Publication Cover
Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 15
270
Views
3
CrossRef citations to date
0
Altmetric
Articles

On implied volatility recovery of a time-fractional Black-Scholes equation for double barrier options

&
Pages 3145-3160 | Received 05 Jun 2019, Accepted 25 Dec 2019, Published online: 19 Jan 2020

References

  • Black F, Scholes M. The pricing of options and coporate liabilities. J Polit Economy. 1973;81:637–654.
  • Dupire B. Pricing with a smile. Risk. 1994;7(1):18–20.
  • Rubinstein M. Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23, 1976 through August 31, 1978. J Finance. 1985;40:455–480.
  • Kilbas AA, Srivastava HM, Trujiilo JJ, Theory and Applications of Fractional Differential Equations. Amsterdam, Netherlands: Elsevier; 2006.
  • Carpinterj A, Mainardi F. Fractals and fractional calculus in continuum mechanics. New York: Springer-Verlag; 1997.
  • Mandelbrot BB. The fractal geometry of nature. New York: W H Freeman; 1982.
  • Wyss W. The fractional Black-Scholes equation. Fract Calc Appl Anal. 2000;3(1):51–61.
  • Cartea A, Del Castillo-Negrete D. Fractional diffusion models of option prices in markets with jumps. Physica A. 2007;2(374):749–763.
  • Jumarie G. Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations. Insur Math Econom. 2008;42(1):271–287.
  • Liang J-R, Wang J, Zhang W-J et al., Option pricing of a bi-fractional Black-Merton-Scholes model with Hurst exponent H in [12,1]. Appl Math Lett. 2010;23:859–863.
  • Chen WT, Xu X, Zhu SP. Analytically pricing double barrier options based on a time-fractional Black-Scholes equation. Comput Math Appl. 2015;69:1407–1419.
  • Zhang H, Liu F, Turner I et al., Numerical solution of the time fractional Black-Scholes model governing European option. Comput Math Appl. 2016;71:1772–1783.
  • Cen Z, Huang J, Xu A et al., Numerical approximation of a time-fractional Black-Scholes equation. Comput Math Appl. 2018;75:2874–2887.
  • Chen C, Wang Z, Yang Y. A new operator splitting method for American options under fractional Black-Scholes models. Comput Math Appl. 2019;77:2130–2144.
  • Chen S, Liu F, Zhuang P et al., Finite difference approximations for the fractional Fokker-Planck equation. Appl Math Mod. 2009;33:256–273.
  • Bouchouev I, Isakov V. Uniqueness, stability and numerical methods for the inverse problem that arises in financial markets. Inverse Prob. 1999;15:R95–R116.
  • Bouchouev I, Isakov V. Recovery of volatility coefficient by linearization. Quant Finance. 2002;2:257–263.
  • Deng ZC, Hon YC, Isakov V. Recovery of time-dependent volatility in option pricing model. Inverse Probl. 2016;32:115010 (30pp).
  • Egger H, Engl HW. Tikhonov regularization applied to the inverse problem of option pricing: convergence anaalysis and rates. Inverse Probl. 2005;21:1027–45.
  • Jiang LS, Chen Q, Wang LJ et al., A new well-posed algorithm to recover implied local volatility. Quant Finance. 2003;3:451–457.
  • Egger H, Hein H, Hofmann B. On decoupling of volatility smile and term structure in inverse option pricing. Inverse Probl. 2006;22:1247–59.
  • Deng ZC, Yu JN, Yang L. An inverse problem of determining the implied volatility in option pricing. J Math Anal Appl. 2008;80:212–32.
  • Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339:1–77.
  • Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999.
  • Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974.
  • Varga RS. Matrix iterative analysis. Berlin: Springer-Verlag; 2000.
  • Agarwal OP. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 2002;29:145–55.
  • Jin B, Rundell W. A turorial on inverse problems for anomalous diffusion processes. Inverse Probl. 2015;31:035003. doi:10.1088/0266-5611/31/3/035003
  • Podlubny I. Mittag-Leffler function, The MATLAB routine http://www.mathworks.com/matlabcentral/fileexchange.
  • Engl H, Hanke M, Neubauer A. Regularization of inverse problems. Kluwer: London; 1996.
  • Tikhonov A, Glasko V. The approximate solution of Fredholm integral equations of the first kind. Vychisl Mat Mat Fiz. 1964;4:564C71.
  • Tikhonov A, Glasko V. Use of the regularization method in non-linear problems. USSR Comput Math Math Phys. 1965;5:93C107.
  • Bauer F, Kindermann S. The quasi-optimality criterion for classical inverse problems. Inverse Prob. 2008;24(P5):035002. (20pp).

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.