Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Semi closed-form pricing autocallable ELS using Brownian Bridge

  • Lee, Minha (Department of Mathematics, Sungkyunkwan University) ;
  • Hong, Jimin (Department of Statistics and Actuarial Science, Soongsil University)
  • Received : 2020.12.10
  • Accepted : 2021.02.20
  • Published : 2021.05.31
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Abstract

This paper discusses the pricing of autocallable structured product with knock-in (KI) feature using the exit probability with the Brownian Bridge technique. The explicit pricing formula of autocallable ELS derived in the existing paper handles the part including the minimum of the Brownian motion using the inclusion-exclusion principle. This has the disadvantage that the pricing formula is complicate because of the probability with minimum value and the computational volume increases dramatically as the number of autocall chances increases. To solve this problem, we applied an efficient and robust simulation method called the Brownian Bridge technique, which provides the probability of touching the predetermined barrier when the initial and terminal values of the process following the Brownian motion in a certain interval are specified. We rewrite the existing pricing formula and provide a brief theoretical background and computational algorithm for the technique. We also provide several numerical examples computed in three different ways: explicit pricing formula, the Crude Monte Carlo simulation method and the Brownian Bridge technique.

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References

  1. Deng G, Husson T, and McCann C (2014). Valuation of structured products, The Journal of Alternative Investments, 16, 71-87. https://doi.org/10.3905/jai.2014.16.4.071
  2. Deng G, Mallett J, and McCann C (2016). Modeling autocallable structured products. In Derivatives and Hedge Funds (pp. 323-334), Palgrave Macmillan, United Kingdom.
  3. Gerber HU and Shiu ESW (1994). Option pricing by esscher transforms, Transactions of the Society of Actuaries, 46, 99-191.
  4. Gobet E (2009). Advanced Monte Carlo methods for barrier and related exotic options, In Handbook of Numerical Analysis, 15, 497-528.
  5. Guillaume T (2015). Autocallable structured products, The Journal of Derivatives, 22, 73-94. https://doi.org/10.3905/jod.2015.22.3.073
  6. Huh J and Kolkiewicz A (2008). Computation of multivariate barrier crossing probability and its applications in credit risk models, North American Actuarial Journal, 12, 263-291. https://doi.org/10.1080/10920277.2008.10597521
  7. Jeong DR, Wee IS, and Kim JS (2010). An operator splitting method for pricing the ELS option, Journal of the Korean Society for Industrial and Applied Mathematics, 14, 175-187. https://doi.org/10.12941/JKSIAM.2010.14.3.175
  8. Jang H, Kim S, Han J, et al. (2019). Fast Monte Carlo simulation for pricing equity-linked securities, Computational Economics, 56, 1-18.
  9. Kim KK and Lim DY (2019). A recursive method for static replication of autocallable structured products, Quantitative Finance, 19, 647-661. https://doi.org/10.1080/14697688.2018.1523546
  10. Kim Y, Bae HO, and Roh HS (2011). FDM Algorithm for pricing of ELS with exit probability, Journal of Derivatives and Quantitative Studies, 19, 427-446. https://doi.org/10.1108/JDQS-04-2011-B0004
  11. Korn R, Korn E, and Kroisandt G (2010). Monte Carlo methods and models in finance and insurance, CRC Press, United States
  12. Lee C, Lyu J, Park E, Lee W, Kim S, Jeong D, and Kim J (2020). Super-fast computation for the three-asset equity-linked securities using the finite difference method, Mathematics, 8, 307. https://doi.org/10.3390/math8030307
  13. Lee H, Ahn S, and Ko B (2019). Generalizing the reflection principle of Brownian motion, and closedform pricing of barrier options and autocallable investments, The North American Journal of Economics and Finance, 50, 101014. https://doi.org/10.1016/j.najef.2019.101014