# Option Prices under the Fractional Black-Scholes Model

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This Demonstration shows the values of vanilla European options in a model based on fractional Brownian motion and on ordinary geometric Brownian motion (the Black–Scholes model). The strike price is fixed at 100. Options values in this model generally overprice Black–Scholes values.

Contributed by: Andrzej Kozlowski (March 2011)

Open content licensed under CC BY-NC-SA

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In spite of its having attractive properties as a model for the stock exchange, the suitability of fractional Brownian motion for option pricing is controversial.

There is some evidence that certain stock returns may exhibit the phenomenon of "long memory" (slowly decreasing covariance between returns at different times) [2], though this seems to be fairly weak. It is also generally accepted that stock returns display the phenomenon of "clustering". None of these phenomena appear in semi-martingale models, such as the Black–Scholes model. They do appear, however, if we consider the analogue of the Black–Scholes model based on fractional Brownian motion with Hurst index , where .

Since fractional Brownian motion is not a semi-martingale, the Itô theory of stochastic integrals cannot be directly applied to it. One can try to replace the Itô integral by a version of the pathwise Riemann–Stieltjes integral, but then, as has been shown by Rogers [6], the resulting model of option values admits arbitrage. As this is contrary to empirical evidence, it has been generally thought that models based on fractional Brownian are not usable for option pricing. Hu and Øksendal [4] defined a new stochastic integral based on the Skorokhod integral and the Wick product and showed that a model based on this integral does not admit arbitrage. Unfortunately, it was shown in [1] that this model does not make economic sense. However, recently Guasoni [3] showed that as soon as one introduces proportional transaction costs in the fractional Black-Scholes model, arbitrage opportunities vanish.

In this Demonstration we adopt a different approach, based on the work of Norros, Valkeila, and Virtamo [3]. They have shown that one can define a centered Gaussian martingale (called "the fundamental martingale") that generates the same filtration as the fractional Brownian motion. Since it is the filtration rather than the stochastic process itself that represents information provided by the market, it seems reasonable to use this martingale for option pricing. It is then easy to obtain formulas analogous to the classical Black–Scholes formulas (and coinciding with them when the Hurst index is ).

[1] T. Bjork and H. Hult, "A Note on Wick Products and the Fractional Black-Scholes
Model," *Finance Stoch.,* 9(2), 2005 pp. 197–209.

[2] N. J. Cutland, P. E. Kopp, and W. Willinger, "Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model," in *Proceedings of the Former Ascona Conferences on Stochastic Analysis, Random Fields and Applications, Progress in Probability*, Vol. 36, Basel: Birkhauser, 1995 pp. 327–351.

[3] P. Guasoni, "No Arbitrage under Transaction Costs, with Fractional Brownian Motion and Beyond," *Math. Finance,* 16(3), 2006 pp. 569–582.

[4] Y. Hu and B. Øksendal, "Fractional White Noise Calculus and Applications to Finance," *Inf. Dim. Anal. Quantum Probab. Rel. Top.,* 6(1), 2003 pp. 1–32.

[5] I. Norros, E. Valkeila, and J. Virtamo, "A Girsanov-Type Formula for the Fractional Brownian Motion," in* Proceedings of the* *First Nordic-Russian Symposium on Stochastics*, Helsinki, Finland, 1996.

[6] L. C. G. Rogers, "Arbitrage with Fractional Brownian Motion," *Math. Finance,* 7(1), 1997 pp. 95–105.

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