Fractional calculus and continuous-time finance

Abstract

In this paper we present a rather general phenomenological theory of tick-by-tick dynamics in financial markets. Many well-known aspects, such as the L\'evy scaling form, follow as particular cases of the theory. The theory fully takes into account the non-Markovian and non-local character of financial time series. Predictions on the long-time behaviour of the waiting-time probability density are presented. Finally, a general scaling form is given, based on the solution of the fractional diffusion equation.Comment: 11 pages, no figures, LaTeX2e, submitted to Physica

Short note on the emergence of fractional kinetics

“…Let p(r, t) be the pd f for a particle to be at r at the time t. Moreover, let λ(δr) be the pd f for a particle to make a jump of length δr after a waiting time τ whose pd f is denoted by ψ(τ). Since the integral τ 0 ψ(ξ) dξ represents the probability that at least one step is made in the temporal interval (0, τ) [8,18], the probability that a given waiting interval between two consecutive jumps is greater than or equal to τ…”

“…holds [8,18]. Hence Ψ(t) is the probability that, after a jump, the diffusing quantity does not change during the temporal interval of duration τ and it is the survival probability at the initial position [6].…”

Response to “Worrying Trends in Econophysics”

“…Papers have been published by very respectable finance researchers claiming market correlations on the basis of a Hurst exponent H≠1/2, see e.g. [29], but a Hurst exponent H≠1/2 does not imply long time correlations unless it has been first established that the time series has stationary increments [27] (see also [30,31] for a discussion of the scaling dynamics and lack of autocorrelations in Levy models). These different ideas are usually confused together into a thick soup by physicists and economists alike.…”

Power law for the calm-time interval of price changes

“…Several researchers have recently investigated the statistical properties of waiting times of high-frequency financial data [17][18][19][20][21][22][23][24], and Scalas et al [17][18][19][20][21] in particular have applied the theory of continuous time random walk (CTRW) to financial data. They also found that the waiting-time survival probability for high-frequency data of the 30 DJIA stocks is non-exponential [21].…”