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Equity-linked security pricing and Greeks at arbitrary intermediate times using Brownian bridge

  • Hanbyeol Jang , Jian Wang and Junseok Kim ORCID logo EMAIL logo

Abstract

We develop a numerical algorithm for predicting prices and Greeks of equity-linked securities (ELS) with a knock-in barrier at any time over the total time period from issue date to maturity by using Monte Carlo simulation (MCS). The ELS is one of the most important financial derivatives in Korea. In the proposed algorithm, first we calculate the probability (0p1) that underlying asset price never hits the knock-in barrier up to the intermediate evaluation date. Second, we compute two option prices Vnk and Vk, where Vnk is the option value which knock-in event does not occur and Vk is the option value which knock-in event occurs. Finally, we predict the option value with a weighted average. We apply the proposed algorithm to two- and three-asset ELS. We provide the pseudo-numerical algorithm and computational results to demonstrate the usefulness of the proposed method.

Funding statement: The second author (Jian Wang) was supported by the China Scholarship Council (201808260026). The corresponding author (Junseok Kim) was supported by the Brain Korea 21 Plus (BK 21) fellowship from the Ministry of Education of Korea.

References

[1] A. G. Barone and R. E. Whaley, Efficient analytic approximation of American option values, J. Finance 42 (1987), no. 2, 301–320. 10.1111/j.1540-6261.1987.tb02569.xSearch in Google Scholar

[2] 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

[3] D.-M. Dang, D. Nguyen and G. Sewell, Numerical schemes for pricing Asian options under state-dependent regime-switching jump-diffusion models, Comput. Math. Appl. 71 (2016), no. 1, 443–458. 10.1016/j.camwa.2015.12.017Search in Google Scholar

[4] D. Davydov and V. Linetsky, Pricing and hedging path-dependent options under the CEV process, Management Sci. 47 (2001), no. 7, 949–965. 10.1287/mnsc.47.7.949.9804Search in Google Scholar

[5] C. P. Fries and M. S. Joshi, Conditional analytic Monte-Carlo pricing scheme of auto-callable products, SSRN Electronic J. (2008), 10.2139/ssrn.1125725. 10.2139/ssrn.1125725Search in Google Scholar

[6] J. E. Gentle, Random Number Generation and Monte Carlo Methods, 2nd ed., Springer, New York, 2003. Search in Google Scholar

[7] 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

[8] J. E. Handschin, Monte Carlo techniques for prediction and filtering of non-linear stochastic processes, Automatica J. IFAC 6 (1970), 555–563. 10.1016/0005-1098(70)90010-5Search in Google Scholar

[9] S. Harase, Comparison of Sobol’ sequences in financial applications, Monte Carlo Methods Appl. 25 (2019), no. 1, 61–74. 10.1515/mcma-2019-2029Search in Google Scholar

[10] D. Jeong, I. S. Wee and J. Kim, An operator splitting method for pricing the ELS option, J. Korean Soc. Ind. Appl. Math. 14 (2010), no. 3, 175–187. Search in Google Scholar

[11] J. Jo and Y. Kim, Comparison of numerical schemes on multi-dimensional Black–Scholes equations, Bull. Korean Math. Soc. 50 (2013), no. 6, 2035–2051. 10.4134/BKMS.2013.50.6.2035Search in Google Scholar

[12] J. Kim, T. Kim, J. Jo, Y. Choi, S. Lee, H. Hwang, M. Yoo and D. Jeong, A practical finite difference method for the three-dimensional Black–Scholes equation, European J. Oper. Res. 252 (2016), no. 1, 183–190. 10.1016/j.ejor.2015.12.012Search in Google Scholar

[13] Y. Kim, H. O. Bae and H. Roh, FDM algorithm for pricing of ELS with exit-probability, Korea Derivative Ass. 19 (2011), 428–446. 10.1108/JDQS-04-2011-B0004Search in Google Scholar

[14] F. Mehrdoust, S. Babaei and S. Fallah, Efficient Monte Carlo option pricing under CEV model, Comm. Statist. Simulation Comput. 46 (2017), no. 3, 2254–2266. 10.1080/03610918.2015.1040497Search in Google Scholar

[15] A. R. Najafi, F. Mehrdoust and S. Shirinpour, Pricing American put option on zero-coupon bond under fractional CIR model with transaction cost, Comm. Statist. Simulation Comput. 47 (2018), no. 3, 864–870. 10.1080/03610918.2017.1295153Search in Google Scholar

[16] L. C. G. Rogers and Z. Shi, The value of an Asian option, J. Appl. Probab. 32 (1995), no. 4, 1077–1088. 10.2307/3215221Search in Google Scholar

[17] S. E. Shreve, Stochastic calculus for finance. II: Continuous-time Models, Springer, New York, 2004. 10.1007/978-1-4757-4296-1Search in Google Scholar

[18] V. Todorov, I. Dimov and Y. Dimitrov, Efficient quasi-Monte Carlo methods for multiple integrals in option pricing, AIP Conf. Proc. 2018 (2018), Article ID 110007. 10.1063/1.5064950Search in Google Scholar

[19] J. Wang and C. Liu, Generating multivariate mixture of normal distributions using a modified Cholesky decomposition, Proceedings of the 38th conference on Winter simulation, IEEE Press, Piscataway (2006), 342–347. 10.1109/WSC.2006.323100Search in Google Scholar

[20] Z. Yang, Optimal exercise boundary of American fractional lookback option in a mixed jump-diffusion fractional Brownian motion environment, Math. Probl. Eng. 2017 (2017), Article ID 5904125. 10.1155/2017/5904125Search in Google Scholar

[21] K. Zhang and S. Wang, Pricing American bond options using a penalty method, Automatica J. IFAC 48 (2012), no. 3, 472–479. 10.1016/j.automatica.2012.01.009Search in Google Scholar

[22] G. Zhao, Y. Zhou and P. Vakili, A new efficient simulation strategy for pricing path-dependent options, Proceedings of the 2006 Winter Simulation Conference, IEEE Press, Piscataway (2006), 703–710. 10.1109/WSC.2006.323149Search in Google Scholar

[23] 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

Received: 2019-05-21
Revised: 2019-08-31
Accepted: 2019-09-10
Published Online: 2019-10-01
Published in Print: 2019-12-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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