doi: 10.3934/jimo.2021077

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 Published  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1997), 575–626. doi: 10.1137/0707048.  Google Scholar

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L. C. Evans, Partial Differential Equations, Second Edition. AMS, R.I., USA. 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11] P. WilmottJ. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993.   Google Scholar
[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1997), 575–626. doi: 10.1137/0707048.  Google Scholar

[5]

L. C. Evans, Partial Differential Equations, Second Edition. AMS, R.I., USA. 2010. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11] P. WilmottJ. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993.   Google Scholar
[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.  Google Scholar

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
$ 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
$ 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
$ 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
$ 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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