Continuous-time random walks and fractional calculus:

Theory and applicationsEnrico Scalas (DISTA East-Piedmont University)

www.econophysics.org

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

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

Summary• Continuous-time random walks as models of

market price dynamics

• Limit theorem

• Link to other models

• Some applications

Tick-by-tick price dynamics

0 20 40 60 80 100

12,0

12,2

12,4

12,6

12,8

13,0

Price variations as a function of time

S

t

Pric

e

Time

Theory (I)Continuous-time random walk in finance

(basic quantities)

tS : price of an asset at time t

tStx log : log price

, : joint probability density of jumps and of waiting times

iii txtx 1 iii tt 1

txp , : probability density function of finding the log price x at time t

Theory (II): Master equation

0

, d

, d

Marginal jump pdf

Marginal waiting-time pdf

,

Permanence in x,t Jump into x,t

In case of independence:

0

' '1Pr d Survival probability

' ' ','',' ,0

dxdttxpttxxtxtxpt

This is the characteristic

function of the log-price process

subordinated to a generalised

Poisson process.

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

0 1

1

n

nn

n

ttE

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

Mittag-Leffler function

10

0 !

,n

nn

n

xtEn

ttxp

1ˆ,ˆ tEtp

Subordination: see Clark, Econometrica, 41, 135-156 (1973).

Theory (IV): Limit theorem, uncoupled case (II)(Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004)

1ˆ,ˆ ,

hr

tEtp rh

rssrttr r ~~ , ,

hh hˆˆ ,

1 ,1ˆ 0,0 rhh

r

hhh

tEtp rh

rh

,ˆlim ,

0,

This is the characteristic function for the Greenfunction of the fractionaldiffusion equation.

20

Scaling of probabilitydensity functions

Asymptotic behaviour

Theory (V): Fractional diffusion(Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004)

tEtu ,ˆ

txWt

txu ,

1,

tuyidyW ,ˆexp2

1,

1,~ˆ,

~ˆ ssussu

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 processesas models of high-frequency

financial data

(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)

Normal and anomalous

diffusion in physicalsystems

Subordinatedprocesses

Fractionalcalculus

Diffusionprocesses

Mathematics

PhysicsFinance andEconomics

Example: The normal compound Poisson process (=1)

0

00 !

exp,n

n

n

xn

tttxp

22 2exp2

1

nnxn

xn Convolution of n Gaussians

ttn

tnkktxttxx

1

)(

txpxp ,

0 txE 0

22var tx

The distribution of x is leptokurtic

04224

4 36 tK

(S.J. Press, Journal of Business, 40, 317-335, 1967)

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:

dy

dxTtygpTyq

Txgy

,,,

,

1

(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): 16063 data; 0 = 7 sInterval 2 (11-14): 20214 data; 0 = 11.3 sInterval 3 (14-17): 19372 data; 0 =7.9 s

nnn

iA iin

n

i

6.011lnln12

11

2

where 1 2 … n A1

2= 352; A22= 285; A3

2= 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!

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.