ON CONTINUOUS-TIME RANDOM WALKS IN FINANCE Enrico Scalas (1) with: Maurizio Mantelli (1) Marco Raberto (2) Rudolf Gorenflo (3) Francesco Mainardi (4) (1) DISTA, Università del Piemonte Orientale, Alessandria, Italy. (2) DIBE, Università di Genova, Italy. (3) Erstes Matematisches Institut, Freie Universität Berlin, Germany. (4) Dipartimento di Fisica, Università di Bologna, Italy.

Summary Theory Empirical Analysis Conclusions In financial markets, not only prices and returns can be considered as random variables, but also the waiting time between two transactions varies randomly. In the following, we analyze the DJIA stocks traded in October The empirical properties of these time series are compared to theoretical prediction based on a continuous time random walk model. Outline

Tick-by-tick price dynamics

Theory (I) Continuous-time random walk in finance (basic quantities) : price of an asset at time t : log price : joint probability density of jumps and of waiting times : probability density function of finding the log price x at time t

Theory (II) Master equation Jump pdf Waiting-time pdf Permanence in x,t Jump into x,t Factorisation in case of independence: Survival probability

Theory (III) Fractional diffusion For a given joint density, the Fourier-Laplace transform of is given by: where: is the waiting time probability density function. Assumption (asymptotic scaling and independence): Caputo fractional derivative Riesz fractional derivative

Theory (IV) Waiting-time distribution Simple assumption (compatible with asymptotic independence): : is the Mittag-Leffler function of order :for large waiting times; for small waiting times. :

Empirical analysis (I) Summary The data set Old results. Are jumps and waiting-times really independent? What about autocorrelations of jumps and waiting times? Scaling of the waiting-time distribution.

Empirical analysis (II) The data set

Empirical analysis (III) Old results LIFFE Bund futures (maturity: June 1997) red line: Mittag-Leffler with blue circles: experimental points reduced chi square:

Empirical analysis (IV) Old results LIFFE Bund futures (maturity: September 1997) red line: Mittag-Leffler with blue circles: experimental points reduced chi square:

Empirical results (V) (In)dependence

Empirical results (VI) Autocorrelations lag 0 =3 min 1-day periodicity

Empirical results (VII) Waiting-times Fit of the cdf with a two-parameter stretched exponential

Empirical results (VIII) Waiting-times: fit quality

Empirical results (IX) Waiting-times: Average value: 0.81 Std: 0.05

Empirical results (X) Waiting-times:

Empirical results (XI) Waiting-times: scaling Green curve: scaling variable with parameters extracted from the previous empirical study.

Conclusions Continuos-time random walk has been used as a phenomenological model for high-frequency price dynamics in financial markets; it naturally leads to the fractional diffusion equation in the hydrodynamic limit; an extensive study on DJIA stocks has been performed. Main results: 1. log-returns and waiting times are not independent random variables; 2. the autocorrelation of absolute log-returns exhibits a power-law behaviour with a non-universal exponent; the autocorrelation of waiting times shows a daily periodicity; 3. the waiting-time cdf is well fitted by a stretched exponential function, leading to a simple scaling transformation.