are power-laws with a non-unique scaling exponent. This kind of multiscale phenomenology, which is well known to physicists since it is relevant in fully developed turbulence and in disordered systems, is recently pointed out for financial series. Starting from historical returns series, we have also derived the volatility distribution, and the results are in agreement with a log-normal shape. In our study we consider the New York Stock Exchange (NYSE) daily composite index closes (January 1966 to June 1998) and the US Dollar/Deutsch Mark (USD-DM) noon buying rates certified by the Federal Reserve Bank of New York (October 1989 to September 1998).
The method of iterated conformal maps allows to study the harmonic measure of Diffusion Limited Aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as $10^{-35}$, and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions $D_q$ are infinite for $q<q^*$, where $q^*$ is of the order of -0.2. In the language of $f(alpha)$ this means that $alpha_{max}$ is finite. The $f(alpha)$ curve loses analyticity (the phenomenon of phase transition) at $alpha_{max}$ and a finite value of $f(alpha_{max})$. We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of $q^*$ and $f(alpha_{max})$. We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.
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