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doi: 10.3934/jimo.2021077
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## A modification of Galerkin's method for option pricing

 School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Western Australia, Australia

* Corresponding author: Mikhail Dokuchaev

Received  June 2020 Revised  January 2021 Early access April 2021

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.

Citation: Mikhail Dokuchaev, Guanglu Zhou, Song Wang. A modification of Galerkin's method for option pricing. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021077
##### References:
 [1] D. N. de G. Allen and R. V. Southwell, Relaxation methods applied to determine the motion, in two dimensions, of a viscous fluid past a fixed cylinder, Quart. J. Mech. Appl. Math., 8 (1955), 129-145.  doi: 10.1093/qjmam/8.2.129. [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. [3] M. Broadie, P. Glasserman and G. Jain, Enhanced Monte Carlo estimates for American option prices, J. Derivatives, 4, 25–44. doi: 10.3905/jod.1997.407983. [4] J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1997), 575–626. doi: 10.1137/0707048. [5] L. C. Evans, Partial Differential Equations, Second Edition. AMS, R.I., USA. 2010. [6] L. C. Evans, Partial Differential Equations (PDF), Graduate Studies in Mathematics, 19 (2nd ed.), Providence, R.I.: American Mathematical Society, 2010. doi: 10.1090/gsm/019. [7] L. C. G. Rogers and Z. Shi, The value of an Asian option, J. Appl. Probab., 32 (1995), 1077-1088.  doi: 10.2307/3215221. [8] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal., 24 (2004), 699-720.  doi: 10.1093/imanum/24.4.699. [9] S. Wang, X. 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. [10] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Oper. Res. Lett., 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006. [11] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [12] K. Zhang and S. Wang, Pricing American bond options using a penalty method, Automatica J. IFAC, 48 (2012), 472-479.  doi: 10.1016/j.automatica.2012.01.009.

show all references

##### References:
 [1] D. N. de G. Allen and R. V. Southwell, Relaxation methods applied to determine the motion, in two dimensions, of a viscous fluid past a fixed cylinder, Quart. J. Mech. Appl. Math., 8 (1955), 129-145.  doi: 10.1093/qjmam/8.2.129. [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. [3] M. Broadie, P. Glasserman and G. Jain, Enhanced Monte Carlo estimates for American option prices, J. Derivatives, 4, 25–44. doi: 10.3905/jod.1997.407983. [4] J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1997), 575–626. doi: 10.1137/0707048. [5] L. C. Evans, Partial Differential Equations, Second Edition. AMS, R.I., USA. 2010. [6] L. C. Evans, Partial Differential Equations (PDF), Graduate Studies in Mathematics, 19 (2nd ed.), Providence, R.I.: American Mathematical Society, 2010. doi: 10.1090/gsm/019. [7] L. C. G. Rogers and Z. Shi, The value of an Asian option, J. Appl. Probab., 32 (1995), 1077-1088.  doi: 10.2307/3215221. [8] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal., 24 (2004), 699-720.  doi: 10.1093/imanum/24.4.699. [9] S. Wang, X. 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. [10] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Oper. Res. Lett., 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006. [11] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [12] K. Zhang and S. Wang, Pricing American bond options using a penalty method, Automatica J. IFAC, 48 (2012), 472-479.  doi: 10.1016/j.automatica.2012.01.009.
Basis function $\phi_k(y)$ for $y_{k-1} = -1$, $y_k = 0$, $y_{k+1} = 1$, $\rho = 0.045$, and $\eta = -0.045$
Comparison of exact solution U and numerical solution V
Comparison of exact function U and numerical solution V for the case of non-constant $\sigma$
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
 $N$, $N_t$ 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
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
 $N$, $N_t$ 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
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
 $N$, $N_t$ 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
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
 $N$, $N_t$ 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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