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Continuous-time random walks and fractional calculus: Theory and applications

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Continuous-time random walks and fractional calculus: Theory and applications

Enrico Scalas (DISTA East-Piedmont University)

www.econophysics.org

DIFI Genoa (IT) 20 October 2004

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

- Introduction to CTRW and applications to Finance

1999-2004: Five years of continuous-time random walks in Econophysics

Enrico Scalas (DISTA East-Piedmont University)

www.econophysics.org

WEHIA 2004 - Kyoto (JP) 27-29 May 2004

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

Theory (I)

Continuous-time random walk in finance

(basic quantities)

: price of an asset at timet

: log price

: joint probability density of jumps

and of waiting times

: probability density function of

finding the log price xat timet

Theory (II): Master equation

Permanence in x,t

Jump into x,t

Marginal jump pdf

Marginal waiting-time pdf

In case of independence:

Survival probability

Theory (III): Limit theorem, uncoupled case (I)

(Scalas, Mainardi, Gorenflo, PRE, 69, 011107, 2004)

Mittag-Leffler function

This is the characteristic

function of the log-price process

subordinated to a generalised

Poisson process.

Subordination: see Clark, Econometrica,

41, 135-156 (1973).

Theory (IV): Limit theorem, uncoupled case (II)

(Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004)

Scaling of probability

density functions

Asymptotic behaviour

This is the characteristic

function for the Green

function of the fractional

diffusion equation.

Theory (V): Fractional diffusion

(Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004)

Green function of the pseudo-differential equation (fractional diffusion equation):

Normal diffusion

for =2, =1.

Continuous-time random walks (CTRWs)

(Scalas, Gorenflo, Luckock, Mainardi, Mantelli, Raberto QF, submitted,

preliminary version cond-mat/0310305, or preprint:

www.maths.usyd.edu.au:8000/u/pubs/publist/publist.html?preprints/2004/scalas-14.html)

Diffusion

processes

Mathematics

Compound Poisson processes

as models of high-frequency

financial data

Fractional

calculus

Subordinated

processes

CTRWs

Physics

Finance and

Economics

Normal and

anomalous

diffusion in physical

systems

Cràmer-Lundberg

ruin theory for

insurance companies

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, 011107, 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/0310351);
- 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.: www.mfn.unipmn.it/scalas/~wehia2003.html).
- 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.

Interval 1 (9-11): 16063 data; 0 = 7 s

Interval 2 (11-14): 20214 data; 0 = 11.3 s

Interval 3 (14-17): 19372 data; 0 =7.9 s

where 1 2 … n

A12= 352; A22= 285; A32= 446>>1.957 (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/0312560)
- 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!

- 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

- 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.