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									Extreme Times in Finance


  J. Masoliver, M. Montero and J. Perelló
    Departament de Fisica Fonamental
        Universitat de Barcelona


                                            1
             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

                                                      2
           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

                                             3
                            CTRW dynamics

    • The log-return and the zero-mean return:
                        S (t  t0 ) 
           Z (t )  ln                          X (t )  Z (t )   Z (t )
                           S (t )  

 X (t ) changes at random times to , t1 , t2 ,..., tn ,...
 Sojourns, Tn  tn  tn 1 , are iid random variables with pdf  (t )
 At each sojourn X (t ) suffers a random change X (t )  X (tn )  X (tn 1 )
 with pdf h( x)
 Waiting times and increments are governed by a joint pdf  ( x, t )

       J. Masoliver, M. Montero, G.H. Weiss Phys. Rev E 67, 021112 (2003)
                                                                               4
CTRW dynamics
(cont.)




                5
                     Return distribution
• Objective
                   p( x, t )dx  Probx  X (t )  x  dx       
• Renewal equation
                                    t      
        p( x, t )  (t ) ( x)   dt '   ( x ', t ') p( x  x ', t  t ')dx '
                                    0     


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

                                          [1  ( s)]/ s
• Formal solution              p( , s) 
                                           1   ( , s)
                                                                                6
                Are jumps and
                waiting times related
                to each other?
a) If they are               b) If they are positively
  independent:                 correlated. Some
                               choices:
  ( x, t )  h( x) (t )                   t 
                             ( , t )          
                                            h( ) 
                             ( , s)   ( s)h  ( s) 
                                                           
                                                                  1/ 
                             ( , s)   h ( )   ( s)  1
                                            

                                        
                                                               
                                                                
                                                                        7
                                    General Results

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

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

• Mean Exit Time
  (MET):
   Ta,b ( x0 )  ta,b ( x0 )


             J. Masoliver, M. Montero, J. Perelló Phys. Rev. E 71, 056130 (2005)
                                                                                   9
                           Integral Equation
                           for the MET
                                         b
                     T ( x0 )     h( x  x0 )T ( x)dx
                                         a

       •  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:
Tc ( x0 )  lim a  Ta , xc ( x0 ) if x0  xc or Tc ( x0 )  limb Txc ,b ( x0 ) if x0  xc

                                                                                          10
                    An exact solution
                                                                         
• Laplace (exponential) distribution:                         h( x )        e | x|
                                                                         2
       jump variance:                2/
                                    2        2



• Exact solution:
                      
          T ( x0 )     1  1   L 2    2 ( x0  (a  b) / 2) 2 
                                        2

                     2                                             

• Symmetrical interval              (b  a  L / 2)
                            
                   T (0)     1  1   L 2  
                                              2
                                                       It is also quadratic in L
                             2                    
                      • For the Laplace pdf the approximate and
                        the exact MET coincide                                          11
Exponential jumps




                    12
                   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  jump variance)
                            

                       L                         2 L
                                                          2
                                                              L 
                                                                 3

  T (0)   1  H (0)     H '(0) / 4  H (0)     O   
            
                                                        
                                                                       13
                    Models and data


                                                               1.70 104 ,
                                                             H (0)  4.45 10 3 ,
                                                             H '(0)  1.54




                     L                         2 L 
                                                       2

T (0)    1  H (0)     H '(0) / 4  H (0)    
          
                                                 
                                                                                14
                     Some
                     Generalizations
• Introduction of correlations by a Markov-chain model.
  Assuming jumps are correlated:

          h( x | x ')dx  Probx  X n  x  dx | X n1  x '


• Integral equation for the MET:
                                        b
         T  x0 | x0    | x0   h( x  x0 | x0 )T  x | x  dx
                                        a


      M. Montero, J. Perelló, J. Masoliver, F. Lillo, S. Miccichè, R.N. Mantegna
      Phys. Rev. E, 72, 056101 (2005).
                                                                                   15
                    A two-state Markov chain model

                           c  ry              c  ry
               h x | y           x  c           x  c
                             2c                  2c
                                                    cov  X n , X n 1 
r = correlation between the magnitude        r
    of two consequtive jumps                       var  X n  var  X n 1 


 Integral equation                            Difference equations

                     1
  T  x0 | c     1  c  T  x0  c | c   1 c  T  x0  c | c  
                     2                                                     


    T  x0  c | c   0 if x0  b  c   T  x0  c | c   0 if x0  a  c
                                                                                  16
        • Solution mid-point: ( L  b  a)

                                           L  1 r 
                                                             2
                    1             2r                   L
                      T  L 2        1        1  
                                1  r  2c  1  r  2c 


        • Scaling time
                                        1  r  T  L 2
                           Tsc  L          
                                        1 r  
Large values of L


                                 Tsc  L 
      (L   c)
                                             L2      Stock independent

                                                                         17
    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
         y    x2h  x | y  dx    c2
               



                                            18
                   II – Stochastic
                   Volatility models

• “Low frequency” data (daily, weekly,...)
  Diffusion models        Geometric Browinian Motion
                                 (Einstein-Bachelier model)

                  dS
                       dt   dW (t )
                   S

• The assumption of constant volatility does not properly
  account for important features of the market
             Stochastic Volatility Models
                                                              19
                      Two-dimensional
                      diffusions
                        dS
                             dt   (t )dW1 (t )
                         S

                              (t )   Y (t ) 

                     dY  F (Y )dt  G (Y )dW2 (t )


Wi (t ) (i  1, 2)   Wiener processes                       dWi (t )  i (t )dt

               i (t ) j (t ')  ij (t  t '),   ii  1, ij  
                                                                                   20
 1. The Ornstein-Uhlenbeck model

        Y , F (Y )   (Y  m), G (Y )  k
         d (t )   (Y  m)dt  kdW2 (t )
                              t
             (t )  m  k  e  (t t ') dW2 (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).



                                                                           21
      2. The CIR-Heston model


      Y , F (Y )   (Y  m), G (Y )  k Y
         dY (t )   (Y  m 2 )dt  k Y dW2 (t )
                            t
          Y(t )  m 2  k  e  (t t ') Y (t ')dW2 (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).


                                                                              22
    3. The Exponential Ornstein-Uhlenbeck model

                  meY , F (Y )   (Y  m), G (Y )  k
                                             k
                        dY (t )   Ydt  dW2 (t )
                                            m
                                    t
                                 k
                        Y(t )   e  ( t t ') dW2 (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).



                                                                                        23
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 )

                                            1      m
• The CIR-Heston model:          d (t )        dt  kdW (t )
                                            2      

• The ExpOU model:        d (t )   ln  m  dt  k dW (t )
                                                                    24
                Extreme times for the
                volatility process

• The MFPT to certain level    (  0 reflecting)
                 
                                       e ( y )                          f ( x) 
                                      y

      T ( )  2 e    ( x )
                                  dx  2 dy                   ( x)  2 2 dx 
                                    0
                                       g ( y)                           g ( x) 

                                                         
                                                     1
• Averaged MFPT                           T ( ) 
                                                       T ( )d
                                                         0

                                  
                                        e ( y )
                                                     y
                           2
             T ( )   xe ( x ) dx  2 dy
                     0               0
                                        g ( y)
                                                                                     25
                               
• Scaling     L           st    pst  d
                  st           


          st  normal level of the volatility


   1 - OU model            st  m
                                            1 2m 
                                        k   2 
   2 - CIR-Heston model         st       2    k 
                                                2m 
                                       2    2 
                                            12

                                                k 
   3- ExpOU model          st  me   k 2 4


                                                         26
                     Some analytical results

1 - OU model           m    k       
                  L 
       T ( L)       xe
                                  ( x 1)2
                                             erf     erf   ( x  1)  dx
                              2


                 L 0                                                       


  Assymptotics
                      2m 2 2
        •   T ( L)       2
                           L                     L     1
                      3k
                      m2 L   2 L2
        •   T ( L)
                       2 k 2
                              e                   L     1
                                                                                   27
2 - CIR-Heston model                      m  k ,   1  m k 2  2   
                       2 2 
                                2 L2 2
                       m                 x1 2 F 1;1   ; x  dx
       T ( L) 
                    1 2 2          
                           L
                                      0

                   2 m 
                F  a; c; x   Kummer’s function of first kind

  Assymptotics
                             st 2
                              2

            •   T ( L)
                           3k 
                             2
                                 L                  L     1

                           m st   2 L2           L      1
            •    T ( L)
                            2 k 2
                                   Le           2
                                                                               28
                                          k    1 2          
3 - ExpOU model                   x   1 2  2 ln  x m  
                                         2    k             
                                      
                     m k 2 2                      1 1 2
                                 L
                                              
            T ( L)     e             ekx        U  ; ; x  dx
                     L                         2 2    

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

  Assymptotics
                            k 2 2

        •     T ( L)
                         e
                        ln  L m 
                                                    L   1


                       L    ln 2  L m          L   1
        •     T ( L)
                        2km 3
                              e               k2

                                                                   29
                  Empirical Data

                                       Nomal Level (daily volatility)
        Financial Indices
                                       1- DJIA:              0.71 %
1- DJIA: 1900-2004 (28545 points)
                                       2- S&P-500:           0.62 %
2- S&P 500: 1943-2003 (15152 points)
                                       3- DAX:               0.84 %
3- DAX: 1959-2003 (11024 points)
                                       4- NIKKEI:            0.96 %
4- NIKKEI: 1970-2003 (8359 points)
                                       5- NASDAQ:            0.78 %
5- NASDAQ: 1971-2004 (8359 points)
                                       6- FTSE-100:          0.77 %
6- FTSE-100: 1984-2004 (5191 points)
                                       7- IBEX-35:           0.96 %
7- IBEX-35: 1987-2004 (4375 points)
                                       8- CAC-40:            1.02 %
8- CAC-40: 1983-2003 (4100 points)



                                                                    30
31
32
33
34
35
               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.

                                                         36
               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.

                                                        37
38
              Comparison with the
              Wiener Process
• The Laplace MET is larger than the MET
  when return follows a Wiener process:
                                          1     1
             T   *
                         (0)  T RW (0)  2 2 
                                 *
                  CTRW
                                          L 2 L

•   We conjecture that this is true in any situation:
                         TCTRW ( x0 )  TRW ( x0 )

• The    Wiener process underestimates the MET.
    Practical consequences for risk control and
    pricing exotic derivatives.
                                                     39
                                                   The Wiener process
                                                   underestimates the MET




                              1     1
T    *
             (0)  T RW (0)  2 2 
                   *
      CTRW
                              L 2 L

    *         T
             CTRW 2                    2
T                      T   *
                                            TRW
     CTRW
               2L            RW        2
                                        2L

T * RW (0)  1 8

                                                                        40

								
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