In this paper, we consider discretely monitored double barrier option based on the Black-Scholes partial differential equation. In this scenario, the option price can be computed recursively upon the heat equation solution. Thus we propose a numerical solution by projection method. We implement this method by considering the Chebyshev polynomials of the second kind. Finally, numerical examples are carried out to show accuracy of the presented method and demonstrate acceptable accordance of our method with other benchmark methods.
Citation: Kazem Nouri, Milad Fahimi, Leila Torkzadeh, Dumitru Baleanu. Numerical method for pricing discretely monitored double barrier option by orthogonal projection method[J]. AIMS Mathematics, 2021, 6(6): 5750-5761. doi: 10.3934/math.2021339
In this paper, we consider discretely monitored double barrier option based on the Black-Scholes partial differential equation. In this scenario, the option price can be computed recursively upon the heat equation solution. Thus we propose a numerical solution by projection method. We implement this method by considering the Chebyshev polynomials of the second kind. Finally, numerical examples are carried out to show accuracy of the presented method and demonstrate acceptable accordance of our method with other benchmark methods.
[1] | H. Geman, M. Yor, Pricing and hedging double-barrier options: A probabilistic approach, Math. Financ., 6 (1996), 365–378. doi: 10.1111/j.1467-9965.1996.tb00122.x |
[2] | A. Pelsser, Pricing double barrier options using Laplace transforms, Financ. Stoch., 4 (2000), 95–104. doi: 10.1007/s007800050005 |
[3] | M. Rubinstein, E. Reiner, Breaking down the barriers, Risk, 4 (1991), 28–35. |
[4] | F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637–654. doi: 10.1086/260062 |
[5] | G. Fusai, I. D. Abraham, C. Sgarra, An exact analytical solution for discrete barrier options, Financ. Stoch., 10 (2006), 1–26. doi: 10.1007/s00780-005-0170-y |
[6] | A. Awasthi, R. TK, An accurate solution for the generalized Black-Scholes equations governing option pricing, AIMS Mathematics, 5 (2020), 2226–2243. doi: 10.3934/math.2020147 |
[7] | J. Biazar, F. Goldoust, Multi-dimensional Legendre wavelets approach on the Black-Scholes and Heston Cox Ingersoll Ross equations, AIMS Mathematics, 4 (2019), 1046–1064. doi: 10.3934/math.2019.4.1046 |
[8] | W. L. Wan. Multigrid method for pricing European options under the CGMY process, AIMS Mathematics, 4 (2019), 1745–1767. |
[9] | Z. Guo, X. Wang, Y. Zhang, Option pricing of geometric Asian options in a sub-diffusive Brownian motion regime, AIMS Mathematics, 4 (2020), 5332–5343. |
[10] | R. Merton, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141–183. doi: 10.2307/3003143 |
[11] | B. Kamrad, P. Ritchken, Multinomial approximating models for options with k state variables, Manage. Sci., 37 (1991), 1640–1652. |
[12] | Y. K. Kwok, Mathematical models of financial derivatives, Springer, 1998. |
[13] | Y. Hong, S. Lee, T. Li, Numerical method of pricing discretely monitored barrier option, J. Comput. Appl. Math., 278 (2015), 149–161. doi: 10.1016/j.cam.2014.08.022 |
[14] | R. Farnoosh, H. Fezazadeh, A. M. Sobhani, A numerical method for discrete single barrier option pricing with time-dependent parameters, Comput. Econ., 48 (2016), 131–145. doi: 10.1007/s10614-015-9506-7 |
[15] | R. Farnoosh, A. Sobhani, M. Beheshti, Efficient and fast numerical method for pricing discrete double barrier option by projection method, Comput. Econ., 73 (2017), 1539–1545. |
[16] | A. Golbabai, L. Ballestra, D. Ahmadian, A highly accurate finite element method to price discrete double barrier options, Comput. Econ., 44 (2014), 153–173. doi: 10.1007/s10614-013-9388-5 |
[17] | M. Milev, A. Tagliani, Numerical valuation of discrete double barrier options, J. Comput. Appl. Math., 233 (2010), 2468–2480. doi: 10.1016/j.cam.2009.10.029 |
[18] | P. A. Samoelson, Rational theory of warrant pricing, Industrial Management Review, 6 (1965), 13–39. |
[19] | J. C. Hull, Options futures and other derivatives, Upper Saddle River, NJ: Prentice Hall, 2009. |
[20] | P. G. Zhang, Exotic options: A guide to second generation options, World Scientific, 1997. |
[21] | W. A. Strauss, Partial Differential Equations: An introduction, New York, Wiley, 1992. |
[22] | K. Maleknejad, K. Nouri, L. Torkzadeh, Operational matrix of fractional integration based on the shifted second kind Chebyshev polynomials for solving fractional differential equations, Mediterr. J. Math., 13 (2016), 1377–1390. doi: 10.1007/s00009-015-0563-x |
[23] | J. P. Boyd, Chebyshev and fourier spectral methods, Courier Corporation, 2001. |
[24] | W. Gautschi, Orthogonal polynomials: computation and approximation, Oxford University Press on Demand, 2004. |
[25] | A. Galantai, Projectors and Projection Methods, Springer Science & Business Media, 2004. |
[26] | T. S. Dai, Y. D. Lyuu, The bino-trinomial tree: A simple model for efficient and accurate option pricing, J. Deriv., 17 (2010), 7–24. doi: 10.3905/jod.2010.17.4.007 |
[27] | K. L. Judd, Numerical methods in economics, MIT Press, 1998. |