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# A modification of Galerkin's method for option pricing

• * Corresponding author: Mikhail Dokuchaev
• 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: • • 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

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

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

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
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