Stochastic Volatility Models. 3. I - CTRW formalism. First developed by Montroll and Weiss (1965); Aimed to study the microstructure of random processe

1. 1
Extreme Times in Finance
J. Masoliver, M. Montero and J. Perelló
Departament de Fisica Fonamental
Universitat de Barcelona

2. 2
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

3. 3
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

4. 4
CTRW dynamics
• The log-return and the zero-mean return:
0
( )
( ) ln
( )
S t t
Z 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)

5. 5
CTRW dynamics
(cont.)

6. 6
Return distribution
0
( , ) ( ) ( ) ' ( ', ') ( ', ') '
t
p x t t x dt x t p x x t t dx
 


    
 
[1 ( )]/
( , )
1 ( , )
s s
p 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

7. 7
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
t
t
h
s s h s
s h s



  

   
   



 
  
 
 

 
 
  
 
 

8. 8
General Results
• Approach to the
Gaussian density
• Long-tailed jump density:
Lévy distribution
• At intermediate times:
the tail behavior is given by
extreme jumps
2
2 / 2
( , ) e (t )
t
p t
  
 


• Normal diffusion
2 2
( ) ( )
X t t t



( ) 1
h k

 

| | /
( , ) e (t )
k t
p t

 
 


( , ) ( ) ( | | )
t
p x t h x x

 
,
t 

9. 9
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 b
T x t x

J. Masoliver, M. Montero, J. Perelló Phys. Rev. E 71, 056130 (2005)

10. 10
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 c
c a a x c c b x b c
T x T x x x T x T x x x
 
   

11. 11
An exact solution
• Laplace (exponential) distribution:
jump variance:
• Exact solution:
| |
( )
2
x
h x e 
 

2 2
2/
 

 
2 2 2
0 0
( ) 1 1 2 ( ( )/ 2)
2
T x L x a b

 
 
     
 
 
2
(0) 1 1 2
2
T L


 
  
 
( / 2)
b a L
  
• Symmetrical interval
• For the Laplace pdf the approximate and
the exact MET coincide
It is also quadratic in L

12. 12
Exponential jumps

13. 13
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
( )
x
h x H
 
 
  
 
 
2 3
2
(0) 1 (0) '(0 ) / 4 (0)
L L L
T H H H O

  
 
     
     
 
     
     
 
 
2
( jump variance)
 

14. 14
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 L
T H H H

 
 
   
   
 
   
   
 
 

15. 15
Some
Generalizations
• Introduction of correlations by a Markov-chain model.
Assuming jumps are correlated:
 
1
( | ') Prob | '
n n
h 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. Mantegna
Phys. Rev. E, 72, 056101 (2005).

16. 16
A two-state Markov chain model
     
|
2 2
c ry c ry
h 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 X
r
X X


 

 
Integral equation Difference equations
         
0 0 0
1
| 1 | 1 |
2
T 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
    

17. 17
 
2
1 2 1
2 1 1
1 2 1 2
r L r L
T L
r c r c


   
   
   
 
   
( )
L b a
 
• Solution mid-point:
• Scaling time
 
 
2
1
1
sc
T L
r
T L
r 

 
  

 
Large values of L
( )
L c
  2
sc
T L L Stock independent

18. 18
tick-by-tick data
of 20 highly capita-
lized stocks traded
at the NYSE in the
4 year period 95-98;
more than 12 milion
transactions.
L b a
 
   
2 2 2
|
y x h x y dx c



 


19. 19
• “Low frequency” data (daily, weekly,...)
Diffusion models
( )
dS
dt dW t
S
 
 
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

20. 20
2
( ) ( ) ( )
dY F Y dt G Y dW t
 
i ii
( ) ( ') ( '), 1,
j ij ij
t t t t
      
   
1
( ) ( )
dS
dt t dW t
S
 
 
 
( ) ( )
t Y t
 

Wiener processes
( ) ( 1,2)
i
W t i  ( ) ( )
i i
dW t t dt


Two-dimensional
diffusions

21. 21
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).

22. 22
2. The CIR-Heston model
2
2
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).

23. 23
3. The Exponential Ornstein-Uhlenbeck model
2
( ')
2
, ( ) ( ), ( )
( ) ( )
Y( ) ( ')
Y
t
t t
me F Y Y m G Y k
k
dY t Ydt dW t
m
k
t 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).

24. 24
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
m
d t dt kdW t
  

 
   
 
 
• The ExpOU model:  
( ) ln ( )
d t dt k dW t
m

  
  

25. 25
Extreme times for the
volatility process
• The MFPT to certain level  
 0
 
( reflecting)
( )
( )
2
0
( ) 2
( )
y y
x e
T e dx dy
g y
 



 
  
• Averaged MFPT
0
1
( ) ( )
T T d


  

 
2
( )
( ) 2
( )
f x
x dx
g x

 

 
 

( )
( )
2
0 0
2
( )
( )
y y
x e
T xe dx dy
g y
 




  

26. 26
• Scaling
st
L


  
st st
p d
   


 
1 - OU model st m
 
2 - CIR-Heston model
 
2
1 2
2
1 2
2
2
2
st
m
k
k
m
k




 
 
 
 

 
 
 
3- ExpOU model
2
4
k
st me 
 
st
  normal level of the volatility

27. 27
Some analytical results
1 - OU model
   
2 2
( 1)
0
( ) erf erf ( 1)
L
x
T L xe x dx
L

 
 


  
 
 

Assymptotics
2
2
2
2
( )
3
m
T L L
k
 
1
L
•
2 2
2
2
( )
2
L
m L
T L e
k



 
1
L
•
 
m k
 


28. 28
2 - CIR-Heston model
 
2 2
2
2
1 2
2
0
2
( ) F 1;1 ;
1 2
2
L
m
T L x x dx
L
m





 
 
 
 
 
 
 
 

 
 
2
, 1 2
m k m k
   
  
Assymptotics 2
2
2
( )
3
st
T L L
k


 
1
L
•
2 2
2
2
( )
2
L
st
m
T L Le
k

 

 
1
L
•
 
F ; ;
a c x  Kummer’s function of first kind

29. 29
3 - ExpOU model    
1 2
1 2 2
ln
2
k
x x m
k



 
 
 
 
Assymptotics
 
2
2
( )
ln
k
e
T L
L m



 
1
L
•
 
2 2
ln
3
( )
2
L m k
L
T L e
km

   
1
L
•
 
F ; ;
a c x  Kummer’s function of second kind
 
2
2 2
1 1
( ) ; ;
2 2
k kx
L
m
T L e e U x dx
L
 



 

 
  
 


30. 30
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 %

31. 31

32. 32

33. 33

34. 34

35. 35

36. 36
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.

37. 37
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.

38. 38

39. 39
• The Laplace MET is larger than the MET
when return follows a Wiener process:
0 0
( ) ( )
CTRW RW
T x T x

* *
2 2
1 1
(0) (0)
2
CTRW RW
T T
L 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

40. 40
* *
2 2
1 1
(0) (0)
2
CTRW RW
T T
L L
 
  
*
2 2
CTRW
CTRW
T
T
L
 

2
*
2
2
RW RW
T T
L


*
(0) 1 8
RW
T 
The Wiener process
underestimates the MET