Speculative option valuation and the fractional diffusion equation

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Speculative option valuation and the fractional
diffusion equation

• Enrico Scalas (DISTA East-Piedmont University)
• www.econophysics.org
• www.fracalmo.org

FDA04 - Bordeaux (FR) 18-21 July 2004
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In collaboration with

• Rudolf Gorenflo
• Francesco Mainardi
• Mark M. Meerschaert

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Summary

• Continuous-time random walks as models of market
  price dynamics
• Limit theorems
• Link to other models
• Application to speculative option valuation
• 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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Theory (VI) The coupled case (I)
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Theory (VII) The coupled case (II)
Basic message Under suitable hypotheses,
the fractional diffusion equation is the
diffusive limit of CTRWs also in the coupled case!
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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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Application to speculative option valuation

• 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

• Large scale MC simulations of synthetic markets
  with
• supercomputers are being performed (with G.
  Germano,
• P. Dagna, and A. Vivoli http//www1.fee.uva.nl/ce
  ndef/sce2004/sce_ams.htm).

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Results (I)
Fig. 1 Simulated log-price as a function of
time. This simulation includes 10000 log-prices.
It takes a few minutes to run on an old Pentium
II processor at 349 MHz.
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Results (II)
Fig 2 Power to factorial ratio for T5000s and
?010s. Only evidenced values of n have been used
to compute p(x,T).
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Results (III)
Fig. 3 Theoretical PDF (solid line) and
simulated PDF (circles). p(x,T) is computed for
T5000s, ?010s, ?0.005. The simulated PDF is
computed from the histogram of 1000 realisations.
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Results (IV)
Fig. 4 Payoff histogram for a very short-term
(T5000 s) plain vanilla call European option
with initial price S(0)100 and strike price
E100. The evolution of the underlying has been
simulated 1000 times times by means of a NCPP,
with parameters given in Fig. 3.
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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
  including speculative option valuation.