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.

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 80 100

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

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

x

txp

t

txp

,,

Theory (III)Fractional diffusion

For a given joint density, the Fourier-Laplacetransform of is given by:

sks

sskp

,~ˆ1

1~1,

~ˆ

where: , d

is the waiting time probability density function.

Assumption (asymptotic scaling and independence):

0 ,0 1,~ˆ sksksk

t

x

Caputo fractional derivative

Riesz fractional derivative

10

21

txp ,

Theory (IV)Waiting-time distribution

s

s

1

1~

Simple assumption (compatible withasymptotic independence):

0

' '1Pr d

E

: is the Mittag-Leffler function of order

E

:for large waiting times;

1 exp 0

for small waiting times.

:

0 1

Empirical analysis (I)Summary

• The data set

• Old results.

• Are jumps and waiting-times really independent?

• What about autocorrelations of jumps and waiting times?

• Scaling of the waiting-time distribution.

Empirical analysis (II)The data set

AAALDAXPBAC

CATCHVDDDISEKGEGMGT

HWPIBMIP

JNJJPMKO

MCDMMMMO

MRKPGST

UKUTXWMTXON

0 10000 20000 30000 40000 50000 60000

Total number of data: 779216

N

Sto

ck

Empirical analysis (III)Old results

LIFFE Bund futures(maturity: June 1997)

red line: Mittag-Leffler with blue circles: experimental points

75.0

5.1~2

(s)

~

reduced chi square:

~

Empirical analysis (IV)Old results

LIFFE Bund futures(maturity: September 1997)

(s)

red line: Mittag-Leffler with blue circles: experimental points

76.0

0.1~2 reduced chi square:

Empirical results (V)(In)dependence

i

0 ÷ 10 10 ÷ 20 > 20< -0.002 25 (38.9) 21 (10.1) 9 (6.0)-0.002 ÷ -0.001 516 (613.6) 230 (159.5) 122 (94.9)-0.001÷ 0 6641 (7114.3) 2085 (1849.1) 1338 (1100.6)0 ÷ 0.001 31661 (31008.0) 7683 (8059.2) 4520 (4797.0)0.001 ÷ 0.002 398 (464.4) 179 (120.7) 80 (71.9)

i

> 0.002 34 (36.1) 10 (9.4) 7 (5.6)

2.27~2

Empirical results (VI)Autocorrelations

lag0=3 min

1-day periodicity

Empirical results (VII)Waiting-times

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

Bet

a

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/G

amm

a

N

BAN 0

0024.0

10042.2 6

B

A

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.