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Extreme Times in Finance

J. Masoliver, M. Montero and J. PerellóDepartament de Fisica Fonamental

Universitat de Barcelona

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Financial Makets:two levels of description

• “Microscopic” description Tick-by-tick data

Continuous Time Random Walk

• “Mesoscopic” description Daily, weekly... data

Diffusion processes

Stochastic Volatility Models

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I - CTRW formalism

• First developed by Montroll and Weiss (1965)

• Aimed to study the microstructure of random processes

• Applications: transport in random media, random networks, self-organized criticallity, earthquake modeling, and… now in financial markets

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CTRW dynamics

• The log-return and the zero-mean return:

0( )( ) ln

( )

S t tZ t

S t

( ) ( ) ( )X t Z t Z t

1 2

1

1

( ) changes at random times , , ,..., ,...

Sojourns, , are iid random variables with pdf ( )

At each sojourn ( ) suffers a random change ( ) ( ) ( )

with pdf ( )

Waiti

o n

n n n

n n

X t t t t t

T t t t

X t X t X t X t

h x

ng times and increments are governed by a joint pdf ( , )x t

J. Masoliver, M. Montero, G.H. Weiss Phys. Rev E 67, 021112 (2003)

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CTRW dynamics (cont.)

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Return distribution

0

( , ) ( ) ( ) ' ( ', ') ( ', ') 't

p x t t x dt x t p x x t t dx

[1 ( )] /

( , )1 ( , )

s sp s

s

( , ) Prob ( )p x t dx x X t x dx

• Renewal equation

• Formal solution

joint distribution of increments and waiting times( , )x t

• Objective

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Are jumps and waiting times related to each other?

a) If they are independent:

b) If they are positively correlated. Some choices:

( , ) ( ) ( )x t h x t

1/

( , )( )

( , ) ( ) ( )

( , ) ( ) ( ) 1

tt

h

s s h s

s h s

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General Results

• Approach to the Gaussian density

• Long-tailed jump density:

Lévy distribution

• At intermediate times:

the tail behavior is given by extreme jumps

22 / 2( , ) e (t )tp t

• Normal diffusion

2 2( ) ( )X t t t

( ) 1h k

| | /( , ) e (t )k tp t

( , ) ( ) ( | | )t

p x t h x x

,t

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Extreme Times

• At which time the return leaves a given interval [a,b] for the first time?

• Mean Exit Time (MET):

, 0 , 0( ) ( )a b a bT x t x

J. Masoliver, M. Montero, J. Perelló Phys. Rev. E 71, 056130 (2005)

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Integral Equation for the MET

• is the mean time between jumps. • The MET does NOT depend on

– the whole time distribution – the coupling between jumps and waiting times

• Mean First Passage Time (MFPT) to a certain critical value:

0 0( ) ( ) ( )b

a

T x h x x T x dx

0 , 0 0 0 , 0 0( ) lim ( ) if or ( ) lim ( ) if c cc a a x c c b x b cT x T x x x T x T x x x

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An exact solution

• Laplace (exponential) distribution: jump variance:

• Exact solution:

| |( )2

xh x e 2 22 /

2 2 20 0( ) 1 1 2 ( ( ) / 2)

2T x L x a b

2(0) 1 1 2

2T L

( / 2)b a L • Symmetrical interval

• For the Laplace pdf the approximate and

the exact MET coincide

It is also quadratic in L

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Exponential jumps

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Approximate solution• We need to specify the jump pdf• We want to get a solution as much general as

possible• We get an approximate solution when:

– the interval L is smaller than the jump variance– jump pdf is an even function and zero-mean with scaling:

1( )

xh x H

2 3

2(0) 1 (0) '(0 ) / 4 (0)

L L LT H H H O

2( jump variance)

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Models and data

4

3

1.70 10 ,

(0) 4.45 10 ,

'(0 ) 1.54

H

H

2

2(0) 1 (0) '(0 ) / 4 (0)

L LT H H H

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Some Generalizations

• Introduction of correlations by a Markov-chain model. Assuming jumps are correlated:

1( | ') Prob | 'n nh x x dx x X x dx X x

• Integral equation for the MET:

0 0 0 0 0| | ( | ) |b

a

T x x x h x x x T x x dx

M. Montero, J. Perelló, J. Masoliver, F. Lillo, S. Miccichè, R.N. MantegnaPhys. Rev. E, 72, 056101 (2005).

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A two-state Markov chain model

|2 2

c ry c ryh x y x c x c

c c

r = correlation between the magnitude of two consequtive jumps

1

1

cov ,

var var

n n

n n

X Xr

X X

Integral equation Difference equations

0 0 0

1| 1 | 1 |

2T x c c T x c c c T x c c

0 0| 0 if T x c c x b c 0 0| 0 if T x c c x a c

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2

1 2 12 1 1

1 2 1 2

r L r LT L

r c r c

( )L b a • Solution mid-point:

• Scaling time

21

1sc

T LrT L

r

Large values of L

( )L c 2scT L L Stock independent

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tick-by-tick data of 20 highly capita-lized stocks traded at the NYSE in the 4 year period 95-98;more than 12 miliontransactions.

L b a

2 2 2| y x h x y dx c

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• “Low frequency” data (daily, weekly,...)

Diffusion models

( )dS

dt dW tS

Geometric Browinian Motion(Einstein-Bachelier model)

• The assumption of constant volatility does not properly account for important features of the market

Stochastic Volatility Models

II – Stochastic Volatility models

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2( ) ( ) ( )dY F Y dt G Y dW t

i ii( ) ( ') ( '), 1, j ij ijt t t t

1( ) ( )dS

dt t dW tS

( ) ( )t Y t

Wiener processes( ) ( 1,2)iW t i ( ) ( )i idW t t dt

Two-dimensional diffusions

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1. The Ornstein-Uhlenbeck model

2

( ')2

, ( ) ( ), ( )

( ) ( ) ( )

( ) ( ')t

t t

Y F Y Y m G Y k

d t Y m dt kdW t

t m k e dW t

E. Stein and J. Stein, Rev. Fin. Studies 4, 727 (1991).J. Masoliver and J. Perelló, Int. J. Theor. Appl. Finance 5, 541 (2002).

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2. The CIR-Heston model

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2 ( ')2

, ( ) ( ), ( )

( ) ( ) ( )

Y( ) ( ') ( ')t

t t

Y F Y Y m G Y k Y

dY t Y m dt k YdW t

t m k e Y t dW t

Cox, J., Ingersoll, J., and S. Ross, (1985a), Econometrica, 53, 385 (1985). S. Heston, Rev. Fin. Studies 6, 327 (1993).A. Dragulescu and V. Yakovenko, Quant. Finance 2, 443 (2002).

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3. The Exponential Ornstein-Uhlenbeck model

2

( ')2

, ( ) ( ), ( )

( ) ( )

Y( ) ( ')

Y

tt t

me F Y Y m G Y k

kdY t Ydt dW t

m

kt e dW t

m

J.-P. Fouque, G. Papanicolau and K. R. Sircar, Int. J. Theor. Appl. Finance 3, 101 (2002).J. Masoliver and J. Perelló, Quant. Finance (2006).

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In SV models the volatility proces is described by a one-dimensional diffusion

( ) ( ) ( ) ( )d t f dt g dW t

• The OU model: ( ) ( ) ( )d t Y m dt kdW t

• The CIR-Heston model:1

( ) ( )2

md t dt kdW t

• The ExpOU model: ( ) ln ( )d t dt k dW tm

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Extreme times for the volatility process

• The MFPT to certain level 0 ( reflecting)

( )( )

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( ) 2( )

y yx e

T e dx dyg y

• Averaged MFPT0

1( ) ( )T T d

2

( )( ) 2

( )

f xx dx

g x

( )( )

20 0

2( )

( )

y yx e

T xe dx dyg y

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• Scalingst

L

st stp d

1 - OU model st m

2 - CIR-Heston model

2

1 2

2

1 22

22

st

mk

kmk

3- ExpOU model2 4k

st me

st normal level of the volatility

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Some analytical results

1 - OU model

2 2( 1)

0

( ) erf erf ( 1)L

xT L xe x dxL

Assymptotics 2

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2( )

3

mT L L

k 1L•

2 22

2( )

2Lm L

T L ek

1L•

m k

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2 - CIR-Heston model

2 2

2

21 2

20

2

( ) F 1;1 ;1 22

LmT L x x dx

Lm

2, 1 2m k m k

Assymptotics 2

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( )3

stT L Lk

1L• 2 2 2

2( )

2Lstm

T L Lek

1L•

F ; ;a c x Kummer’s function of first kind

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3 - ExpOU model 1 2

1 2 2ln

2

kx x m

k

Assymptotics

2 2

( )ln

keT L

L m

1L•

2 2ln

3( )

2L m kL

T L ekm

1L•

F ; ;a c x Kummer’s function of second kind

2 2 21 1( ) ; ;

2 2k kx

L

mT L e e U x dx

L

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Empirical Data

Financial Indices

1- DJIA: 1900-2004 (28545 points)2- S&P 500: 1943-2003 (15152 points)3- DAX: 1959-2003 (11024 points)4- NIKKEI: 1970-2003 (8359 points)5- NASDAQ: 1971-2004 (8359 points)6- FTSE-100: 1984-2004 (5191 points)7- IBEX-35: 1987-2004 (4375 points) 8- CAC-40: 1983-2003 (4100 points)

Nomal Level (daily volatility)

1- DJIA: 0.71 %2- S&P-500: 0.62 %3- DAX: 0.84 %4- NIKKEI: 0.96 %5- NASDAQ: 0.78 %6- FTSE-100: 0.77 %7- IBEX-35: 0.96 %8- CAC-40: 1.02 %

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Conclusions (I)• The CTRW provides insight relating the market

microstructure with the distributions of intraday prices and even longer-time prices.

• It is specially suited to treat high frequency data. • It allows a thorough description of extreme times

under a very general setting. • MET’s do not depend on any potential coupling

between waiting times and jumps.• Empirical verification of the analytical estimates

using a very large time series of USD/DEM transaction data.

• The formalism allows for generalizations to include price correlations.

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Conclusions (II)

• The “macroscopic” description of the market is quite well described by SV models.

• Many SV models allow a analytical treatment of the MFPT.

• The MFPT may help to determine a suitable SV model

• OU and CIR-Heston models yield a quadratic behavior of the MFPT for small volatilities that is not conflicting with data. For large volatilities their exponential growth does not agree with data.

• In a first approximation the ExpOU model seems to agree with data for both small and large volatilities.

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• The Laplace MET is larger than the MET when return follows a Wiener process:

0 0( ) ( )CTRW RWT x T x

* *2 2

1 1(0) (0)

2CTRW RWT TL L

• We conjecture that this is true in any situation:

• The Wiener process underestimates the MET. Practical consequences for risk control and pricing exotic derivatives.

Comparison with the Wiener Process

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* *2 2

1 1(0) (0)

2CTRW RWT TL L

*2 2

CTRWCTRW

TT

L

2*

22RW RWT TL

* (0) 1 8RWT

The Wiener process underestimates the MET