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A Gaussian Markov alternative to fractional Brownian motion for pricing financial derivatives

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  • Daniel Conus
  • Mackenzie Wildman

Abstract

Replacing Black-Scholes' driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementation of a simplified alternative to using fractional Brownian motion for pricing derivatives. By relaxing the assumption of past independence of Brownian motion but retaining the Markovian property, we are developing a competing model that retains the mathematical simplicity of the standard Black-Scholes model but also has the improved accuracy of allowing for past dependence. This is achieved by replacing Black-Scholes' underlying process, Brownian motion, with a particular Gaussian Markov process, proposed by Vladimir Dobri\'{c} and Francisco Ojeda.

Suggested Citation

  • Daniel Conus & Mackenzie Wildman, 2016. "A Gaussian Markov alternative to fractional Brownian motion for pricing financial derivatives," Papers 1608.03428, arXiv.org.
    • Handle: RePEc:arx:papers:1608.03428
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    References listed on IDEAS

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    1. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    2. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    3. 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.
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    Cited by:

    1. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, arXiv.org.
    2. Peter Carr & Andrey Itkin, 2019. "ADOL - Markovian approximation of rough lognormal model," Papers 1904.09240, arXiv.org.

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