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Localization of the Black-Scholes equation using transparent

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					         Transparent boundary conditions
     Implementation and numerical results




Localization of the Black-Scholes equation
  using transparent boundary conditions

                               e
     Ekaterina Voltchkova and S´bastien Tordeux

                                        e
                         TSE – Universit´ Toulouse 1

                             IMT – INSA Toulouse




3d Conference on Numerical Methods in Finance
                         e
            Marne La Vall´e, April 15–17, 2009

            E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions
           Implementation and numerical results


Black-Scholes equation for European option prices

                σ2S 2 2
  ∂τ V (S, τ ) +     ∂ V (S, τ ) + rS ∂S V (S, τ ) − r V (S, τ ) = 0,
                 2 S
  V (S, T ) = (S − K )+ ,        S ∈ (0, +∞), τ ∈ [0, T )
                             V (S, τ )




                                              K                    S
  Asymptotic behavior:
           V (S, τ )          S − K e −r (T −τ ) ,              S −→ +∞,
           V (S, τ )          0,                                S −→ 0+ .
                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions
           Implementation and numerical results


Change of variables

        x = ln (S/S0 ),             t = T − τ,            w (x, t) = V (S, τ )

                   σ2 2                σ2
  ∂t w (x, t) =      ∂x w (x, t) + r −                       ∂x w (x, t) − r w (x, t).
                   2                   2
  We suppress the 0-order term:

                            u(x, t) = exp(rt) w (x, t)

  Advection diffusion equation with constant coefficients:
                    σ2 2                                                                     σ2
    ∂t u(x, t) =      ∂ u(x, t) + µ ∂x u(x, t)                      with µ = r −
                    2 x                                                                      2
                       x ∈ (−∞, ∞),                    t ∈ (0, T ]
                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                 Transparent boundary conditions
             Implementation and numerical results


Initial condition
  Put
         V (S, T ) = (K − S)+                  ⇔      u(x, 0) = (K − S0 e x )+ .
  Call
         V (S, T ) = (S − K )+                 ⇔      u(x, 0) = (S0 e x − K )+ .
  Other payoff functions

                   V (S, T )




                                K1                        K2                  S


                    E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                 Transparent boundary conditions
             Implementation and numerical results


Initial condition
  Put
         V (S, T ) = (K − S)+                  ⇔      u(x, 0) = (K − S0 e x )+ .
  Call
         V (S, T ) = (S − K )+                 ⇔      u(x, 0) = (S0 e x − K )+ .
  Other payoff functions

                   V (S, T )




                                K1                        K2                  S


                    E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                 Transparent boundary conditions
             Implementation and numerical results


Initial condition
  Put
         V (S, T ) = (K − S)+                  ⇔      u(x, 0) = (K − S0 e x )+ .
  Call
         V (S, T ) = (S − K )+                 ⇔      u(x, 0) = (S0 e x − K )+ .
  Other payoff functions

                   V (S, T )




                                K1            K2                              S


                    E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions
           Implementation and numerical results


Localization on a bounded computational domain

             x ∈ (−∞, ∞)                          −→          x ∈ (x− , x+ )
 Standard approach
 Dirichlet or Neumann boundary conditions based on the
 asymptotics of the solution.
 Example (call):

 Dirichlet:      V (S− , τ ) = 0                   ⇔          u(x− , t) = 0
 V (S+ , τ ) = S+ − Ke −r (T −τ )                  ⇔          u(x+ , t) = S0 e x+ +rt − K .


 Neumann:              ∂S V (S− , τ ) = 0                ⇔           ∂x u(x− , t) = 0
                       ∂S V (S+ , τ ) = 1                ⇔           ∂x u(x+ , t) = S0 e x+ +rt .

                  E. Voltchkova and S. Tordeux     Transparent boundary conditions for the BS equation
            Transparent boundary conditions
        Implementation and numerical results


Localization on a bounded computational domain

          x ∈ (−∞, ∞)                          −→          x ∈ (x− , x+ )
     Standard approach
        Dirichlet or Neumann?
        How to chose x− and x+ ?
     Transparent boundary conditions
        Work for any interval (x− , x+ )
        (provided it contains the singularities).
        Theoretically exact conditions.
        Numerically almost exact.
        Almost as easy to implement as Neumann or Dirichlet.

               E. Voltchkova and S. Tordeux     Transparent boundary conditions for the BS equation
               Transparent boundary conditions         Exact conditions
           Implementation and numerical results        Approximate conditions


Transparent boundary conditions
for the advection diffusion equation
                                     t




                                                  x+                     x


  Idea: the initial condition on {x > x+ } and boundary values
  on {x = x+ } define completely the solution of the PDE in the
  gray domain. In particular,

             ∂x u(x+ , t) = S[u(x+ , t)t>0 ,                        u(x, 0)x>x+ ]

  Find the operator S in an explicit form and use it as a
  boundary condition when solving for x ≤ x+ .
                  E. Voltchkova and S. Tordeux         Transparent boundary conditions for the BS equation
              Transparent boundary conditions      Exact conditions
          Implementation and numerical results     Approximate conditions


Laplace Transform
                                          +∞
                  u(x, p) =                      u(x, t) exp(−pt)dt
                                      0
 Laplace transform of the time derivative:
                     ∂t u(x, p) = p u(x, p) + u(x, 0)
 We introduce
                        uH (x, t) = u(x, t) − ϕ(x, t)
 where ϕ is a solution of the PDE with ϕ(x, 0) = u(x, 0).
 Important: u(x, 0) has no singularity on x > x+ ,
            and ϕ(x, t) has a simple explicit form, e.g.
     Call: ϕ(x, t) = S0 e x+rt − K ,                          Put: ϕ(x, t) = 0.
                 E. Voltchkova and S. Tordeux      Transparent boundary conditions for the BS equation
                Transparent boundary conditions    Exact conditions
            Implementation and numerical results   Approximate conditions


Solution in the Laplace domain

  In the quadrant {x > x+ } we have a homogeneous problem:

                 σ2 2
   ∂t uH (x, t) =  ∂ uH (x, t) + µ ∂x uH (x, t),                            for x > x+ ,            t≥0
                 2 x
   uH (x, 0) = 0, for x > x+

  Apply the Laplace transform:

                σ2 2
    puH (x, p) = ∂x uH (x, p) + µ ∂x uH (x, p),                               for x > x+ ;
                2

  This is an ODE on uH (·, p).


                   E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                Transparent boundary conditions     Exact conditions
            Implementation and numerical results    Approximate conditions


Solution in the Laplace domain (continued)

                       σ2 2
    puH (x, p) =         ∂ uH (x, p) + µ ∂x uH (x, p),                         for x > x+ ;
                       2 x
  Solving this ODE gives
                         uH (x, p) = A− e λ− x + A+ e λ+ x
  with
                                       µ    |µ|                  2pσ 2
                         λ± = −          2
                                           ± 2           1+
                                       σ    σ                     µ2
  To avoid exploding solutions for x → +∞, set A+ = 0 :

                                                   µ    |µ|                  2pσ 2
         uH (x, p) = A− exp −                        2
                                                       + 2           1+            x .
                                                   σ    σ                     µ2

                   E. Voltchkova and S. Tordeux     Transparent boundary conditions for the BS equation
             Transparent boundary conditions     Exact conditions
         Implementation and numerical results    Approximate conditions


The Dirichlet-to-Neumann Map
 A Dirichlet-to-Neumann operator S maps the Dirichlet value
 to the Neumann value of the solution of a PDE

                       ∂x uH = SuH               at      x = x+

 In the Laplace domain we have for uH

                                                µ    |µ|                  2pσ 2
      uH (x, p) = A exp −                         2
                                                    + 2          1+             x .
                                                σ    σ                     µ2

 and consequently

                                   µ    |µ|                  2pσ 2
      ∂x uH (x+ , p) = −             2
                                       + 2            1+           uH (x+ , p).
                                   σ    σ                     µ2

                E. Voltchkova and S. Tordeux     Transparent boundary conditions for the BS equation
            Transparent boundary conditions         Exact conditions
        Implementation and numerical results        Approximate conditions


Some useful Laplace transforms

                    function                                   Laplace transform
                          1                                                   1
                         √                                                   √
                           πt                                                  p
                         t
         1                    u(s)                                        u(p)
        √                    √      ds                                    √
          π          0         t −s                                         p
                t
       1             e −a(t−s)                                          u(p)
      √              √         u(s)ds                                  √
        π   0           t −s                                             p+a
                                 t
    1                                e −a(t−s)                     √
   √ ∂t + a                          √         u(s)ds                  p + a u(p)
     π                       0          t −s

                    E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
              Transparent boundary conditions      Exact conditions
          Implementation and numerical results     Approximate conditions


The exact DTN map: a non local operator
 The Dirichlet-to-Neumann map:
     a multiplication operator in Laplace domain
     an integro-differential operator in time domain

                           µ
   ∂x uH (x+ , t) = −         uH (x+ , t)−
                           σ2
                                              µ2
    2     1
          2        µ2                 t
                                          e − 2σ2 (t−s) uH (x+ , s)
 − 2          ∂t + 2                               √                ds          =: S+ uH (x+ , t)
   σ π            2σ              0                  t −s
 Coming back to the non-homogeneous solution we obtain

     ∂x u(x+ , t) = S+ u(x+ , t) + (∂x ϕ(x+ , t) − S+ ϕ(x+ , t))

                 E. Voltchkova and S. Tordeux      Transparent boundary conditions for the BS equation
                 Transparent boundary conditions            Exact conditions
             Implementation and numerical results           Approximate conditions


The Dirichlet-to-Neumann map at x = x−
                                                        t




                                           x−                                   x

 Using the same arguments for {x < x− }, we obtain

        ∂x u(x− , t) = S− u(x− , t) + (∂x ψ(x− , t) − S− ψ(x− , t))

 with
                                                                                          µ2
             µ         2                            1
                                                    2            µ2               t
                                                                                      e − 2σ2 (t−s) u(s)
 S− u(t) := − 2 u(t)+ 2                                     ∂t + 2                         √             ds.
             σ        σ π                                       2σ            0               t −s


                    E. Voltchkova and S. Tordeux            Transparent boundary conditions for the BS equation
                   Transparent boundary conditions    Exact conditions
               Implementation and numerical results   Approximate conditions


Implementation of the exact transparent conditions

        PDE: usual Finite Difference scheme (Cranck-Nicolson).
        Boundary: integration by parts to remove the singularity
        + trapezoidal rule.
   Numerical experiments


        Advantages
                Very good precision. The error is only due to the
                discretization of the integral which may be improved.
        Drawbacks
                Messy formulas to implement.
                At each time iteration, boundary conditions involve all
                previous values of u on the boundary (not only u n−1 ).


                      E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions        Exact conditions
           Implementation and numerical results       Approximate conditions


Approximation of the transparent conditions
  First idea: approximate the integro-differential
  Dirichlet-to-Neumann operator by a differential operator
  by approximating its Laplace symbol by a polynomial.

                                     µ   |µ|                      2pσ 2
       ∂x uH (x+ , p) = −               + 2               1+            uH (x+ , p).
                                     σ2  σ                         µ2

                                                  ⇓
      ∂x uH (x+ , p) = − a0 + a1 p + · · · + am p m uH (x+ , p).
                                                  ⇓
                                                  m
      ∂x uH (x+ , t) = − a0 + a1 ∂t + · · · + am ∂t uH (x+ , t).

                  E. Voltchkova and S. Tordeux        Transparent boundary conditions for the BS equation
               Transparent boundary conditions    Exact conditions
           Implementation and numerical results   Approximate conditions


Numerical results for polynomial approximation


  We have performed numerical experiments using Taylor
  expansion of the symbol at p = 0.

  Unfortunately, this approximation does not work well:
      The precision is not better than for Dirichlet or Neumann.
      Increasing m does not improve the precision.

                                    (see next slide)




                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions    Exact conditions
           Implementation and numerical results   Approximate conditions



r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T =1
                                                  (Loading movie...)
Nx = 101
Nt = 100

Dirichlet
Neumann
Taylor



                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions    Exact conditions
           Implementation and numerical results   Approximate conditions



r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T =1
                                                  (Loading movie...)
Nx = 101
Nt = 100

Dirichlet
Neumann
Taylor



                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
              Transparent boundary conditions    Exact conditions
          Implementation and numerical results   Approximate conditions


Comments on the polynomial approximation

 The polynomial approximation does not give good results.
 The reason is that polynomials are not a good approximation
 for 1 + (2σ 2 /µ2 )p ( see graphics ):
     the radius of convergence is small
     ⇒ bad approximation for p > rconv = µ2 /2σ 2 ;
 Remark: small p (low frequencies) correspond to large t,
          large p (high frequencies) correspond to small t.

 Example: if r = 0.05, σ = 0.2, then rconv ≈ 0.01.
 Heuristically, the approximation is bad for t < 100!
 We have also tried Taylor expansion at a different point p > 0.
 The results are globally the same.

                 E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                Transparent boundary conditions    Exact conditions
            Implementation and numerical results   Approximate conditions


Approximation by rational functions
  Alternative approach: approximate by rational functions.

      µ |µ|                  2pσ 2                              β1           βmr
  −     +            1+                   ≈ η0 + η1 p +              +···+
      σ2 σ2                   µ2                              p + γ1       p + γmr

  This leads to the boundary condition

  ∂x uH (x+ , t) = η0 uH (x+ , t) + η1 ρ0 (t) + β1 ρ1 (t) + · · · + βmr ρmr (t)

  where the auxiliary functions ρi satisfy the following ODEs:

      ρ0 (t) = ∂t uH (x+ , t),
      ∂t ρi (t) + γi ρi (t) = uH (x+ , t),               ρi (0) = 0,                i = 1, . . . , mr

                   E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                    Transparent boundary conditions    Exact conditions
                Implementation and numerical results   Approximate conditions


Approximation by rational functions

        √
            z
        Approximation with 2 interpolation points
3




2




1                                                                                 8

                                                                                  4


                                                                    z                                             z
    1   2          3      4       5      6       7     8     9                            20      40         60



                       E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                    Transparent boundary conditions    Exact conditions
                Implementation and numerical results   Approximate conditions


Approximation by rational functions

        √
            z
        Approximation with 3 interpolation points
3




2




1                                                                                 8

                                                                                  4


                                                                    z                                             z
    1   2          3      4       5      6       7     8     9                            20      40         60



                       E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                    Transparent boundary conditions    Exact conditions
                Implementation and numerical results   Approximate conditions


Approximation by rational functions

        √
            z
        Approximation with 4 interpolation points
3




2




1                                                                                 8

                                                                                  4


                                                                    z                                             z
    1   2          3      4       5      6       7     8     9                            20      40         60



                       E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                    Transparent boundary conditions    Exact conditions
                Implementation and numerical results   Approximate conditions


Approximation by rational functions

        √
            z
        Approximation with 5 interpolation points
3




2




1                                                                                 8

                                                                                  4


                                                                    z                                             z
    1   2          3      4       5      6       7     8     9                            20      40         60



                       E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                    Transparent boundary conditions    Exact conditions
                Implementation and numerical results   Approximate conditions


Approximation by rational functions

        √
            z
        Approximation with 6 interpolation points
3




2




1                                                                                 8

                                                                                  4


                                                                    z                                             z
    1   2          3      4       5      6       7     8     9                            20      40         60



                       E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions
           Implementation and numerical results



r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T =1
                                                  (Loading movie...)
Nx = 101
Nt = 100

Dirichlet
Neumann
Rational



                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions
           Implementation and numerical results



r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T =1
                                                  (Loading movie...)
Nx = 101
Nt = 100

Transparent
Rational




                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions
           Implementation and numerical results



r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T =1
                                                  (Loading movie...)
Nx = 101
Nt = 100

Transparent
Rational




                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
               Transparent boundary conditions
           Implementation and numerical results



r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T =1
                                                  (Loading movie...)
Nx = 101
Nt = 100

Transparent
Rational




                  E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
                 Transparent boundary conditions
             Implementation and numerical results


Comments on the implementation
 The implementation is easy due to the introduction of
 auxiliary functions. The vector of unknowns is
   u n = (λn l , . . . , λn , λn , u0 , u1 , . . . , uN , uN+1 , ρn , ρn , . . . , ρn r )
   ˜       m              1    0
                                    n    n            n    n
                                                                  0    1            m

 Iterations: A˜n+1 = b(˜n ) with constant matrix A
              u        u                                                              show matrix   .
  ...
    n+1              n+1
  aui−1 + buin+1 + cui+1 = dui−1 + euin + fui+1 ,
                             n              n
                                                                             i = 1, . . . , N
   n+1       n+1
  uN+1 −    uN−1
                                                                   ˜
              = η0 u n+1 + η1 ρn+1 + β1 ρn+1 + · · · + βmr ρn+1 + (Sφ)n+1
                                                            mr
                               0         1                            N
      2∆x
   n+1    n
  uN − uN     1
            = (ρn+1 + ρn )0
     ∆t       2 0
  ρn+1 − ρn  1 n+1
   i      i
            = (uN − γi ρn+1 + uN − γi ρn )
                            i
                                   n
                                           i
     ∆t      2

                    E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
             Transparent boundary conditions
         Implementation and numerical results


Conclusion

 Transparent conditions are:
     suitable for any small computational domain
     (containing the singularities)
     easy to implement
     apply to a wide range of initial conditions
     Remark: apply without changes to σ(S) provided that
     σ = const outside of (S− , S+ ).
 ⇒ Should be used instead of Dirichlet and Neumann
 Future work
     PDE with time-dependent coefficients, PIDE.


                E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
             Transparent boundary conditions
         Implementation and numerical results


Bibliography


     Halpern Laurence, Artificial boundary conditions for the
     linear advection diffusion equation. Math. Comp. (86)
     Joly Patrick, Pseudo-transparent boundary conditions for
     the diffusion equation. M2AS (89)
     Dubach Eric, Artificial boundary conditions for diffusion
     equations: numerical study. JCAM (95)
     Halpern L. and Rauch J., Absorbing boundary conditions
     for diffusion equation, Numer. math. (95)




                E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
             Transparent boundary conditions
         Implementation and numerical results


Bibliography


     Halpern Laurence, Artificial boundary conditions for the
     linear advection diffusion equation. Math. Comp. (86)
     Joly Patrick, Pseudo-transparent boundary conditions for
     the diffusion equation. M2AS (89)
     Dubach Eric, Artificial boundary conditions for diffusion
     equations: numerical study. JCAM (95)
     Halpern L. and Rauch J., Absorbing boundary conditions
     for diffusion equation, Numer. math. (95)




                E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
             Transparent boundary conditions
         Implementation and numerical results


Bibliography


     Halpern Laurence, Artificial boundary conditions for the
     linear advection diffusion equation. Math. Comp. (86)
     Joly Patrick, Pseudo-transparent boundary conditions for
     the diffusion equation. M2AS (89)
     Dubach Eric, Artificial boundary conditions for diffusion
     equations: numerical study. JCAM (95)
     Halpern L. and Rauch J., Absorbing boundary conditions
     for diffusion equation, Numer. math. (95)




                E. Voltchkova and S. Tordeux    Transparent boundary conditions for the BS equation
r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T =1
                                             (Loading movie...)
Nx = 101
Nt = 100

Dirichlet
Neumann
Transparent
 next




              E. Voltchkova and S. Tordeux   Transparent boundary conditions for the BS equation
r = 0.05
σ = 0.2
S0 = 100

Put option
K = 100
T = 10
                                             (Loading movie...)
Nx = 101
Nt = 100

Dirichlet
Neumann
Transparent
 back




              E. Voltchkova and S. Tordeux   Transparent boundary conditions for the BS equation
Taylor approximation of the symbol
                              1
         μ   |μ|  2pσ 2       2
            + 2 1+ 2
   6     σ2  σ     μ
         Taylor order 0
   5

   4

   3

   2

   1

   0
                                                            p
                .1                  .2                .3
  −1

  −2

  −3


             E. Voltchkova and S. Tordeux   Transparent boundary conditions for the BS equation
Taylor approximation of the symbol
                              1
         μ   |μ|  2pσ 2       2
            + 2 1+ 2
   6     σ2  σ     μ
         Taylor order 1
   5

   4

   3

   2

   1

   0
                                                            p
                .1                  .2                .3
  −1

  −2

  −3


             E. Voltchkova and S. Tordeux   Transparent boundary conditions for the BS equation
Taylor approximation of the symbol
                              1
         μ   |μ|  2pσ 2       2
            + 2 1+ 2
   6     σ2  σ     μ
         Taylor order 2
   5

   4

   3

   2

   1

   0
                                                            p
                .1                  .2                .3
  −1

  −2

  −3


             E. Voltchkova and S. Tordeux   Transparent boundary conditions for the BS equation
Taylor approximation of the symbol
                              1
         μ   |μ|  2pσ 2       2
            + 2 1+ 2
   6     σ2  σ     μ
         Taylor order 3
   5

   4

   3

   2

   1

   0
                                                            p
                .1                  .2                .3
  −1

  −2

  −3


             E. Voltchkova and S. Tordeux   Transparent boundary conditions for the BS equation
Taylor approximation of the symbol
                              1
         μ   |μ|  2pσ 2       2
            + 2 1+ 2
   6     σ2  σ     μ
         Taylor order 4
   5

   4

   3

   2
                                                                             back
   1

   0
                                                            p
                .1                  .2                .3
  −1

  −2

  −3


             E. Voltchkova and S. Tordeux   Transparent boundary conditions for the BS equation
Matrix A of the linear system A˜n+1 = b n
                               u

  1 + Δt γ1
      2                          − Δt
                                    2
         1+   Δt
              2 γ2               − Δt
                                    2
                1 + Δt γ3
                    2            − Δt
                                    2
                          − Δt
                            2      1
  −2Δxα1 −2Δxα2 −2Δxα3 −2Δxξ1 −1 −2Δxξ0 1
                               a b      c



                                                     a b c
                                                    −1 −2Δxη0 1 −2Δxη1 −2Δxβ1 −2Δxβ2 −2Δxβ3
                                                          1     − Δt
                                                                  2
                                                        −2Δt
                                                                       1 + Δt δ1
                                                                           2
                                                        − Δt
                                                           2                     1 + Δt δ2
                                                                                     2
                                                        −2Δt
                                                                                           1 + Δt δ3
                                                                                               2

                     E. Voltchkova and S. Tordeux     Transparent boundary conditions for the BS equation

				
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