This paper is an empirical study investigating the effect of using a fractional Black–Scholes model for pricing Call options in comparison to the classical Black–Scholes model. We estimate the Hurst parameter by nine different methods available in software R, for a set of fifteen assets from four different industries and for various lengths of used historical data. We use implied volatility for calculation of option price with a one month time to maturity. As a main result, we provide a table of tendencies of suitability for using each of the Hurst exponent estimation methods depending on the length of underlying data set.
BallestraL.V. and CecereL., A numerical method to compute the volatility of the fractional Brownian motion implied by American options, International Journal of Applied Mathematics26(2) (2013), 203–220. doi:10.12732/ijam.v26i2.7.
2.
Barndorff-NielsenO.E.E.MikoschT.E. and ResnickS.I.E., Lévy Processes: Theory and Applications, Springer, 2001.
3.
BenderC. and ViitasaariL., Fractional Brownian Motion in Financial Modeling. Wiley StatsRef: Statistics Reference Online, (2016), 1–5.
4.
BiaginiF.HuY.ØksendalB. and ZhangT., Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, 2008.
5.
BlackF. and ScholesM., The pricing of options and corporate liabilities, The Journal of Political Economy81(3) (1973), 637–654. doi:10.1086/260062.
6.
BraniaK. and GurgulH., The impact of estimation methods and data frequency on the results of long memory assessment, Managerial Economics16(1) (2015), 7–37. doi:10.7494/manage.2015.16.1.7.
7.
CheriditoP., Arbitrage in fractional Brownian motion models, Finance and Stochastics7(4) (2003), 533–553. doi:10.1007/s007800300101.
8.
ComteF.CoutinL. and RenaultE., Affine fractional stochastic volatility models, Annals of Finance8(2) (2012), 337–378. doi:10.1007/s10436-010-0165-3.
9.
ComteF. and RenaultE., Long-memory in continuous-time stochastic volatility models, Mathematical Finance8 (1998), 291–323. doi:10.1111/1467-9965.00057.
10.
ContR. and TankovP., Financial Modelling with Jump Processes, Chapman and Hall/CRC Financial Mathematics Series, 2003.
11.
FlintE. and MareE., Fractional Black–Scholes Option pricing, Volatility Calibration and Implied Hurst Exponents, 2016, 10 June 2016, available at: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2793927.
12.
FouqueJ.P.PapanicolaouG. and SircarK.R., Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, 2000.
13.
GarnierJ. and SølnaK., Correction to Black–Scholes Formula due to Fractional Stochastic Volatility, 2015, 3 Sep 2015 arXiv:1509.01175v1 [q-fin.MF].
14.
GuasoniP., No arbitrage under transaction costs, with fractional Brownian motion and beyond, Mathematical Finance16(3) (2006), 569–582. doi:10.1111/j.1467-9965.2006.00283.x.
15.
HuY. and ØksendalB., Fractional White Noise Calculus and Applications to Finance, Infin. Dimens. Anal. Quantum. Probab. Relat. Top.6(1) (2003). doi:10.1142/S0219025703001110.
16.
KirichenkoL.RadivilovaT. and DeinekoZ., Comparative Analysis for Estimating of the Hurst Exponent for Stationary and Nonstationary Time Series, International Journal Information Technologies & Knowledge5 (2011).
17.
MandelbrotB.B., Fractals and Scaling in Finance, Discontinuity, Concentration, Risk, Springer, 1997.
18.
MandelbrotB.B. and HudsonR.L., The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin and Reward, Profile Books, 2008.
19.
MishuraY., Fractional stochastic integration and Black–Scholes equation for fractional Brownian model with stochastic volatility, Stochastics and Stochastic Reports76(4) (2007).
20.
MitraS.K., Is Hurst Exponent Value Useful in Forecasting Financial Time Series?, Asian Social Science8(8) (2012).
21.
NeculaC., Option pricing in a fractional Brownian motion environment, Mathematical Report6(3) (2004).
22.
QianB. and RasheedK., Hurst exponent and financial market predictability, in: Proceedings of the 2nd IASTED International Conference of Financial Engineering and Applications, Cambridge, MA, USA, 2004, pp. 203–209.
23.
RostekS., Option Pricing in Fractional Brownian Markets, Lecture Notes in Economics and Mathematical Systems, Vol. 622, Springer, 2009.
24.
SarkarS., Fractional Brownian Motion: Long range dependency, Markov property, Simulation strategy, A review, 2008.
25.
ShiryaevA.N., Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific, 1999.
26.
ShokrollahiF. and KiliçmanA., The valuation of currency options by fractional Brownian motion, SpringerPlus5 (2016), 1145. doi:10.1186/s40064-016-2784-2.
27.
SottinenT. and ValkeilaE., On arbitrage and replication in the fractional Black–Scholes pricing model, Statistics & Decisions21 (2003), 137–151.
28.
TaqquM.S. and TeverovskyV., On estimating the intensity of long-range dependence in finite and infinite variance time series, in: A Practical Guide to Heavy Tails: Statistical Techniques and ApplicationsAdlerR.J.et al., eds, Birkhäuser, Boston, 1998, pp. 177–217.
29.
ZhouQ., The application of fractional Brownian motion in option pricing, International Journal of Multimedia and Ubiquitous Engineering10(1) (2015), 173–182. doi:10.14257/ijmue.2015.10.1.16.
