IDEAS home Printed from https://ideas.repec.org/a/eee/ecofin/v61y2022ics1062940822000560.html
   My bibliography  Save this article

A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge

Author

Listed:
  • Lee, Hangsuck
  • Lee, Minha
  • Ko, Bangwon

Abstract

Barrier options based upon the extremum of more than one underlying prices do not allow for closed-form pricing formulas, and thus require numerical methods to evaluate. One example is the autocallable structured product with knock-in feature, which has gained a great deal of popularity in the recent decades. In order to increase numerical efficiency for pricing such products, this paper develops a semi-analytic valuation algorithm which is free from the computational burden and the monitoring bias of the crude Monte Carlo simulation. The basic idea is to combine the simulation of the underlying prices at certain time points and the exit (or non-exit) probability of the Brownian bridge. In the literature, the algorithm was developed to deal with a single-asset barrier option under the Black–Scholes model. Now we extend the framework to cover two-asset barrier options and autocallable product. For the purpose, we explore the non-exit probability of the two-dimensional Brownian bridge, which has not been researched before. Meanwhile, we employ the actuarial method of Esscher transform to simplify our calculation and improve our algorithm via importance sampling. We illustrate our algorithm with numerical examples.

Suggested Citation

  • Lee, Hangsuck & Lee, Minha & Ko, Bangwon, 2022. "A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge," The North American Journal of Economics and Finance, Elsevier, vol. 61(C).
    • Handle: RePEc:eee:ecofin:v:61:y:2022:i:c:s1062940822000560
    DOI: 10.1016/j.najef.2022.101704
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S1062940822000560
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.najef.2022.101704?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Robert J. Elliott & Leunglung Chan & Tak Kuen Siu, 2005. "Option pricing and Esscher transform under regime switching," Annals of Finance, Springer, vol. 1(4), pages 423-432, October.
    2. Gerber, Hans U. & Shiu, Elias S. W., 1996. "Actuarial bridges to dynamic hedging and option pricing," Insurance: Mathematics and Economics, Elsevier, vol. 18(3), pages 183-218, November.
    3. Lee, Hangsuck & Ko, Bangwon & Song, Seongjoo, 2019. "Valuing step barrier options and their icicled variations," The North American Journal of Economics and Finance, Elsevier, vol. 49(C), pages 396-411.
    4. Helen Haworth & Christoph Reisinger & William Shaw, 2008. "Modelling bonds and credit default swaps using a structural model with contagion," Quantitative Finance, Taylor & Francis Journals, vol. 8(7), pages 669-680.
    5. Mary Hardy, 2001. "A Regime-Switching Model of Long-Term Stock Returns," North American Actuarial Journal, Taylor & Francis Journals, vol. 5(2), pages 41-53.
    6. Ng, Andrew Cheuk-Yin & Li, Johnny Siu-Hang, 2011. "Valuing variable annuity guarantees with the multivariate Esscher transform," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 393-400.
    7. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2012. "Valuing equity-linked death benefits and other contingent options: A discounted density approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 73-92.
    8. Zhou, Chunsheng, 2001. "An Analysis of Default Correlations and Multiple Defaults," The Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 555-576.
    9. Serena Tiong, 2000. "Valuing Equity-Indexed Annuities," North American Actuarial Journal, Taylor & Francis Journals, vol. 4(4), pages 149-163.
    10. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.
    11. Lee, Hangsuck & Ahn, Soohan & Ko, Bangwon, 2019. "Generalizing the reflection principle of Brownian motion, and closed-form pricing of barrier options and autocallable investments," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Sangkwon Kim & Jisang Lyu & Wonjin Lee & Eunchae Park & Hanbyeol Jang & Chaeyoung Lee & Junseok Kim, 2024. "A Practical Monte Carlo Method for Pricing Equity-Linked Securities with Time-Dependent Volatility and Interest Rate," Computational Economics, Springer;Society for Computational Economics, vol. 63(5), pages 2069-2086, May.
    2. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo & Lee, Minha, 2024. "Valuing three-asset barrier options and autocallable products via exit probabilities of Brownian bridge," The North American Journal of Economics and Finance, Elsevier, vol. 73(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2022. "Multi‐step reflection principle and barrier options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(4), pages 692-721, April.
    2. Hangsuck Lee & Gaeun Lee & Seongjoo Song, 2021. "Multi-step Reflection Principle and Barrier Options," Papers 2105.15008, arXiv.org.
    3. Godin, Frédéric & Lai, Van Son & Trottier, Denis-Alexandre, 2019. "Option pricing under regime-switching models: Novel approaches removing path-dependence," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 130-142.
    4. Fan, Kun & Shen, Yang & Siu, Tak Kuen & Wang, Rongming, 2015. "Pricing annuity guarantees under a double regime-switching model," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 62-78.
    5. Lee, Hangsuck & Ko, Bangwon & Song, Seongjoo, 2019. "Valuing step barrier options and their icicled variations," The North American Journal of Economics and Finance, Elsevier, vol. 49(C), pages 396-411.
    6. Lee, Hangsuck & Ha, Hongjun & Lee, Minha, 2021. "Valuation of piecewise linear barrier options," The North American Journal of Economics and Finance, Elsevier, vol. 58(C).
    7. Siu, Tak Kuen, 2023. "European option pricing with market frictions, regime switches and model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 233-250.
    8. Hangsuck Lee & Hongjun Ha & Minha Lee, 2022. "Piecewise linear boundary crossing probabilities, barrier options, and variable annuities," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(12), pages 2248-2272, December.
    9. Ng, Andrew C.Y. & Li, Johnny Siu-Hang & Chan, Wai-Sum, 2013. "Pricing options on stocks denominated in different currencies: Theory and illustrations," The North American Journal of Economics and Finance, Elsevier, vol. 26(C), pages 339-354.
    10. Lee, Hangsuck & Lee, Gaeun & Song, Seongjoo, 2023. "Min–max multi-step barrier options and their variants," The North American Journal of Economics and Finance, Elsevier, vol. 67(C).
    11. Li, Johnny Siu-Hang & Ng, Andrew C.Y. & Chan, Wai-Sum, 2015. "Managing financial risk in Chinese stock markets: Option pricing and modeling under a multivariate threshold autoregression," International Review of Economics & Finance, Elsevier, vol. 40(C), pages 217-230.
    12. Lee, Hangsuck & Kim, Eunchae & Ko, Bangwon, 2022. "Valuing lookback options with barrier," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).
    13. Godin, Frédéric & Trottier, Denis-Alexandre, 2021. "Option pricing in regime-switching frameworks with the Extended Girsanov Principle," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 116-129.
    14. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo & Lee, Minha, 2024. "Valuing three-asset barrier options and autocallable products via exit probabilities of Brownian bridge," The North American Journal of Economics and Finance, Elsevier, vol. 73(C).
    15. Hangsuck Lee & Hongjun Ha & Minha Lee, 2022. "Piecewise linear double barrier options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(1), pages 125-151, January.
    16. Yinghui Dong & Guojing Wang & Kam C. Yuen, 2014. "Bilateral Counterparty Risk Valuation on a CDS with a Common Shock Model," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 643-673, September.
    17. Tian, Ping & Zhou, Hang & Zhou, Duotai, 2023. "Analysis about the Black-Scholes asset price under the regime-switching framework," International Review of Financial Analysis, Elsevier, vol. 88(C).
    18. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    19. Kirkby, J. Lars & Nguyen, Duy, 2021. "Equity-linked Guaranteed Minimum Death Benefits with dollar cost averaging," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 408-428.
    20. Shin-Yun Wang & Ming-Che Chuang & Shih-Kuei Lin & So-De Shyu, 2021. "Option pricing under stock market cycles with jump risks: evidence from the S&P 500 index," Review of Quantitative Finance and Accounting, Springer, vol. 56(1), pages 25-51, January.

    More about this item

    Keywords

    Autocallable structured product; Barrier options; Black–Scholes model; Esscher transform; Non-exit probability; Two-dimensional Brownian bridge;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ecofin:v:61:y:2022:i:c:s1062940822000560. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/620163 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.