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The closed-form option pricing formulas under the sub-fractional Poisson volatility models

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  • Wang, XiaoTian
  • Yang, ZiJian
  • Cao, PiYao
  • Wang, ShiLin

Abstract

A new fractional process called the sub-fractional Poisson process NH(t) is proposed, which has continuous sample paths, long- memory, leptokurtosis and heavy tail distribution, is of convenience to price options and simulate the variance process of risk asset return. Based on the sub-fractional Poisson process NH(t) the new fractional variance processes have been proposed, which may capture the skewness and the long-memory as well as mean-reverting in the stock price volatilities. In particular, the characteristic function method for option pricing is given, and the analytical formulas for European option price C(t,St) have been obtained under the risk-neutral probability measure.

Suggested Citation

  • Wang, XiaoTian & Yang, ZiJian & Cao, PiYao & Wang, ShiLin, 2021. "The closed-form option pricing formulas under the sub-fractional Poisson volatility models," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    • Handle: RePEc:eee:chsofr:v:148:y:2021:i:c:s0960077921003660
    DOI: 10.1016/j.chaos.2021.111012
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    1. Alexandra Chronopoulou & Frederi G. Viens, 2012. "Stochastic volatility and option pricing with long-memory in discrete and continuous time," Quantitative Finance, Taylor & Francis Journals, vol. 12(4), pages 635-649, December.
    2. Tomasz Bojdecki & Luis G. Gorostiza & Anna Talarczyk, 2004. "Sub-fractional Brownian motion and its relation to occupation times," RePAd Working Paper Series lrsp-TRS376, Département des sciences administratives, UQO.
    3. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat & Wang, Yintian, 2008. "Option valuation with long-run and short-run volatility components," Journal of Financial Economics, Elsevier, vol. 90(3), pages 272-297, December.
    4. Bojdecki, Tomasz & Gorostiza, Luis G. & Talarczyk, Anna, 2004. "Sub-fractional Brownian motion and its relation to occupation times," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 405-419, October.
    5. Kirman Alan & Teyssière Gilles, 2002. "Microeconomic Models for Long Memory in the Volatility of Financial Time Series," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 5(4), pages 1-23, January.
    6. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    7. Alan Kirman, 1993. "Ants, Rationality, and Recruitment," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(1), pages 137-156.
    8. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    9. Cajueiro, Daniel O. & Tabak, Benjamin M., 2009. "Testing for long-range dependence in the Brazilian term structure of interest rates," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1559-1573.
    10. Araya, Héctor & Bahamonde, Natalia & Torres, Soledad & Viens, Frederi, 2019. "Donsker type theorem for fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 1-8.
    11. Eugene F. Fama, 1963. "Mandelbrot and the Stable Paretian Hypothesis," The Journal of Business, University of Chicago Press, vol. 36, pages 420-420.
    12. Ole Barndorff-Nielsen & Elisa Nicolato & Neil Shephard, 2002. "Some recent developments in stochastic volatility modelling," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 11-23.
    13. Zhigang Tong, 2016. "Option pricing in stochastic volatility models driven by fractional Lévy processes," International Journal of Financial Markets and Derivatives, Inderscience Enterprises Ltd, vol. 5(1), pages 56-75.
    14. Thomas Lux & Michele Marchesi, 2000. "Volatility Clustering In Financial Markets: A Microsimulation Of Interacting Agents," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 675-702.
    15. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    16. Alexandra Chronopoulou & Frederi Viens, 2012. "Estimation and pricing under long-memory stochastic volatility," Annals of Finance, Springer, vol. 8(2), pages 379-403, May.
    17. Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    18. Biagini, Francesca & Fink, Holger & Klüppelberg, Claudia, 2013. "A fractional credit model with long range dependent default rate," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1319-1347.
    19. 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.
    20. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    Cited by:

    1. Axel A. Araneda, 2021. "Price modelling under generalized fractional Brownian motion," Papers 2108.12042, arXiv.org, revised Nov 2023.

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