19992004: Five years of continuoustime random walks in Econophysics

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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
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Summary

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

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Tick-by-tick price dynamics
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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
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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
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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).
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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.
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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.
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Continuous-time random walks (CTRWs)
(Scalas, Gorenflo, Luckock, Mainardi, Mantelli,
Raberto QF, submitted, preliminary version
cond-mat/0310305, or preprint www.maths.usyd.edu
.au8000/u/pubs/publist/publist.html?preprints/200
4/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
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Example The normal compound Poisson process (?1)
Convolution of n Gaussians
The distribution of ?x is leptokurtic
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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).

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Applications

• Portfolio management simulation of a synthetic
  market
• (E. Scalas et al. www.mfn.unipmn.it/scalas/wehia
  2003.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.

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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 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 gtgt 1.957 (1
significance)
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Empirical results on the waiting-time survival
function and their relevance for market models
(Anderson-Darling test) (II)

• 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!

See S. Cincotti presentation Session 6/2!
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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 can be helpful in various applications.