Abstract
In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper.
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: PID2019-105986GB-C21
Award Identifier / Grant number: PID2020-118339GB-I00
Funding source: Departament d’Empresa i Coneixement, Generalitat de Catalunya
Award Identifier / Grant number: 2020-PANDE-00074
Funding statement: A. Aimi, C. Guardasoni and S. Sanfelici are members of the INdAM-GNCS research group, Italy. This work has been partially supported by INdAM-GNCS research projects as well as by grants PID2019-105986GB-C21 and PID2020-118339GB-I00 from the Spanish Ministry of Economy and Competitiveness, and grant 2020-PANDE-00074 from the Secretaria d’Universitats i Recerca del departament d’Empresa i Coneixement de la Generalitat de Catalunya.
A The COS BEM Heston Matlab Code
References
[1] N. Achtsis, R. Cools and D. Nuyens, Conditional sampling for barrier option pricing under the Heston model, Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proc. Math. Stat. 65, Springer, Heidelberg (2013), 253–269. 10.1007/978-3-642-41095-6_9Search in Google Scholar
[2] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ. 81 (1973), no. 3, 637–654. 10.1086/260062Search in Google Scholar
[3] A. Borovykh, A. Pascucci and C. W. Oosterlee, Pricing Bermudan options under local Lévy models with default, J. Math. Anal. Appl. 450 (2017), no. 2, 929–953. 10.1016/j.jmaa.2017.01.071Search in Google Scholar
[4] A. Borovykh, A. Pascucci and C. W. Oosterlee, Efficient computation of various valuation adjustments under local Lévy models, SIAM J. Financial Math. 9 (2018), no. 1, 251–273. 10.1137/16M1099005Search in Google Scholar
[5] P. P. Boyle and S. H. Lau, Bumping up against the barrier with the binomial method, J. Derivatives 1 (1994), 6–14. 10.3905/jod.1994.407891Search in Google Scholar
[6] P. Carr, D. B. Madan and R. H. Smith, Option valuation using the fast fourier transform, J. Comput. Finance 2 (1999), 61–73. 10.21314/JCF.1999.043Search in Google Scholar
[7] C. Chiarella, B. Kang and G. H. Meyer, The evaluation of barrier option prices under stochastic volatility, Comput. Math. Appl. 64 (2012), no. 6, 2034–2048. 10.1016/j.camwa.2012.03.103Search in Google Scholar
[8] S. Coskun and R. Korn, Pricing barrier options in the Heston model using the Heath–Platen estimator, Monte Carlo Methods Appl. 24 (2018), no. 1, 29–41. 10.1515/mcma-2018-0004Search in Google Scholar
[9] D. Duffie, Dynamic Asset Pricing Theory, Princeton University, Princeton, 1996. Search in Google Scholar
[10] F. Fang and C. W. Oosterlee, A novel pricing method for European options based on Fourier-cosine series expansions, SIAM J. Sci. Comput. 31 (2008/09), no. 2, 826–848. 10.1137/080718061Search in Google Scholar
[11] F. Fang and C. W. Oosterlee, Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions, Numer. Math. 114 (2009), no. 1, 27–62. 10.1007/s00211-009-0252-4Search in Google Scholar
[12] F. Fang and C. W. Oosterlee, A Fourier-based valuation method for Bermudan and barrier options under Heston’s model, SIAM J. Financial Math. 2 (2011), no. 1, 439–463. 10.1137/100794158Search in Google Scholar
[13] P. Glasserman, Monte Carlo Methods in Financial Engineering, Appl. Math. (New York) 53, Springer, New York, 2004. 10.1007/978-0-387-21617-1Search in Google Scholar
[14] P. Glasserman and J. Staum, Conditioning on one-step survival for barrier option simulations, Oper. Res. 49 (2001), no. 6, 923–937. 10.1287/opre.49.6.923.10018Search in Google Scholar
[15] A. Golbabai, L. V. Ballestra and D. Ahmadian, A highly accurate finite element method to price discrete double barrier options, Comput. Econ. 44 (2014), 153–173. 10.1007/s10614-013-9388-5Search in Google Scholar
[16] C. Guardasoni, Semi-analytical method for the pricing of barrier options in case of time-dependent parameters (with Matlab) codes, Commun. Appl. Ind. Math. 9 (2018), no. 1, 42–67. 10.1515/caim-2018-0004Search in Google Scholar
[17] C. Guardasoni and S. Sanfelici, Fast numerical pricing of barrier options under stochastic volatility and jumps, SIAM J. Appl. Math. 76 (2016), no. 1, 27–57. 10.1137/15100504XSearch in Google Scholar
[18] S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6 (1993), no. 2, 327–343. 10.1093/rfs/6.2.327Search in Google Scholar
[19] G. Junike and K. Pankrashkin, Precise option pricing by the COS method—how to choose the truncation range, Appl. Math. Comput. 421 (2022), Paper No. 126935. 10.1016/j.amc.2022.126935Search in Google Scholar
[20] A. Lipton and W. McGhee, Universal barriers, Risk 15 (2002), no. 5, 81–85. 10.1016/S1040-6190(02)00281-6Search in Google Scholar
[21] A. Lipton and A. Sepp, Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility, Wilmott 121 (2022), 70–84. 10.54946/wilm.11053Search in Google Scholar
[22] S. C. Maree, L. Ortiz-Gracia and C. W. Oosterlee, Pricing early-exercise and discrete barrier options by Shannon wavelet expansions, Numer. Math. 136 (2017), no. 4, 1035–1070. 10.1007/s00211-016-0858-2Search in Google Scholar
[23] L. Ortiz-Gracia and C. W. Oosterlee, Robust pricing of European options with wavelets and the characteristic function, SIAM J. Sci. Comput. 35 (2013), no. 5, B1055–B1084. 10.1137/130907288Search in Google Scholar
[24] L. Ortiz-Gracia and C. W. Oosterlee, A highly efficient Shannon wavelet inverse Fourier technique for pricing European options, SIAM J. Sci. Comput. 38 (2016), no. 1, B118–B143. 10.1137/15M1014164Search in Google Scholar
[25] P. Ritchken, On pricing barrier options, J. Derivatives 3 (1991), 19–28. 10.3905/jod.1995.407939Search in Google Scholar
[26] L. C. Rogers and O. Zane, Valuing moving barrier options, J. Comput. Finance 1 (1997), 5–11. 10.21314/CF.1997.003Search in Google Scholar
[27] S. Sanfelici, Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff, Decis. Econ. Finance 27 (2004), no. 2, 125–151. 10.1007/s10203-004-0046-1Search in Google Scholar
[28] R. Zvan, K. R. Vetzal and P. A. Forsyth, PDE methods for pricing barrier options, J. Econom. Dynam. Control 24 (2000), 1563–1590. 10.1016/S0165-1889(00)00002-6Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
