Continuous-time random walks and fractional calculus: Theory and applications Enrico Scalas (DISTA East-Piedmont University) DIFI Genoa (IT) 20 October 2004

Summary Introduction to CTRW and applications to Finance Applications to Physics Conclusions

Introduction to CTRW and applications to Finance

: Five years of continuous-time random walks in Econophysics Enrico Scalas (DISTA East-Piedmont University) WEHIA Kyoto (JP) May 2004

Summary Continuous-time random walks as models of market price dynamics Limit theorem Link to other models Some applications

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 Marginal jump pdf Marginal waiting-time pdf Permanence in x,t Jump into x,t In case of independence: Survival probability

This is the characteristic function of the log-price process subordinated to a generalised Poisson process. Theory (III): Limit theorem, uncoupled case (I) (Scalas, Mainardi, Gorenflo, PRE, 69, , 2004) Mittag-Leffler function Subordination: see Clark, Econometrica, 41, (1973).

Theory (IV): Limit theorem, uncoupled case (II) (Scalas, Gorenflo, Mainardi, PRE, 69, , 2004) This is the characteristic function for the Green function of the fractional diffusion equation. Scaling of probability density functions Asymptotic behaviour

Theory (V): Fractional diffusion (Scalas, Gorenflo, Mainardi, PRE, 69, , 2004) Green function of the pseudo-differential equation (fractional diffusion equation): Normal diffusion for  =2,  =1.

Continuous-time random walks (CTRWs) CTRWs Cràmer-Lundberg ruin theory for insurance companies Compound Poisson processes as models of high-frequency financial data ( Scalas, Gorenflo, Luckock, Mainardi, Mantelli, Raberto QF, submitted, preliminary version cond-mat/ , or preprint: Normal and anomalous diffusion in physical systems Subordinated processes Fractional calculus Diffusion processes Mathematics Physics Finance and Economics

Example: The normal compound Poisson process (  =1) Convolution of n Gaussians The distribution of  x is leptokurtic

Generalisations Perturbations of the NCPP: general waiting-time and log-return densities; (with R. Gorenflo, Berlin, Germany and F. Mainardi, Bologna, Italy, PRE, 69, , 2004); variable trading activity (spectrum of rates); (with H.Luckock, Sydney, Australia, QF submitted); link to ACE; (with S. Cincotti, S.M. Focardi, L. Ponta and M. Raberto, Genova, Italy, WEHIA 2004!); dependence between waiting times and log-returns; (with M. Meerschaert, Reno, USA, in preparation, but see P. Repetowicz and P. Richmond, xxx.lanl.gov/abs/cond-mat/ ); other forms of dependence (autoregressive conditional duration models, continuous-time Markov models); (work in progress in connection to bioinformatics activity).

Applications Portfolio management: simulation of a synthetic market (E. Scalas et al.: VaR estimates: e.g. speculative intra-day option pricing. If g(x,T) is the payoff of a European option with delivery time T: (E. Scalas, communication submitted to FDA ‘04). Large scale simulations of synthetic markets with supercomputers are envisaged.

Empirical results on the waiting-time survival function and their relevance for market models (Anderson-Darling test) (I) Interval 1 (9-11): data;  0 = 7 s Interval 2 (11-14): data;  0 = 11.3 s Interval 3 (14-17): data;  0 =7.9 s where  1   2  …   n A 1 2 = 352; A 2 2 = 285; A 3 2 = 446 >> (1% significance)

Non-exponential waiting-time survival function now observed by many groups in many different markets (Mainardi et al. (LIFFE) Sabatelli et al. (Irish market and ), K. Kim & S.-M. Yoon (Korean Future Exchange)), but see also Kaizoji and Kaizoji (cond-mat/ ) Why should we bother? This has to do both with the market price formation mechanism and with the bid-ask process. If the bid-ask process is modelled by means of a Poisson distribution (exponential survival function), its random thinning should yield another Poisson distribution. This is not the case! A clear discussion can be found in a recent contribution by the GASM group. Possible explanation related to variable daily activity! Empirical results on the waiting-time survival function and their relevance for market models (Anderson-Darling test) (II)

Applications to Physics

Problem Understanding the scaling of transport with domain size has become the critical issue in the design of fusion reactors. It is a challenging task due to the overwhelming complexity of magnetically confined plasmas that are typically in a turbulent state. Diffusive models have been used since the beginning.

Focus and method Tracer transport in pressure-gradient- driven plasma turbulence. Variations in pressure gradient trigger instabilities leading to intermittent and avalanchelike transport. Non-linear equations for the motion of tracers are numerically solved. The pdf of tracer position is non-Gaussian with algebraic decaying tails.

Solution I There is tracer trapping due to turbulent eddies. There are large jumps due to avalanchelike events. These two effects are the source of anomalous diffusion.

Solution II Fat tails (nearly three decades)

Fractional diffusion model

Model I

Model II

Conclusions CTRWs are suitable as phenomenological models for high-frequency market dynamics. They are related to and generalise many models already used in econometrics. They are suitable phenomenological models of anomalous diffusion.