Queueing theoretical analysis of first passage processes in foreign by utg65734

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									  Modeling of internet
   trading systems:
     Queueing theoretical approaches
    Jun-ichi Inoue (1) and Naoya Sazuka (2)
        (1) Hokkaido University          (2) Sony Corporation

   To appear in Quantitative Finance (2007) as
 “Queueing theoretical analysis of foreign currency
                 exchange rates”

International Conference on Modeling and Simulation 07
          Kolkata, India 4th December 2007
Fluctuations in intervals of events
                  ISI in a single BUND future Sony bank
                  neuron          (“Bond” in  rate
                               German word)
Average time ~3 [ms]           ~10 [s]          ~ 20 [min]
interval
PDF of       Gamma             Mittag-Leffler       ?
duration
       duration


                                                             time

         Price change, Neuronal spikes, etc..
       Aim of this study
Evaluate time fluctuations in intervals between events:
            duration t


                                         w                             time
     Price change
                          Observation time
                          (A trader checks the rate)

  How long does a trader wait until the next price changes ?
   Waiting time w (residual lifetime)
   Time between observing the rate and the next price change
   <w> is more informative quantity than < t > for on-line trading system

 We evaluate the waiting time by queueing theoretical approach
Data: Sony Bank rate
Sony bank USD/JPY exchange rate:
・Rate for individual customers of the Sony bank (http://moneykit.net/)
・Tradable on the web 24 hours a day
・The rate depends on the market rate, not customers’ order
 Sony bank rate
      124
      123
      122
      121
      120
      119
      118
                                                     ticks
              500    1000 1500 2000 2500 3000
FPP in financial markets
Sony bank rate
   124    http://moneykit.net/
   123                                                How do we estimate
   122
   121
   120
                                                            P(t )
   119                                                    FPT distribution ?
   118
                                                  ticks
            500   1000 1500 2000 2500 3000
                                                             Sony bank rate


Sony bank rate
                                 Market rate                 ε = 0.1 [yen]
                                                             ε = 0.1 [yen]
is regarded as
a first passage
process
                        t        First passage time
Sony Bank rate as a FPP
  The duration gets longer than that of market rate




  X
                               2ε

                                                 market rate

                                                 sony bank rate

                      first passage time




                                      t
FPT pdf of the Sony Bank rate
 Empirical data analysis leads to

       P (t ) =
        W
                      mt m−1
                        a          ( )
                               exp −   tm
                                        a   Weibull distribution



        Sazuka (2005)




          Weibull paper analysis



  m = 0.59, a = 50.855
Lorentz curve and Gini index
Gini index is originally a quantity to measure earning inequality
・ It takes a value between 0 and 1
・ It gives 1 for perfect inequality and 0 for perfect equality
・ Japan (0.314), USA (0.357), Mexico (0.480), Denmark (0.225)        r
                                          r
                             X (r ) = ∫ P(t )dt , Y (r ) = ∞∫0 tP ( t ) dt
                                       0
                                        Income distribution ∫0 tP (t ) dt
                                              We regard P(t) as FPT pdf
       Y = f ( X ) : Lorentz curve

           For a Weibull distribution:

           Y = Q ( m + 1, − log(1 − X ) ) , Q( z , x) = ∫ t z −1e − t dt
                                                                 x
                   1
                                                                 0
Gini index for a Weibull pdf
Gini index is originally a quantity to measure earning inequality
・ It takes a value between 0 and 1
・ It gives 1 for perfect inequality and 0 for perfect equality

                                                           Sazuka and Inoue (2007)
                    Gini index for Weibull FPT pdf


Sony bank rate
                     P (t ) =
                      W
                                 mt m−1
                                   a         ( )
                                          exp −   tm
                                                   a
                                                         Empirical data evaluation
                                                                        N
                                                         G=       1
                                                                        ∑ ( 2i − N − 1)x
                          G = 1 − ( 12 )
                                               1                 N 2μ                   i
                                                   m                    i =1

                                                                 0.693079
                                  Theoretical result
     Poisson arrival process                                       31,000 points from
                                         0.694618
                                                                   September 2002 to
                                                                   May 2004
          0.59
Queueing theoretical analysis
                      Sony bank USD/JPY exchange rates
    Waiting time


                                                                      Time
                                                            FPT
              Login time of customers

How long does the customer wait until the next price change ?
The waiting time should depend on the login time.

The renewal-reward theorem gives

                     ( )                      ∞
                             , E (⋅⋅⋅) = ∫ ( ⋅⋅⋅)P(t )dt
                   E t2
     w=            2 E(t )                    0
                                                         FPT distribution
Results for the Weibull FPT pdf
               P (t ) =                 ( )
 We assume :
                            mt m−1
                W             a      exp −   tm
                                              a

                                        From empirical data analysis
                                                  m = 0.59, a = 50.855
                          Γ( m )
                             2
      w=a
                  1
                      m
                                          w = 42.236 [min]
                          Γ( m )
                             1
                                                        t2
                                         wsampling =    2t
                                                              49 [min]

                                             cf. For Poisson processes

                                          P(t ) = λ e − λt
                                        w= E(t) =20 [m w pling /2
                                                      in] sam
            W versus E(t)
             E (t ) = w gives
             mΓ(2 / m) = {Γ(1/ m)}
                                     2




          w > E (t )


“Inspection paradox”                     w < E (t )
                Summary
• The queueing theoretical approach provides us the average
  waiting time until the next price change after the customer
  checks the price
• The average waiting time obtained by the renewal reward
  theorem assuming Weibull FPT distribution is good
  agreement with empirical evidence
• FPT of Sony bank rate is not exponentially distributed, the
  jump process is non-Markovian
• The deviation from the average waiting time or more general
  formulation is possible. The result is available as the IEEE
  proceedings of fundation of computer intelligence 2007
• Explanation of the gap between theory and empirical data is
  explained: Sazuka, Inoue and Scalas (working paper)

								
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