European Lookback option pricing with floating strike price under fractional Black-Scholes models

[1] Andersen, TG, & Bollerslev, T. (1997). Heterogeneous information arrivals and return volatility dynamics: Uncovering the long-run in high frequency returns. The Journal of Finance, 52(3), 975{1005. https://doi.org/10.1111/j.1540-6261.1997.tb02722.x

[2] Carr, P., & Wu, L. (2003). The  nite moment log stable process and option pricing. The Journal of Finance, 58(2), 753{777. https://doi.org/10.1111/1540-6261.00544

[3] Chen, C., Wang, Z., & Yang, Y., (2019). A new operator splitting method for American options under fractional Black-Scholes models. Computers and Mathematics with Applications, 77(8), 2130{2144. https://doi.org/10.1016/j.camwa.2018.12.007

[4] Chen, Q., Zhang, Q., & Liu, C. (2019). The pricing and numerical analysis of Lookback options for mixed fractional Brownian motion. Chaos, Solitons and Fractals, 128, 123{128. https://doi.org/10.1016/j.chaos.2019.07.038

[5] Cheridito, P. (2003). Arbitrage in fractional Brownian motion models. Finance and Stochastics, 7(4), 533{553.
https://doi.org/10.1007/s007800300101

[6] Chung, SL, Huang, YT, Shih, PT, & Wang, JY (2019). Semistatic hedging and pricing American oating strike Lookback options. Journal of Futures Markets, 39(4), 418{434. https://doi.org/10.1002/fut.21986

[7] Conze, A. (1991). Path dependent options: The case of Lookback options. The Journal of Finance, 46(5), 1893{1907. https://doi.org/10.1111/j.1540-6261.1991.tb04648.x

[8] David, A. (2004). Levy processes and stochastic calculus. Cambridge University Press, London. https://doi.org/10.1017/CBO9780511755323

[9] Ding, Z., Granger, C., & Engle, R. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1), 83{106. https://doi.org/10.1016/0927-5398(93)90006-D

[10] Elliott, RJ, & Hoek, JVD (2003). A general fractional white noise theory and applications to  nance. Mathematical Finance, 13(2), 301{330. https://doi.org/10.1111/1467-9965.00018

[11] Fallah, S., & Mehrdoust, F. (2019). On the existence and uniqueness of the solution to the double Heston model equation and valuing Lookback option. Journal of Computational and Applied Mathematics, 350, 412{422. https://doi.org/10.1016/j.cam.2018.10.045

[12] Farhadi, A., Salehi, M., & Erjaee, GH (2018). A new version of Black-Scholes equation presented by time-fractional derivative. Iranian Journal of Science and Technology, Transactions A: Science, 42(4), 2159{2166. https://doi.org/10.1007/s40995-017-0244-7

[13] Goldman, MB, Sosin, HB, & Gatto, MA (1979). Path dependent options: Buy at the low, sell at the high. The Journal of Finance, 34(5), 1111{1127. https://doi.org/10.2307/2327238

[14] Hashemi, SAS, Saeedi, H., & Bastani, AF (2024). A hybrid Chelyshkov wavelet- nite di erences method for time-fractional black-Scholes equation. Journal of Mahani Mathematical Research, 13(2), 423{452. https://doi.org/10.22103/jmmr.2024.22371.1526

[15] Hu, YZ, & ?ksendal, B. (2003). Fractional white noise calculus and applications to  nance. In nite Dimensional Analysis,
Quantum Probability and Related Topics, 6(1), 1{32. https://doi.org/10.1142/S0219025703001110

[16] Jumarie, G. (2008). Stock exchange fractional dynamics de ned as fractional exponential growth driven by Gaussian white noise. Application to fractional Black-Scholes equations. Insurance: Mathematics and Economics, 42(1), 271{287. https://doi.org/10.1016/j.insmatheco.2007.03.001

[17] Kim, KI, Park, HS, & Qian, XS (2011). A mathematical modeling for the Lookback option with jump-di usion using binomial tree method. Journal of Computational and Applied Mathematics, 235(17), 5140{5154. https://doi.org/10.1016/j.cam.2011.05.002

[18] Leung, KS (2013). An analytic pricing formula for Lookback options under stochastic volatility. Applied Mathematics Letters, 26(1), 145{149. https://doi.org/10.1016/j.aml.2012.07.008

[19] Mandelbrot, BB (1997). Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4757-2763-0

[20] Muller, UA, Dacorogna, MM, & Pictet, OV (1998). Heavy tails in highfrequency  nancial data. A Practical Guide to Heavy Tails: Statistical Techniques and Applications, 55{78. https://dx.doi.org/10.2139/ssrn.939

[21] Park, SH, & Kim, JH (2013). A semi-analytic pricing formula for Lookback options under a general stochastic volatility model. Statistics and Probability Letters, 83(11), 2537{2543. https://doi.org/10.1016/j.spl.2013.08.002

[22] Podlubny, I. (1999). Fractional Di erential Equations. Academic Press.

[23] Rezaei, D., & Izadi, M. (2023). An analytical solution to time-space fractional Black-Scholes option pricing model. University Politehnica of Bucharest Scienti c Bulletin-Series A-Applied Mathematics and Physics, 85(1), 129{40.

[24] Rezaei, M., & Yazdanian, AR (2019). Numerical solution of the time-fractional Black-Scholes equation for European double barrier option with time-dependent parameters under the CEV model. Financial Engineering and Portfolio Management, 10(39), 339{369. https://dorl.net/dor/20.1001.1.22519165.1398.10.39.16.4  [In Persian]

[25] Rezaei, M., Yazdanian, AR, Ashra , A., & Mahmoudi, SM (2021). Numerical pricing based on fractional Black-Scholes equation with timedependent parameters under the CEV model: Double barrier options. Computers and Mathematics with Applications, 90, 104{111. https://doi.org/10.1016/j.camwa.2021.02.021

[26] Rezaei, M., Yazdanian, AR, Ashra , A., & Mahmoudi, SM (2022). Numerically pricing nonlinear time-fractional Black-Scholes equation with timedependent parameters under transaction costs. Computational Economics, 60(1), 243{280. https://doi.org/10.1007/s10614-021-10148-z

[27] Rezaei, M., Yazdanian, AR, Mahmoudi, SM, & Ashra , A. (2021). A compact di erence scheme for time-fractional Black-Scholes equation with time-dependent parameters under the CEV model: American options. Computational Methods for Di erential Equations, 9(2), 523{552. https://doi.org/10.22034/cmde.2020.36000.1623

[28] Shreve, S. (2004). Stochastic calculus for  nance II, Continuous-Time Models. Springer Finance.

[29] Wang, XT, Wu, M., Zhou, ZM, & Jing, WS (2012). Pricing European option with transaction costs under the fractional long memory stochastic volatility model. Physica A, 391(4), 1469{1480. https://doi.org/10.1016/j.physa.2011.11.014

[30] Wong, HY, & Chan, CM (2007). Lookback options and dynamic fund protection under multiscale stochastic volatility.
Insurance: Mathematics and Economics, 40(3), 357{385. https://doi.org/10.1016/j.insmatheco.2006.05.006

[31] Wyss, W. (2000). The fractional Black-Scholes equations. Fractional Calculus and Applied Analysis, 3(1), 51{61.

[32] Xiao, W., Zhang, W., Xu, W., & Zhang, X. (2012). The valuation of equity warrants in a fractional Brownian  environment. Physica A, 391(4), 1742{1752. https://doi.org/10.1016/j.physa.2011.10.024

[33] Xiao, WL, Zhang, WG, Zhang, XL, & Wang, YL (2010). Pricing currency options in a fractional Brownian motion with jumps. Economic Modelling, 27(5), 935{942. https://doi.org/10.1016/j.econmod.2010.05.010

[34] Yang, Z. (2020). Default probability of American Lookback option in a mixed jump-di usion model. Physica A, 540, 123242. https://doi.org/10.1016/j.physa.2019.123242

[35] Zhang, Q., & Taksar, T. (2009). Analytical approximate solutions to American barrier and Lookback option values. Handbook of Numerical Analysis, Elsevier, 15, 665{684. https://doi.org/10.1016/S1570-8659(08)00017-3