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

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  • 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.

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  • 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
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    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.
    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.
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    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

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