Pricing Modified Barrier Options Using the Bino-Trinomial Tree Model: A Strategy for Loss Minimization

Rima Aulia Rahayu, Fitriani Agustina, Kuntjoro Adji Sidarto

Abstract


A particular exotic option that is widely traded in the global financial market is the barrier option. Barrier options are attractive because they have a limit that must be reached to activate the option. These limits may be utilized by investors as a point of reference to minimize potential losses. Accordingly, the researcher attempts to use the bino-trinomial tree model as a new approach to minimize losses. The purpose of this study is to analyze the bino-trinomial tree model to provide investors with more flexible hedging experience. The bino-trinomial tree model is obtained by combining the trinomial tree model at the first stage, then the binomial tree model at a further stage. This analysis was conducted by calculating the type of knock-out european call options. The results demonstrate that this model can effectively, accurately and flexibly manage the complex options required by modern investors, including multi-step single moving barrier options and single window barrier options.


Keywords


barrier options; bino-trinomial tree; trinomial tree

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References


[1] L. Miljković. “The role of financial derivatives in financial risks management”. MEST Journal 11.1 (2023), pp. 97–104. DOI: https://doi.org/10.12709/mest.11.11.01.09.

[2] P. Jalaludin, A. Rahman, A. Nuraini, and R. Amigo. “Perhitungan harga opsi eropa dengan metode trinomial pada perusahaan mitsubishi”. Jurnal Lentera Akuntansi 9.1 (2024), pp. 116–124. DOI: https://doi.org/10.34127/jrakt.v9i1.1179.

[3] S. R. Gottimukkala. “Optimizing exotic option pricing: Monte carlo simulation and variance reduction techniques”. Preprints 2024.9 (2024), pp. 1–20. DOI: https://doi.org/10.20944/preprints202409.2256.v1.

[4] K. Yao and Z. Qin. “Barrier option pricing formulas of an uncertain stock model”. Fuzzy Optimization and Decision Making 20.1 (Mar. 2021), pp. 81–100. DOI: https://doi.org/10.1007/s10700-020-09333-w.

[5] M. Gao and Z. Wei. “The barrier binary options”. Journal of Mathematical Finance 10.1 (2020), pp. 140–156. DOI: https://doi.org/10.4236/jmf.2020.101010.

[6] N. Ratanov. “Jump-telegraph market model: Barrier binary options”. Mathematical and Statistical Methods for Actuarial Sciences and Finance. MAF 2022. M. Corazza, C. Perna, and C. Pizzi, Eds. Springer, Cham, 2022, pp. 390–396. DOI: https://doi.org/10.1007/978-3-030-99638-3_63.

[7] S. M. Nuugulu, F. Gideon, and K. C. Patidar. “An efficient numerical method for pricing double-barrier options on an underlying stock governed by a fractal stochastic process”. Fractal and Fractional 7.5 (May 2023), p. 389. DOI: https://doi.org/10.3390/fractalfract7050389.

[8] C. Zhou, Y. Wang, and M. Zhang. “Pricing barrier options under stochastic volatility: Improvements on the black-scholes framework”. Journal of Computational and Applied Mathematics 405 (2022), p. 113041. DOI: https://doi.org/10.1016/j.cam.2022.113041.

[9] A. Itkin and D. Muravey. “Semi-closed form prices of barrier options in the Hull–White model”. The Journal of Derivatives 29.1 (2021), pp. 9–26. Based on arXiv:2004.09591; Risk.net Dec 2020. DOI: https://doi.org/10.48550/arXiv.2004.09591.

[10] P. Roul. “A high accuracy numerical method and its convergence for time-fractional black-scholes equation governing european options”. Applied Numerical Mathematics 151 (2020), pp. 472–493. DOI: https://doi.org/10.1016/j.apnum.2019.11.004.

[11] Y. Jerbi and R. Bouzid. “Barrier options pricing under stochastic volatility using monte carlo simulation”. International Journal of Finance & Banking Studies 12.3 (2023), pp. 32–39. DOI: https://doi.org/10.20525/ijfbs.v12i3.2851.

[12] L. V. Ballestra. “Enhancing finite difference approximations for double barrier options: Mesh optimization and repeated richardson extrapolation”. Computational Management Science 18.2 (2021), pp. 239–263. DOI: https://doi.org/10.1007/s10287-021-00394-9.

[13] J.-Y. Wang, C.-J. Wang, T.-S. Dai, T.-C. Chen, L.-C. Liu, and L. Zhou. “Efficient and robust combinatorial option pricing algorithms on the trinomial lattice for polynomial and barrier options”. Mathematical Problems in Engineering 2022, Article 5843491 (2022). DOI: https://doi.org/10.1155/2022/5843491.

[14] J. Talponen and M. Turunen. “Option pricing: A yet simpler approach”. Decisions in Economics and Finance 45 (2022), pp. 57–81. DOI: https://doi.org/10.1007/s10203-021-00338-7.

[15] G. Leduc and K. J. Palmer. “What a difference one probability makes in the convergence of binomial trees”. International Journal of Theoretical and Applied Finance 23.6 (2020), pp. 1–26. DOI: https://doi.org/10.1142/S0219024920500405.

[16] D. Josheski and M. Apostolov. “A review of the binomial and trinomial models for option pricing and their convergence to the black-scholes model determined option prices”. Econometrics. Ekonometria. Advances in Applied Data Analysis 24.2 (Jul. 2020), pp. 53–85. DOI: https://doi.org/10.15611/eada.2020.2.05.

[17] M. H. Asnawi and A. Aziz. “Metode enhanced trinomial pada aproksimasi numerik pada barrier options pricing”. Jurnal Riset Mahasiswa Matematika 1.1 (Oct. 2021), pp. 23–31. DOI: https://doi.org/10.18860/jrmm.v1i1.13412.

[18] T.-S. Dai and Y.-D. Lyuu. “The bino-trinomial tree: A simple model for efficient and accurate option pricing”. Journal of Derivatives 17.4 (Jun. 2010), pp. 7–24. DOI: https://doi.org/10.3905/jod.2010.17.4.007.

[19] F. Agustina, N. Sumarti, and K. A. Sidarto. “Construction of the bino-trinomial method using the fuzzy set approach for option pricing”. Journal of the Indonesian Mathematical Society 30.2 (Aug. 2024), pp. 179–204. DOI: https://doi.org/10.22342/jims.30.2.1775.179-204.

[20] Y.-M. Lu and Y.-D. Lyuu. “Very fast algorithms for implied barriers and moving-barrier options pricing”. Mathematics and Computers in Simulation 205 (Mar. 2023), pp. 251–271. DOI: https://doi.org/10.1016/j.matcom.2022.09.018.

[21] H. Lee, Y. Kye, B. Kong, and S. Song. “Multi-step double barrier options under time-varying interest rates”. The North American Journal of Economics and Finance (2025), p. 102372. DOI: https://doi.org/10.1016/j.najef.2025.102372.

[22] H. Lee, E. Kim, and B. Ko. “Valuing lookback options with barrier”. The North American Journal of Economics and Finance 60 (2022), p. 101660. DOI: https://doi.org/10.1016/j.najef.2022.101660.

[23] H. Lee, G. Lee, and S. Song. “Multi-step reflection principle and barrier options”. Journal of Futures Markets 42.4 (2022), pp. 692–721. DOI: https://doi.org/10.1002/fut.22306.

[24] Riaman, K. Parmikanti, I. Irianingsih, K. Joebaedi, and S. Supian. “Convergence of binomial tree methods to black–scholes model on determining stock option prices”. IOP Conference Series: Materials Science and Engineering. Ser. ISE, Vol. 567, 2019, p. 012013. DOI: https://doi.org/10.1088/1757-899X/567/1/012013.

[25] D. Lilyana, B. Subartini, A. Supriatna, et al. “Calculation of call option using trinomial tree method and black-scholes method case study of microsoft corporation”. Journal of Physics: Conference Series. IOP Publishing, Vol. 1722, 2021, p. 012064. DOI: https://doi.org/10.1088/1742-6596/1722/1/012064.

[26] S. A. Jumana and A. S. Hossain. “Pricing exotic options using some lattice procedures”. GANIT: Journal of Bangladesh Mathematical Society 41.1 (2021), pp. 26–40. DOI: https://doi.org/10.3329/ganit.v41i1.55024.

[27] R. Mendoza-Ramírez, R. Silva, R. Domínguez-Mora, E. Juan-Diego, and E. Carrizosa-Elizondo. “Comparison of two convergence criterion in the optimization process using a recursive method in a multi-reservoir system”. Water 14.19 (2022), p. 2952. DOI: https://doi.org/10.3390/w14192952.

[28] J. Stoklosa and A. Xia. “Studies of barrier options and their sensitivities”. Supervisor: Kostya Borovkov; Second Examiner: Aihua Xia. Honours thesis. The University of Melbourne, Nov. 2007. Available online.




DOI: https://doi.org/10.18860/cauchy.v10i2.33239

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