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Abstract

The purpose of this paper is to assess the risk premium of a fractional financial lognormal (Black–Scholes) process relative to a non-fractional and complete financial markets pricing model. The intents of this paper are two-fold. On the one hand, provide a definition of the risk premium implied by the discount rate applied to future fractional returns relative to that of a non-fractional financial (and complete market) model. To do so, an insurance rationale is used to define a no-arbitrage risk neutral probability measure. On the other, highlight the effects of a model granularity and its Hurst index on financial risk models and their implications to risk management. In particular, we argue that fractional Brownian motion (BM) does not define a normal probability distribution but a fractional volatility model. To present simply the ideas underlying this paper, we price an elementary fractional risk free bond and its risk premium relative to a known spot interest rate. Similarly, the Black–Scholes no arbitrage model is presented in both its non-fractional conventional form and in its fractional framework. The granularity risk premium is then calculated.

Keywords

Granularity fractional finance risk premium Black–Scholes model pricing

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References

[1] Baillie

R.T.

, Long memory processes and fractional integration in econometrics, Journal of Econometrics 73 (1996), 5–59.

[2] Barnett

and Lewis

, Outliers in Statistical Data, 3rd edn, Wiley, New York, 1994.

[3] Baum

C.F.

Barkoulas

J.T.

and Caglayan

, Persistence in international inflation rates, Southern Economic Journal 65 (1999), 900–913.

[4] Baum

C.F.

Barkoulas

J.T.

and Caglayan

, Fractional monetary dynamics, Applied Economics 31 (1999), 1393–1400.

[5] Bender

, Integration with respect to fractional Brownian motion and related market models, PhD thesis, Department of Mathematics and Statistics, University of Konstanz, 2003.

[6] Bender

Sottinen

and Valkeila

, Arbitrage with fractional Brownian motion?, Theory of Stochastic Processes 12 (2006), 28.

[7] Benth

F.E.

, On arbitrage-free pricing of weather derivatives based on fractional Brownian motion, Applied Mathematical Finance 10(4) (2003), 303–324.

[8] Beran

, Statistical methods for data with long-range dependence, Stat. Sci. 7 (1992), 404–427.

[9] Bjork

and Hult

, A note on Wick products and the fractional Black–Scholes model, Finance and Stochastics 9(2) (2005), 197–209.

10.

[10] Black

and Scholes

, The pricing of options and corporate liabilities, Journal of Political Economy 81(3) (1973), 637–654.

11.

[11] Bouchaud

J.-P.

and Sornette

, The Black–Scholes option pricing problem in mathematical finance: Generalization and extensions for a large class of stochastic process, Journal of Physics (France) 4 (1994), 863–881.

12.

[12] Cajueiro

D.O.

and Tabak

B.M.

, Test for time-varying long-range dependence in volatility for emerging markets, Physica A 346(3,4) (2005), 577–588.

13.

[13] Cajueiro

D.O.

and Tabak

B.M.

, Are the crude oil markets becoming weakly efficient over time? A test for time-varying long-range dependence in prices and volatility, Energy Economics 29(1) (2007), 28–36.

14.

[14] Cajueiro

D.O.

and Tabak

B.M.

, Testing for time-varying long-range dependence in real state equity returns, Chaos, Solitons & Fractals 38(1) (2008), 293–307.

15.

[15] Cheridito

, Regularizing fractional Brownian motion with a view towards stock price modelling, PhD thesis, Eidgenöss Technische Hochschule Zürich, Swiss Federal Institute of Technology, 2001.

16.

[16] Cheridito

, Mixed fractional Brownian motion, Bernoulli 7 (2001), 913–934.

17.

[17] Cheridito

, Arbitrage in fractional Brownian motion models, Finance and Stochastics 7(4) (2003), 533–553.

18.

[18] Colemana

and Sirichand

, Fractional integration and the volatility of UK interest rates, Economics Letters 116 (2012), 381–384.

19.

[19] Comte

and Renault

, Long memory in continuous-time stochastic volatility models, Mathematical Finance 8(4) (1998), 291–323.

20.

[20] Dasgupta

and Kallianpur

, Arbitrage opportunities for a class of Gladyshev processes, Applied Mathematics and Optimization 41 (2000), 377–385.

21.

[21] Doukhan

Oppenheim

and Taqqu

M.S.

(eds), Theory and Applications of Long-Range Dependence, Birkhäuser, Boston, 2003.

22.

[22] Duncan

T.E.

and Pasik-Duncan

, Stochastic calculus for fractional Brownian motion, SIAM Journal on Control and Optimization 38(2) (2000), 582–612.

23.

[23] Dung

N.T.

, A class of fractional stochastic differential equations, Vietnam J. Math. 36(3) (2008), 271–279.

24.

[24] Dung

N.T.

, Fractional geometric mean-reversion processes, J. Math. Anal. Appl. 380(1) (2011), 396–402.

25.

[25] Dung

N.T.

and Tien

, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl. 397 (2013), 334–348.

26.

[26] Elliott

R.J.

and Van Der Hoek

, A general fractional white noise theory and applications to finance, Mathematical Finance 13(2) (2003), 301–330.

27.

[27] Feller

, The asymptotic distribution of the range of sums of independent random variables, Annals of Math. Stat. 22 (1951), 427–432.

28.

[28] Fox

and Taqqu

M.S.

, Non-central limit theorems for quadratic forms in random variables having long range dependence, Ann. Probab. 13 (1985), 428–446.

29.

[29] Fung

H.-G.

W.-C.

and Peterson

J.E.

, Examining the dependency in intra-day stock index futures, The Journal of Futures Markets 14 (1994), 405–419.

30.

[30] Fung

H.G.

and Lo

W.C.

, Memory in interest rate futures, The Journal of Futures Markets 13 (1993), 865–873.

31.

[31] Garman

M.B.

and Klass

M.J.

, On the estimation of security price volatility from historical data, Journal of Business 53(1) (1980), 67–78.

32.

[32] Geweke

and Porter-Hudak

, The estimation and application of long time series models, J. Time Series Anal. 4 (1984), 221–238.

33.

[33] Granger

, Long memory relationships and the aggregation of dynamic models, Journal of Econometrics 14 (1980), 227–238.

34.

[34] Granger

C.W.

and Joyeux

, An introduction to long memory time series models and fractional differencing, J. Time Series Anal. 1(1) (1980), 15–29.

35.

[35] Gray

H.L.

Zhang

N.R.

and Woodward

W.A.

, On generalized fractional processes, J. Time Series Anal. 10 (1989), 233–257.

36.

[36] Green

M.T.

and Fielitz

, Long term dependence in common stock returns, J. of Fin. Econ. 4 (1977), 339–349.

37.

[37] Green

M.T.

and Fielitz

, Long term dependence and least squares regression in investment analysis, Management Science 26(10) (1980), 1031–1038.

38.

[38] Gu

Liang

and Zhang

, Time-changed geometric fractional Brownian motion and option pricing with transaction costs, Physica A: Statistical Mechanics and its Applications 391(15) (2012), 3971–3977.

39.

[39] Helms

Kaen

and Rosenman

, Memory in commodity futures contracts, Journal of Futures Markets 4 (1984), 559–567.

40.

[40] Hu

and Øksendal

, Fractional white noise calculus and applications to finance, Infinite Dimensional Analysis, Quantum Probability and Related Topics 6(1) (2003), 1–32.

41.

[41] Hurst

H.E.

, Long terms storage of reservoirs, Transaction of the American Society of Civil Engineers 116 (1951).

42.

[42] Imhoff

J.P.

, On the range of Brownian motion and its inverse process, Ann. Probab. 13(3) (1985), 1011–1017.

43.

[43] Irwin

J.O.

, On a criterion for the rejection of outlying observations, Biometrika 17 (1925), 237–250.

44.

[44] Jumarie

, Stochastic differential equations with fractional Brownian motion input, Int. J. Syst. Sc. 24(6) (1993), 1113–1132.

45.

[45] Jumarie

, On the representation of fractional Brownian motion as an integral with respect to (dt), Appl. Math. Lett. 18 (2005), 739–748.

46.

[46] Jumarie

, On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. Math. Lett. 18 (2005), 817–826.

47.

[47] Jumarie

, Merton’s model of optimal portfolio in a Black–Scholes market driven by a fractional Brownian motion with short-range dependence, Insurance: Mathematics and Economics 37 (2005), 585–598.

48.

[48] Jumarie

, Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51 (2006), 1367–1376.

49.

[49] Jumarie

, New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Math. Comput. Modelling 44 (2006), 231–254.

50.

[50] Jumarie

, Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black–Scholes equations, Insurance: Mathematics and Economics 42 (2008), 271–287.

51.

[51] Jumarie

, Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions, Applied Mathematics Letters 22 (2009), 378–385.

52.

[52] Jumarie

, Fractional multiple birth–death processes with birth probabilities

λ_{i} {(Δ t)}^{α} + o ({(Δ t)}^{α})

, Journal of the Franklin Institute 347 (2010), 1797–1813.

53.

[53] Jumarie

, Fractional Differential Calculus for Non-differentiable Functions, Lambert Academic Publishing, Saarbrucken, 2013.

54.

[54] Klüppelberg

and Kühn

, Fractional Brownian motion as a weak limit of Poisson shot noise processes – With applications to finance, Stochastic Processes and their Applications 113(2) (2004), 333–351.

55.

[55] Kumar

and Nivedita

, Multifractal properties of the Indian financial market, Physica A 388 (2009), 1593–1602.

56.

[56] Liouville

, Sur le calcul des differentielles à indices quelconques, J. Ecole Polytechnique 13 (1832), 71 (in French).

57.

[57] Lo

A.W.

, Long term memory in stock market prices, Econometrica 59(5) (1991), 1279–1313.

58.

[58] Lo

A.W.

, Fat tails, long memory and the stock market since 1960’s, Economic Notes 26 (1997), 213–245.

59.

[59] Mandelbrot

, When can price be arbitraged efficiently? A limit to the validity of the random walk and martingale models, Review of Economics and Statistics 53 (1971), 225–236.

60.

[60] Mandelbrot

, Statistical methodology for non-periodic cycles: From the covariance to R/S analysis, Annals of Economic and Social Measurement 1 (1972), 259–290.

61.

[61] Mandelbrot

, Three fractal models in finance: Discontinuity, concentration, risk, Economic Notes 26 (1997), 171–212.

62.

[62] Mandelbrot

and Wallis

, Noah, Joseph and operational hydrology, Water Resources Research 4 (1968), 909–918.

63.

[63] Mandelbrot

and Wallis

J.R.

, Computer experiments with fractional noises, Water Resources Research 5 (1969), 228–267.

64.

[64] Mandelbrot

B.B.

, The variation of certain speculative price, The Journal of Business 36 (1963), 394–419.

65.

[65] Mandelbrot

B.B.

and Taqqu

M.S.

, Robust R/S analysis of long run serial correlation, in: Proc. 42nd Session of the Internat. Statistical Institute, Bull. ISI, Vol. 48, Manila, 1979, pp. 69–104.

66.

[66] Mandelbrot

B.B.

and Van Ness

J.W.

, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968), 422–437.

67.

[67] Mariani

M.C.

Florescu

Beccar Varela

M.P.

and Ncheuguim

, Long correlations and Levy models applied to the study of memory effects in high frequency data, Physica A 388 (2009), 1659–1664.

68.

[68] Meng

and Wang

, Comparison of Black–Scholes formula with fractional Black–Scholes formula in the foreign exchange option market with changing volatility, Asia-Pacific Financial Markets 17(2) (2010), 99–111.

69.

[69] Merton

, An intertemporal capital asset pricing model, Econometrica 41 (1973), 867–887.

70.

[70] Neyman

and Scott

E.L.

, Outliers proneness of phenomena and of related distributions, in: Optimizing Methods in Statistics Rustagi

, ed., Academic Press, New York, 1971.

71.

[71] Nualart

and Taqqu

M.S.

, Some issues concerning Wick integrals and the Black and Scholes formula, 2006.

72.

[72] O

H.-C.

Kim

D.-H.

and Pak

C.-H.

, Analytical pricing of defaultable discrete coupon bonds in unified two-factor model of structural and reduced form models, J. Math. Anal. Appl. 416 (2014), 314–334.

73.

[73] Osler

T.J.

, Taylor’s series generalized for fractional derivatives and applications, SIAM Journal of Mathematical Analysis 2(1) (1971), 37–47.

74.

[74] Otway

T.H.

, Records of the Florentine proveditori degli cambiatori: An example of an antipersistent time series in economics, Chaos, Solitons & Fractals 5 (1995), 103–107.

75.

[75] Ozdemir

Z.A.

, Linkages between international stock markets: A multivariate long-memory approach, Physica A 388 (2009), 2461–2468.

76.

[76] Parkinson

, The extreme value method for estimating the variance of the rate of return, Journal of Business 53(1) (1980), 61–65.

77.

[77] Ramirez

J.A.

Alvarez

Rodriguez

and Anaya

G.F.

, Time-varying Hurst exponent for US stock markets, Physica A 387 (2008), 6159–6569.

78.

[78] Rogers

L.C.G.

, Arbitrage with fractional Brownian motion, Mathematical Finance 7 (1997), 95–105.

79.

[79] Rostek

, Option Pricing in Fractional Brownian Markets, Springer, Berlin, 2009.

80.

[80] Rostek

and Schöbel

, A note on the use of fractional Brownian motion for financial modeling, Economic Modelling 30 (2013), 30–35.

81.

[81] Sottinen

, Fractional Brownian motion, random walks and binary market models, Finance and Stochastics 5(3) (2001), 343–355.

82.

[82] Sottinen

and Valkeila

, On arbitrage and replication in the fractional Black–Scholes pricing model, Statistics and Decisions 21 (2003), 93–107.

83.

[83] Stratonovich

R.L.

, A new form of representing stochastic integrals and equations, Journal of SIAM Control 4 (1966), 362–371.

84.

[84] Tapiero

C.S.

, A financial CCAPM and economic inequalities, Quantitative Finance 15 (2015), 521–534.

85.

[85] Tapiero

C.S.

, Fractional Finance, Lecture Notes, NYU School of Engineering, 2015.

86.

[86] Tapiero

C.S.

and Vallois

, Run length statistics and the Hurst exponent in random and birth–death random walks, Chaos, Solitons and Fractals (1996).

87.

[87] Tapiero

C.S.

and Vallois

, Range reliability in random walks, Mathematics Methods of Operations Research 45(3) (1997), 325–345.

88.

[88] Tapiero

C.S.

and Vallois

, The inter-event range process and testing for chaos in time series, Neural Network World 10(1,2) (2000), 89–99.

89.

[89] Tapiero

C.S.

and Vallois

, Fractional randomness and risk finance, the extreme risk initiative, Working paper, New York University School of Engineering, Brooklyn, New York, 2015.

90.

[90] Taqqu

M.S.

, Fractional Brownian motion and long-range dependence, in: Theory and Applications of Long-Range Dependence Doukhan

Oppenheim

and Taqqu

M.S.

, eds, Birkhäuser, 2003, pp. 5–38.

91.

[91] Tsay

W.J.

, The long memory story of the real interest rate, Economics Letters 67 (2000), 325–330.

92.

[92] Vallois

and Tapiero

C.S.

, Moments of an amplitude process in a random walk, Recherche Operationnelle/Operation Research (RAIRO) 29(1) (1995), 1–17.

93.

[93] Vallois

and Tapiero

C.S.

, The range process in random walks: Theoretical results and applications, in: Advances in Computational Economics Ammans

Rustem

and Whinston

, eds, Kluwer Publications, 1996.

94.

[94] Vallois

and Tapiero

C.S.

, The range inter-event process in a symmetric birth death random walks, Applied Stochastic Models in Business and Industry (2001).

95.

[95] Vallois

and Tapiero

C.S.

, Volatility estimators and the inverse range process in a random volatility random walk and Wiener processes, Physica A (2007).

96.

[96] Viano

M.C.

Deniau

and Oppenheim

, Continuous-time fractional ARMA processes, Statistics & Probability Letters 21 (1994), 323–336.

97.

[97] Wang

X.-T.

, Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model, Physica A 389 (2010), 438–444.

98.

[98] Wang

X.-T.

Zhu

E.-H.

Tang

M.-M.

and Yan

H.-G.

, Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian fractional Brownian model, Physica A 389 (2010), 445–451.

99.

[99] Willinger

Taqqu

M.S.

and Teverovsky

, Stock market prices and long-range dependence, Finance and Stochastics 3 (1999), 1–13.

100.

[100] Xiao

Zhang

and Wang

, Pricing currency options in a fractional Brownian motion with jumps, Economic Modelling 27(5) (2010), 935–942.

The price of granularity and fractional finance

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