IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v69y2016icp168-193.html
   My bibliography  Save this article

A self-exciting threshold jump–diffusion model for option valuation

Author

Listed:
  • Siu, Tak Kuen

Abstract

A self-exciting threshold jump–diffusion model for option valuation is studied. This model can incorporate regime switches without introducing an exogenous stochastic factor process. A generalized version of the Esscher transform is used to select a pricing kernel. The valuation of both the European and American contingent claims is considered. A piecewise linear partial-differential–integral equation governing a price of a standard European contingent claim is derived. For an American contingent claim, a formula decomposing a price of the American claim into the sum of its European counterpart and the early exercise premium is provided. An approximate solution to the early exercise premium based on the quadratic approximation technique is derived for a particular case where the jump component is absent. Numerical results for both European and American options are presented for the case without jumps.

Suggested Citation

  • Siu, Tak Kuen, 2016. "A self-exciting threshold jump–diffusion model for option valuation," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 168-193.
    • Handle: RePEc:eee:insuma:v:69:y:2016:i:c:p:168-193
    DOI: 10.1016/j.insmatheco.2016.05.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668715302389
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2016.05.008?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. Walter Allegretto & Giovanni Barone-Adesi & Robert Elliott, 1995. "Numerical evaluation of the critical price and American options," The European Journal of Finance, Taylor & Francis Journals, vol. 1(1), pages 69-78.
    2. 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.
    3. Liew, Chuin Ching & Siu, Tak Kuen, 2010. "A hidden Markov regime-switching model for option valuation," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 374-384, December.
    4. Albert N. Shiryaev & Jan Kallsen, 2002. "The cumulant process and Esscher's change of measure," Finance and Stochastics, Springer, vol. 6(4), pages 397-428.
    5. Robert Elliott & Tak Siu, 2011. "A risk-based approach for pricing American options under a generalized Markov regime-switching model," Quantitative Finance, Taylor & Francis Journals, vol. 11(11), pages 1633-1646.
    6. Brockwell, P. J. & Hyndman, R. J., 1992. "On continuous-time threshold autoregression," International Journal of Forecasting, Elsevier, vol. 8(2), pages 157-173, October.
    7. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    8. X. Guo, 2001. "Information and option pricings," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 38-44.
    9. Naik, Vasanttilak, 1993. "Option Valuation and Hedging Strategies with Jumps in the Volatility of Asset Returns," Journal of Finance, American Finance Association, vol. 48(5), pages 1969-1984, December.
    10. 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.
    11. Siu, Tak Kuen & Yang, Hailiang & Lau, John W., 2008. "Pricing currency options under two-factor Markov-modulated stochastic volatility models," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 295-302, December.
    12. Wai-Sum Chan & Albert Wong & Howell Tong, 2004. "Some Nonlinear Threshold Autoregressive Time Series Models for Actuarial Use," North American Actuarial Journal, Taylor & Francis Journals, vol. 8(4), pages 37-61.
    13. Robert Elliott & Tak Siu, 2015. "Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(2), pages 133-149, May.
    14. Franses,Philip Hans & Dijk,Dick van, 2000. "Non-Linear Time Series Models in Empirical Finance," Cambridge Books, Cambridge University Press, number 9780521770415, October.
    15. Tak Kuen Siu, 2014. "A Hidden Markov-Modulated Jump Diffusion Model for European Option Pricing," International Series in Operations Research & Management Science, in: Rogemar S. Mamon & Robert J. Elliott (ed.), Hidden Markov Models in Finance, edition 127, chapter 0, pages 185-209, Springer.
    16. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    17. Whitten, S.P. & Thomas, R.G., 1999. "A Non-Linear Stochastic Asset Model for Actuarial Use," British Actuarial Journal, Cambridge University Press, vol. 5(5), pages 919-953, December.
    18. Yang Shen & Kun Fan & Tak Kuen Siu, 2014. "Option Valuation Under a Double Regime‐Switching Model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 34(5), pages 451-478, May.
    19. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    20. Krager, Horst & Kugler, Peter, 1993. "Non-linearities in foreign exchange markets: a different perspective," Journal of International Money and Finance, Elsevier, vol. 12(2), pages 195-208, April.
    21. Hsieh, David A, 1991. "Chaos and Nonlinear Dynamics: Application to Financial Markets," Journal of Finance, American Finance Association, vol. 46(5), pages 1839-1877, December.
    22. Barone-Adesi, Giovanni & Whaley, Robert E, 1987. "Efficient Analytic Approximation of American Option Values," Journal of Finance, American Finance Association, vol. 42(2), pages 301-320, June.
    23. Goldfeld, Stephen M. & Quandt, Richard E., 1973. "A Markov model for switching regressions," Journal of Econometrics, Elsevier, vol. 1(1), pages 3-15, March.
    24. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    25. Yuen, Fei Lung & Yang, Hailiang, 2009. "Option Pricing in a Jump-Diffusion Model with Regime Switching," ASTIN Bulletin, Cambridge University Press, vol. 39(2), pages 515-539, November.
    26. Hansen Bruce E., 1997. "Inference in TAR Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 2(1), pages 1-16, April.
    27. Enders, Walter & Granger, Clive W J, 1998. "Unit-Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 304-311, July.
    28. Siu, Tak Kuen, 2005. "Fair valuation of participating policies with surrender options and regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 533-552, December.
    29. Elliott, Robert J. & Siu, Tak Kuen & Badescu, Alexandru, 2011. "On pricing and hedging options in regime-switching models with feedback effect," Journal of Economic Dynamics and Control, Elsevier, vol. 35(5), pages 694-713, May.
    30. Pradeep K. Yadav & Peter F. Pope & Krishna Paudyal, 1994. "Threshold Autoregressive Modeling In Finance: The Price Differences Of Equivalent Assets1," Mathematical Finance, Wiley Blackwell, vol. 4(2), pages 205-221, April.
    31. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14, April.
    32. Robert J. Elliott & Tak Kuen Siu, 2013. "Option Pricing and Filtering with Hidden Markov-Modulated Pure-Jump Processes," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(1), pages 1-25, March.
    33. Fei Su & Kung-Sik Chan, 2016. "Option Pricing with Threshold Diffusion Processes," North American Actuarial Journal, Taylor & Francis Journals, vol. 20(2), pages 133-141, April.
    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. Donatien Hainaut & Franck Moraux, 2019. "A switching self-exciting jump diffusion process for stock prices," Annals of Finance, Springer, vol. 15(2), pages 267-306, June.
    2. Antoine Lejay & Paolo Pigato, 2017. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Working Papers hal-01669082, HAL.
    3. 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.
    4. Yang Shen & Tak Kuen Siu, 2018. "A Risk-Based Approach for Asset Allocation with A Defaultable Share," Risks, MDPI, vol. 6(1), pages 1-27, February.
    5. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.

    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. Dong-Mei Zhu & Jiejun Lu & Wai-Ki Ching & Tak-Kuen Siu, 2019. "Option Pricing Under a Stochastic Interest Rate and Volatility Model with Hidden Markovian Regime-Switching," Computational Economics, Springer;Society for Computational Economics, vol. 53(2), pages 555-586, February.
    2. Tak Kuen Siu & Robert J. Elliott, 2019. "Hedging Options In A Doubly Markov-Modulated Financial Market Via Stochastic Flows," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-41, December.
    3. 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.
    4. 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.
    5. Shen, Yang & Siu, Tak Kuen, 2013. "Stochastic differential game, Esscher transform and general equilibrium under a Markovian regime-switching Lévy model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 757-768.
    6. 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.
    7. Pious Asiimwe & Charles Wilson Mahera & Olivier Menoukeu-Pamen, 2016. "On the Price of Risk Under a Regime Switching CGMY Process," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 23(4), pages 305-335, December.
    8. 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.
    9. Leunglung Chan & Song-Ping Zhu, 2014. "An exact and explicit formula for pricing Asian options with regime switching," Papers 1407.5091, arXiv.org.
    10. Liew, Chuin Ching & Siu, Tak Kuen, 2010. "A hidden Markov regime-switching model for option valuation," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 374-384, December.
    11. Su-Lien Lu & Kuo-Jung Lee, 2021. "Investigating the Determinants of Credit Spread Using a Markov Regime-Switching Model: Evidence from Banks in Taiwan," Sustainability, MDPI, vol. 13(17), pages 1-25, August.
    12. Farzad Fard & Ning Rong, 2014. "Pricing and managing risks of ruin contingent life annuities under regime switching variance gamma process," Annals of Finance, Springer, vol. 10(2), pages 315-332, May.
    13. Siu, Tak Kuen, 2008. "A game theoretic approach to option valuation under Markovian regime-switching models," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1146-1158, June.
    14. LeBaron, Blake, 2003. "Non-Linear Time Series Models in Empirical Finance,: Philip Hans Franses and Dick van Dijk, Cambridge University Press, Cambridge, 2000, 296 pp., Paperback, ISBN 0-521-77965-0, $33, [UK pound]22.95, [," International Journal of Forecasting, Elsevier, vol. 19(4), pages 751-752.
    15. Forsyth, Peter & Vetzal, Kenneth, 2014. "An optimal stochastic control framework for determining the cost of hedging of variable annuities," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 29-53.
    16. Singh, Tarlok, 2014. "On the regime-switching and asymmetric dynamics of economic growth in the OECD countries," Research in Economics, Elsevier, vol. 68(2), pages 169-192.
    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. Robert Elliott & Tak Siu, 2015. "Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(2), pages 133-149, May.
    19. Fard, Farzad Alavi & Siu, Tak Kuen, 2013. "Pricing participating products with Markov-modulated jump–diffusion process: An efficient numerical PIDE approach," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 712-721.
    20. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.

    More about this item

    Keywords

    Option valuation; Self-exciting threshold model; Generalized Esscher transform; Piecewise linear partial differential equation; Quadratic approximation;
    All these keywords.

    JEL classification:

    • 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:insuma:v:69:y:2016:i:c:p:168-193. 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/505554 .

    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.