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A modification of Galerkin's method for option pricing

  • * Corresponding author: Mikhail Dokuchaev

    * Corresponding author: Mikhail Dokuchaev 
Abstract Full Text(HTML) Figure(3) / Table(4) Related Papers Cited by
  • We present a novel method for solving a complicated form of a partial differential equation called the Black-Scholes equation arising from pricing European options. The novelty of this method is that we consider two terms of the equation, namely the volatility and dividend, as variables dependent on the state price. We develop a Galerkin finite element method to solve the problem. More specifically, we discretize the system along the state variable and build new basis functions which we use to approximate the solution. We establish convergence of the proposed method and numerical results are reported to show the proposed method is accurate and efficient.

    Mathematics Subject Classification: Primary: 15A18, 90C30, 90C33.

    Citation:

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  • Figure 1.  Basis function $ \phi_k(y) $ for $ y_{k-1} = -1 $, $ y_k = 0 $, $ y_{k+1} = 1 $, $ \rho = 0.045 $, and $ \eta = -0.045 $

    Figure 2.  Comparison of exact solution U and numerical solution V

    Figure 3.  Comparison of exact function U and numerical solution V for the case of non-constant $ \sigma $

    Table 1.  Error of calculation of the put option for r = 0

    $ N $, $ N_t $ E
    20, 20 0.003902114
    40, 40 0.006529201
    80, 80 0.007018533
    160,160 0.003593509
    320,320 0.0003050627
    640,640 7.043937e-05
     | Show Table
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    Table 2.  Error of calculation of the put option for r = 0.025

    $ N $, $ N_t $ E
    20, 20 0.003319792
    40, 40 0.005971542
    80, 80 0.006657621
    160,160 0.003484265
    320,320 0.001151376
    640,640 0.0003031325
     | Show Table
    DownLoad: CSV

    Table 3.  Error of calculation of the put option for r = 0.05

    $ N $, $ N_t $ E
    20, 20 0.002830843
    40, 40 0.00547276
    80, 80 0.006323301
    160,160 0.003382576
    320,320 0.001133573
    640,640 0.0003015444
     | Show Table
    DownLoad: CSV

    Table 4.  Error of calculation of the case of state-dependent volatility

    $ N $, $ N_t $ E
    20, 20 8.60202
    40, 40 0.1838133
    80, 80 0.09596427
    160,160 0.04898591
    320,320 0.02474154
    640,640 0.01243255
     | Show Table
    DownLoad: CSV
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    [2] L. Angermann and S. Wang, Convergence of a fitted finite volume method for the penalized Black-Scholes equation governing European and American option pricing, Numer. Math., 106 (2007), 1-40.  doi: 10.1007/s00211-006-0057-7.
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    [9] S. WangX. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, J. Optim. Theory Appl., 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3.
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