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Discretization of continuous-time arbitrage strategies in financial markets with fractional Brownian motion

Author

Listed:
  • Kerstin Lamert

    (Brandenburg University of Technology Cottbus-Senftenberg)

  • Benjamin R. Auer

    (Friedrich Schiller University Jena
    Department of Finance
    Money and International Finance)

  • Ralf Wunderlich

    (Brandenburg University of Technology Cottbus-Senftenberg)

Abstract

This study evaluates the practical usefulness of continuous-time arbitrage strategies designed to exploit serial correlation in fractional financial markets. Specifically, we revisit the strategies of Shiryaev (On arbitrage and replication for fractal models, 1998) and Salopek (Stoch Process Appl 76:217–230, 1998) and transfer them to a real-world setting by distretizing their dynamics and introducing transaction costs. In Monte Carlo simulations with various market and trading parameter settings as well as a formal analysis of discretization error, we show that both are promising with respect to terminal portfolio values and loss probabilities. These features and complementary sparsity make them worth serious consideration in the toolkit of quantitative investors.

Suggested Citation

  • Kerstin Lamert & Benjamin R. Auer & Ralf Wunderlich, 2025. "Discretization of continuous-time arbitrage strategies in financial markets with fractional Brownian motion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 101(2), pages 163-218, April.
  • Handle: RePEc:spr:mathme:v:101:y:2025:i:2:d:10.1007_s00186-025-00889-0
    DOI: 10.1007/s00186-025-00889-0
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    as
    1. Sottinen Tommi & Valkeila Esko, 2003. "On arbitrage and replication in the fractional Black–Scholes pricing model," Statistics & Risk Modeling, De Gruyter, vol. 21(2), pages 93-108, February.
    2. Jonathan A. Batten & Cetin Ciner & Brian M. Lucey & Peter G. Szilagyi, 2013. "The structure of gold and silver spread returns," Quantitative Finance, Taylor & Francis Journals, vol. 13(4), pages 561-570, March.
    3. Badrinath, S.G. & Gubellini, S., 2011. "On the characteristics and performance of long-short, market-neutral and bear mutual funds," Journal of Banking & Finance, Elsevier, vol. 35(7), pages 1762-1776, July.
    4. K. J. Hong & S. Satchell, 2015. "Time series momentum trading strategy and autocorrelation amplification," Quantitative Finance, Taylor & Francis Journals, vol. 15(9), pages 1471-1487, September.
    5. Tarek A Hassan & Rui C Mano, 2019. "Forward and Spot Exchange Rates in a Multi-Currency World," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 134(1), pages 397-450.
    6. Songzi Du & Haoxiang Zhu, 2017. "What is the Optimal Trading Frequency in Financial Markets?," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 84(4), pages 1606-1651.
    7. Frazzini, Andrea & Pedersen, Lasse Heje, 2014. "Betting against beta," Journal of Financial Economics, Elsevier, vol. 111(1), pages 1-25.
    8. Diane Wilcox & Tim Gebbie, 2008. "Serial Correlation, Periodicity And Scaling Of Eigenmodes In An Emerging Market," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(07), pages 739-760.
    9. Valeriy Zakamulin & Javier Giner, 2024. "Optimal trend-following rules in two-state regime-switching models," Journal of Asset Management, Palgrave Macmillan, vol. 25(4), pages 327-348, July.
    10. Benjamin J. C. Kim & David Karemera, 2006. "Assessing the forecasting accuracy of alternative nominal exchange rate models: the case of long memory," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 25(5), pages 369-380.
    11. 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.
    12. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    13. Weron, Rafał, 2002. "Estimating long-range dependence: finite sample properties and confidence intervals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 312(1), pages 285-299.
    14. 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.
    15. Sikora, Grzegorz, 2018. "Statistical test for fractional Brownian motion based on detrending moving average algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 54-62.
    16. Yun Xiang & Shijie Deng, 2024. "Statistical arbitrage under a fractal price model," Annals of Operations Research, Springer, vol. 335(1), pages 425-439, April.
    17. Oleg Bondarenko, 2003. "Statistical Arbitrage and Securities Prices," The Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 875-919, July.
    18. Salopek, D. M., 1998. "Tolerance to arbitrage," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 217-230, August.
    19. Ben R. Marshall & Nhut H. Nguyen & Nuttawat Visaltanachoti, 2017. "Time series momentum and moving average trading rules," Quantitative Finance, Taylor & Francis Journals, vol. 17(3), pages 405-421, March.
    20. Rostek, S. & Schöbel, R., 2013. "A note on the use of fractional Brownian motion for financial modeling," Economic Modelling, Elsevier, vol. 30(C), pages 30-35.
    21. Pan, Ming-Shiun & Liano, Kartono & Huang, Gow-Cheng, 2004. "Industry momentum strategies and autocorrelations in stock returns," Journal of Empirical Finance, Elsevier, vol. 11(2), pages 185-202, March.
    22. Narasimhan Jegadeesh & Sheridan Titman, 2001. "Profitability of Momentum Strategies: An Evaluation of Alternative Explanations," Journal of Finance, American Finance Association, vol. 56(2), pages 699-720, April.
    23. Coeurjolly, Jean-Francois, 2000. "Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 5(i07).
    24. Eling, Martin & Schuhmacher, Frank, 2007. "Does the choice of performance measure influence the evaluation of hedge funds?," Journal of Banking & Finance, Elsevier, vol. 31(9), pages 2632-2647, September.
    25. Moskowitz, Tobias J. & Ooi, Yao Hua & Pedersen, Lasse Heje, 2012. "Time series momentum," Journal of Financial Economics, Elsevier, vol. 104(2), pages 228-250.
    26. Claude B. Erb & Campbell R. Harvey, 2015. "The Strategic and Tactical Value of Commodity Futures," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 6, pages 125-178, World Scientific Publishing Co. Pte. Ltd..
    27. Jegadeesh, Narasimhan & Titman, Sheridan, 1993. "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency," Journal of Finance, American Finance Association, vol. 48(1), pages 65-91, March.
    28. Brock, William & Lakonishok, Josef & LeBaron, Blake, 1992. "Simple Technical Trading Rules and the Stochastic Properties of Stock Returns," Journal of Finance, American Finance Association, vol. 47(5), pages 1731-1764, December.
    29. Harrison, J Michael & Pitbladdo, Richard & Schaefer, Stephen M, 1984. "Continuous Price Processes in Frictionless Markets Have Infinite Variation," The Journal of Business, University of Chicago Press, vol. 57(3), pages 353-365, July.
    30. Wang, Xiao-Tian & Wu, Min & Zhou, Ze-Min & Jing, Wei-Shu, 2012. "Pricing European option with transaction costs under the fractional long memory stochastic volatility model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1469-1480.
    31. Julia S. Mehlitz & Benjamin R. Auer, 2024. "Memory-enhanced momentum in commodity futures markets," The European Journal of Finance, Taylor & Francis Journals, vol. 30(8), pages 773-802, May.
    32. Benjamin R Auer, 2016. "Pure return persistence, Hurst exponents and hedge fund selection – A practical note," Journal of Asset Management, Palgrave Macmillan, vol. 17(5), pages 319-330, September.
    33. Christopher Krauss, 2017. "Statistical Arbitrage Pairs Trading Strategies: Review And Outlook," Journal of Economic Surveys, Wiley Blackwell, vol. 31(2), pages 513-545, April.
    34. Jerry Coakley & Neil Kellard & Jian Wang, 2016. "Commodity futures returns: more memory than you might think!," The European Journal of Finance, Taylor & Francis Journals, vol. 22(14), pages 1457-1483, November.
    35. Kristoufek, Ladislav, 2012. "How are rescaled range analyses affected by different memory and distributional properties? A Monte Carlo study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(17), pages 4252-4260.
    36. López-García, M.N. & Trinidad-Segovia, J.E. & Sánchez-Granero, M.A. & Pouchkarev, I., 2021. "Extending the Fama and French model with a long term memory factor," European Journal of Operational Research, Elsevier, vol. 291(2), pages 421-426.
    37. Cumova, Denisa & Nawrocki, David, 2014. "Portfolio optimization in an upside potential and downside risk framework," Journal of Economics and Business, Elsevier, vol. 71(C), pages 68-89.
    38. Hiemstra, Craig & Jones, Jonathan D., 1997. "Another look at long memory in common stock returns," Journal of Empirical Finance, Elsevier, vol. 4(4), pages 373-401, December.
    39. Benjamin R. Auer & Frank Schuhmacher, 2015. "Liquid Betting against Beta in Dow Jones Industrial Average Stocks," Financial Analysts Journal, Taylor & Francis Journals, vol. 71(6), pages 30-43, November.
    40. Malceniece, Laura & Malcenieks, Kārlis & Putniņš, Tālis J., 2019. "High frequency trading and comovement in financial markets," Journal of Financial Economics, Elsevier, vol. 134(2), pages 381-399.
    41. Chordia, Tarun & Subrahmanyam, Avanidhar & Tong, Qing, 2014. "Have capital market anomalies attenuated in the recent era of high liquidity and trading activity?," Journal of Accounting and Economics, Elsevier, vol. 58(1), pages 41-58.
    42. Rachev, Svetlozar & Jasic, Teo & Stoyanov, Stoyan & Fabozzi, Frank J., 2007. "Momentum strategies based on reward-risk stock selection criteria," Journal of Banking & Finance, Elsevier, vol. 31(8), pages 2325-2346, August.
    43. Jean-Philippe Bouchaud, 2021. "The Inelastic Market Hypothesis: A Microstructural Interpretation," Papers 2108.00242, arXiv.org, revised Jan 2022.
    44. Erhan Bayraktar & H. Vincent Poor, 2005. "Arbitrage In Fractal Modulated Black–Scholes Models When The Volatility Is Stochastic," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 283-300.
    45. Hogan, Steve & Jarrow, Robert & Teo, Melvyn & Warachka, Mitch, 2004. "Testing market efficiency using statistical arbitrage with applications to momentum and value strategies," Journal of Financial Economics, Elsevier, vol. 73(3), pages 525-565, September.
    46. Barroso, Pedro & Santa-Clara, Pedro, 2015. "Momentum has its moments," Journal of Financial Economics, Elsevier, vol. 116(1), pages 111-120.
    47. Schuhmacher, Frank & Eling, Martin, 2011. "Sufficient conditions for expected utility to imply drawdown-based performance rankings," Journal of Banking & Finance, Elsevier, vol. 35(9), pages 2311-2318, September.
    48. 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.
    49. Jean-Philippe Bouchaud, 2022. "The inelastic market hypothesis: a microstructural interpretation," Quantitative Finance, Taylor & Francis Journals, vol. 22(10), pages 1785-1795, October.
    50. Paolo Guasoni & Yuliya Mishura & Miklós Rásonyi, 2021. "High-frequency trading with fractional Brownian motion," Finance and Stochastics, Springer, vol. 25(2), pages 277-310, April.
    51. Juliana Nascimento & Warren Powell, 2010. "Dynamic Programming Models and Algorithms for the Mutual Fund Cash Balance Problem," Management Science, INFORMS, vol. 56(5), pages 801-815, May.
    52. Bessembinder, Hendrik, 2018. "Do stocks outperform Treasury bills?," Journal of Financial Economics, Elsevier, vol. 129(3), pages 440-457.
    53. Matthieu Garcin, 2022. "Forecasting with fractional Brownian motion: a financial perspective," Quantitative Finance, Taylor & Francis Journals, vol. 22(8), pages 1495-1512, August.
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    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

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