The sub-fractional CEV model. (English) Zbl 1527.60029
Summary: The sub-fractional Brownian motion (sfBm) is a stochastic process, characterized by non-stationarity in their increments and long-range dependence, considered as an intermediate step between the standard Brownian motion (Bm) and the fractional Brownian motion (fBm). The mixed process, a linear combination between a Bm and an independent sfBm, called mixed sub-fractional Brownian motion (msfBm), keeps the features of the sfBm adding the semi-martingale property for \(H > 3/4\), is a suitable candidate to use in price fluctuation modeling, in particular for option pricing. In this note, we arrive at the European Call price under the Constant Elasticity of Variance (CEV) model driven by a mixed sub-fractional Brownian motion. Empirical tests show the capacity of the proposed model to capture the temporal structure of option prices across different maturities.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 91G30 | Interest rates, asset pricing, etc. (stochastic models) |
Keywords:
sub-fractional Brownian motion; CEV model; option pricing; sub-fractional Fokker-Planck; long-range dependence; econophysicsSoftware:
DLMFReferences:
| [1] | Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., 116, 1, 1-18 (2006) · Zbl 1082.60024 |
| [2] | Bojdecki, Tomasz; Gorostiza, Luis; Talarczyk, Anna, Particle systems with quasi-homogeneous initial states and their occupation time fluctuations, Electron. Commun. Probab., 15, 191-202 (2010) · Zbl 1226.60048 |
| [3] | Dzhaparidze, Kacha; Van Zanten, Harry, A series expansion of fractional Brownian motion, Probab. Theory Related Fields, 130, 1, 39-55 (2004) · Zbl 1059.60048 |
| [4] | Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna, Sub-fractional Brownian motion and its relation to occupation times, Statist. Probab. Lett., 69, 4, 405-419 (2004) · Zbl 1076.60027 |
| [5] | Mandelbrot, Benoit B.; Van Ness, John W., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 4, 422-437 (1968) · Zbl 0179.47801 |
| [6] | Tudor, Constantin, Some properties of the sub-fractional Brownian motion, Stochastics, 79, 5, 431-448 (2007) · Zbl 1124.60038 |
| [7] | Mishura, Yuliya; Zili, Mounir, Stochastic Analysis of Mixed Fractional Gaussian Processes (2018), ISTE Press - Elsevier · Zbl 1455.60004 |
| [8] | Bassler, Kevin E.; McCauley, Joseph L.; Gunaratne, Gemunu H., Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets, Proc. Natl. Acad. Sci., 104, 44, 17287-17290 (2007) |
| [9] | Morales, Raffaello; Di Matteo, Tiziana; Aste, Tomaso, Non-stationary multifractality in stock returns, Physica A, 392, 24, 6470-6483 (2013) |
| [10] | Costa, Rogério L.; Vasconcelos, G. L., Long-range correlations and nonstationarity in the Brazilian stock market, Physica A, 329, 1-2, 231-248 (2003) · Zbl 1056.91506 |
| [11] | Baldovin, Fulvio; Bovina, Dario; Camana, Francesco; Stella, Attilio L., Modeling the non-Markovian, non-stationary scaling dynamics of financial markets, (Abergel, F.; Chakrabarti, B. K.; Chakraborti, A.; Mitra, M., Econophysics of Order-Driven Markets. Econophysics of Order-Driven Markets, New Economic Windows (2011), Springer), 239-252 |
| [12] | Zhang, Xili; Xiao, Weilin, Arbitrage with fractional Gaussian processes, Physica A, 471, 620-628 (2017) · Zbl 1400.91692 |
| [13] | El-Nouty, Charles; Zili, Mounir, On the sub-mixed fractional Brownian motion, Appl. Math. J. Chinese Univ., 30, 1, 27-43 (2015) · Zbl 1340.60054 |
| [14] | Tudor, Constantin, Sub-fractional Brownian Motion as a Model in FinanceWorking paper (2008), University of Bucharest |
| [15] | Liu, Junfeng; Li, Li; Yan, Litan, Sub-fractional model for credit risk pricing, Int. J. Nonlinear Sci. Numer. Simul., 11, 4, 231-236 (2010) |
| [16] | Xu, Feng; Li, Runze, The pricing formulas of compound option based on the sub-fractional Brownian motion model, (Journal of Physics: Conference Series, vol. 1053 (2018), IOP Publishing), Article 012027 pp. |
| [17] | Xu, Feng; Zhou, Shengwu, Pricing of perpetual American put option with sub-mixed fractional Brownian motion, Fract. Calc. Appl. Anal., 22, 4, 1145-1154 (2019) · Zbl 1439.91035 |
| [18] | Black, Fischer; Scholes, Myron, The pricing of options and corporate liabilities, J. Political Econ., 81, 3, 637-654 (1973) · Zbl 1092.91524 |
| [19] | Cox, John C., Notes on Option Pricing I: Constant Elasticity of Variance DiffusionsWorking paper (1975), Stanford University |
| [20] | Cox, John C., The constant elasticity of variance option pricing model, J. Portfolio Manag., 23, 5, 15-17 (1996) |
| [21] | Linetsky, Vadim; Mendoza, Rafael, Constant elasticity of variance (CEV) diffusion model, (Cont, Rama, Encyclopedia of Quantitative Finance (2010), John Wiley & Sons, Ltd) |
| [22] | Araneda, Axel A., The fractional and mixed-fractional CEV model, J. Comput. Appl. Math., 363, 106-123 (2020) · Zbl 1422.91677 |
| [23] | Yan, Litan; Shen, Guangjun; He, Kun, Itô’s formula for a sub-fractional Brownian motion, Commun. Stoch. Anal., 5, 1, 9 (2011) · Zbl 1331.60068 |
| [24] | Ünal, Gazanfer, Fokker-Plank-Kolmogorov equation for fBm: Derivation and analytical solutions, (Mathematical Physics (2007), World Scientific), 53-60 · Zbl 1131.82034 |
| [25] | Cheridito, Patrick, Mixed fractional Brownian motion, Bernoulli, 7, 6, 913-934 (2001) · Zbl 1005.60053 |
| [26] | Delbaen, Freddy; Shirakawa, Hiroshi, A note on option pricing for the constant elasticity of variance model, Asia-Pacific Financial Mark., 9, 2, 85-99 (2002) · Zbl 1072.91020 |
| [27] | Feller, William, Two singular diffusion problems, Ann. of Math., 173-182 (1951) · Zbl 0045.04901 |
| [28] | Hsu, Ying-Lin; Lin, T. I.; Lee, C. F., Constant elasticity of variance (CEV) option pricing model: Integration and detailed derivation, Math. Comput. Simulation, 79, 1, 60-71 (2008) · Zbl 1144.91325 |
| [29] | Schroder, Mark, Computing the constant elasticity of variance option pricing formula, J. Finance, 44, 1, 211-219 (1989) |
| [30] | Masoliver, Jaume, Nonstationary feller process with time-varying coefficients, Phys. Rev. E, 93, 1, Article 012122 pp. (2016) |
| [31] | Olde Daalhuis, Adri, Confluent hypergeometric functions, (Olver, Frank; Lozier, Daniel; Boisvert, Ronald; Clark, Charles, NIST Handbook of Mathematical Functions, 321-349 (2010), Cambridge University Press) · Zbl 1198.00002 |
| [32] | Wharton Research Data Services, “WRDS” wrds.wharton.upenn.edu (Accessed 31 October 2020). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
