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Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model

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  • Robert Jarrow
  • Jayen Tan

Abstract

Fractional Brownian motion (fBm) exhibits attractive features for financial modeling, including long-range dependence, path roughness, and anomalous diffusion. However, its non-semimartingale nature precludes the use of conventional no-arbitrage approaches to option pricing. We address this limitation by introducing a time-changed fBm, obtained by evaluating fBm at stochastic gamma activity time, where activity time represents cumulative executed trading time. The resulting process retains the defining properties of fBm while recovering the semimartingale structure. Building on this construction, we develop the fractional Variance Gamma (fVG) model and propose a generalized method of moments (GMM) estimation procedure for option pricing. An empirical analysis of the S\&P 500 yields an estimated Hurst exponent of approximately 0.45, consistent with mildly sublinear temporal scaling of return moments.

Suggested Citation

  • Robert Jarrow & Jayen Tan, 2026. "Option Pricing with Time-Changed Fractional Brownian Motion: A Fractional Variance Gamma Model," Papers 2608.03925, arXiv.org.
    • Handle: RePEc:arx:papers:2608.03925
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    1. David Heath & Sidney Resnick & Gennady Samorodnitsky, 1998. "Heavy Tails and Long Range Dependence in On/Off Processes and Associated Fluid Models," Mathematics of Operations Research, INFORMS, vol. 23(1), pages 145-165, February.
    2. Christos Christodoulou-Volos & Fotios Siokis, 2006. "Long range dependence in stock market returns," Applied Financial Economics, Taylor & Francis Journals, vol. 16(18), pages 1331-1338.
    3. Robert A. Jarrow, 2021. "Continuous-Time Asset Pricing Theory," Springer Finance, Springer, edition 2, number 978-3-030-74410-6, February.
    4. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    5. Hyun-Jung Ryoo & Graham Smith, 2002. "Korean stock prices under price limits: variance ratio tests of random walks," Applied Financial Economics, Taylor & Francis Journals, vol. 12(8), pages 545-553.
    6. Dilip B. Madan & King Wang, 2023. "The economics of time as it is embedded in the prices of options§," Quantitative Finance, Taylor & Francis Journals, vol. 23(4), pages 579-593, April.
    7. Andrew W. Lo, A. Craig MacKinlay, 1988. "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test," The Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 41-66.
    8. Laurent Calvet & Adlai Fisher, 2002. "Multifractality In Asset Returns: Theory And Evidence," The Review of Economics and Statistics, MIT Press, vol. 84(3), pages 381-406, August.
    9. Cheolwoo Park & F�lix Hernández-Campos & Long Le & J. S. Marron & Juhyun Park & Vladas Pipiras & F. D. Smith & Richard L. Smith & Michele Trovero & Zhengyuan Zhu, 2011. "Long-range dependence analysis of Internet traffic," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(7), pages 1407-1433, June.
    10. Graham Smith & Keith Jefferis & Hyun-Jung Ryoo, 2002. "African stock markets: multiple variance ratio tests of random walks," Applied Financial Economics, Taylor & Francis Journals, vol. 12(7), pages 475-484.
    11. Hélyette Geman & Dilip B. Madan & Marc Yor, 2001. "Time Changes for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 79-96, January.
    12. Hoque, Hafiz A.A.B. & Kim, Jae H. & Pyun, Chong Soo, 2007. "A comparison of variance ratio tests of random walk: A case of Asian emerging stock markets," International Review of Economics & Finance, Elsevier, vol. 16(4), pages 488-502.
    13. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    14. Christian Bender & Tommi Sottinen & Esko Valkeila, 2008. "Pricing by hedging and no-arbitrage beyond semimartingales," Finance and Stochastics, Springer, vol. 12(4), pages 441-468, October.
    15. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
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