ON CONTINUOUS-TIME RANDOM WALKS IN FINANCE Enrico Scalas (1) with: Maurizio Mantelli (1) Marco Raberto (2) Rudolf Gorenflo (3) Francesco Mainardi (4) (1) DISTA, Università del Piemonte Orientale, Alessandria, Italy. (2) DIBE, Università di Genova, Italy. (3) Erstes Matematisches Institut, Freie Universität Berlin, Germany. (4) Dipartimento di Fisica, Università di Bologna, Italy.
ON CONTINUOUS-TIME RANDOM WALKS IN FINANCE. Enrico Scalas (1) with : Maurizio Mantelli (1) Marco Raberto (2) Rudolf Gorenflo (3) Francesco Mainardi (4). (1) DISTA, Università del Piemonte Orientale, Alessandria, Italy. (2) DIBE, Università di Genova, Italy. - PowerPoint PPT Presentation
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(1) DISTA, Università del Piemonte Orientale, Alessandria, Italy.(2) DIBE, Università di Genova, Italy.(3) Erstes Matematisches Institut, Freie Universität Berlin, Germany.(4) Dipartimento di Fisica, Università di Bologna, Italy.
Summary
•Theory
•Empirical Analysis
•Conclusions
In financial markets, not only prices and returns can be considered as random variables, but also the waiting time between two transactions varies randomly. In the following, we analyze the DJIA stocks traded in October 1999. The empirical properties of these time series arecompared to theoretical prediction based on a continuous time random walk model.
Outline
Tick-by-tick price dynamics
0 20 40 60 80 10012,012,212,412,612,813,0
Price variations as a function of time
S
t
Price
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
Jump pdf
Waiting-time pdf
,
Permanence in x,t Jump into x,t
Factorisation in case of independence:
0
' '1Pr d
Survival probability
' ' ',''' ,0
dtdxtxpxxtttxtxpt
xtxp
ttxp
,,
Theory (III)Fractional diffusion
For a given joint density, the Fourier-Laplacetransform of is given by:
sks
sskp,
~̂1
1~1,~̂
where: , d
is the waiting time probability density function.Assumption (asymptotic scaling and independence):
Fit of the cdf with a two-parameterstretched exponential
Empirical results (VIII)Waiting-times: fit quality
2~
0
0
10000 20000 30000 40000 50000 600000,0
0,5
1,0
Beta = 0.81
Beta
N
Empirical results (IX)Waiting-times:
Average value: 0.81Std: 0.05
Empirical results (X)Waiting-times: 0
10000 20000 30000 40000 50000 60000
0,02
0,04
0,06
0,08
0,10
0,12
1/Gam
ma
N
BAN 0
0024.010042.2 6
BA
Empirical results (XI)Waiting-times: scaling
Green curve: scaling variable 10u
with parameters extracted from theprevious empirical study.
Conclusions• Continuos-time random walk has been used as a phenomenological model for high-frequency price dynamics in financial markets;
• it naturally leads to the fractional diffusion equation in the hydrodynamic limit;
• an extensive study on DJIA stocks has been performed.
Main results:
1. log-returns and waiting times are not independent random variables;2. the autocorrelation of absolute log-returns exhibits a power-law behaviour with a non-universal exponent; the autocorrelation of waiting times shows a daily periodicity;3. the waiting-time cdf is well fitted by a stretched exponential function, leading to a simple scaling transformation.