+ All Categories
Home > Documents > ON CONTINUOUS-TIME RANDOM WALKS IN FINANCE

ON CONTINUOUS-TIME RANDOM WALKS IN FINANCE

Date post: 25-Feb-2016
Category:
Upload: lawson
View: 53 times
Download: 1 times
Share this document with a friend
Description:
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
Popular Tags:
19
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.
Transcript
Page 1: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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.

Page 2: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 3: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 4: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 5: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 6: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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):

0 ,0 1,~ˆ sksksk

t

x

Caputo fractional derivative

Riesz fractional derivative10

21

txp ,

Page 7: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 8: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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.

Page 9: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Stoc

k

Page 10: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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:

Page 11: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

~

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:

Page 12: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 13: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

Empirical results (VI)Autocorrelations

lag0=3 min

1-day periodicity

Page 14: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

Empirical results (VII)Waiting-times

Fit of the cdf with a two-parameterstretched exponential

Page 15: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

Empirical results (VIII)Waiting-times: fit quality

2~

0

0

Page 16: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 17: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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

Page 18: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

Empirical results (XI)Waiting-times: scaling

Green curve: scaling variable 10u

with parameters extracted from theprevious empirical study.

Page 19: ON CONTINUOUS-TIME RANDOM  WALKS IN FINANCE

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.


Recommended