Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-19T16:00:13.626Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  23 August 2023

Department of Business, Jiangnan University, Wuxi 214100, Jiangsu, China; e-mail:
Department of Mathematics, Zhejiang A&F University, Hangzhou 310027, China; e-mail:


In this paper, the pricing of equity warrants under a class of fractional Brownian motion models is investigated numerically. By establishing a new nonlinear partial differential equation (PDE) system governing the price in terms of the observable stock price, we solve the pricing system effectively by a robust implicit-explicit numerical method. This is fundamentally different from the documented methods, which first solve the price with respect to the firm value analytically, by assuming that the volatility of the firm is constant, and then compute the price with respect to the stock price and estimate the firm volatility numerically. It is shown that the proposed method is stable in the maximum-norm sense. Furthermore, a sharp theoretical error estimate for the current method is provided, which is also verified numerically. Numerical examples suggest that the current method is efficient and can produce results that are, overall, closer to real market prices than other existing approaches. A great advantage of the current method is that it can be extended easily to price equity warrants under other complicated models.

Research Article
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Agliardi, R., Popivanov, P. and Slavona, A., “Nonhypoellipticity and comparison principle for partial differential equations of Black–Scholes type”, Nonlinear Anal. 12 (2011) 14291436; doi:10.1016/j.nonrwa.2010.10.003.CrossRefGoogle Scholar
Almani, H. M., Hosseini, S. M. and Tahmasebi, M., “Fractional Brownian motion with two variable Hurst exponent”, J. Comput. Appl. Math. 388 (2021) Article ID 113262; doi:10.1016/ Scholar
Araneda, A. A. and Bertschinger, N., “The sub-fractional CEV model”, Phys. A 573 (2021) Article ID 125974; doi:10.1016/j.physa.2021.125974.CrossRefGoogle Scholar
Björk, T. and Hult, H., “A note on Wick products and the fractional Black–Scholes model”, Finance Stoch. 9 (2005) 197209; doi:10.1007/s00780–004–0144–5.CrossRefGoogle Scholar
Black, F. and Scholes, M., “The pricing of options and corporate liabilities”, J. Polit. Econ. 81 (1973) 637659; doi:10.1142/97898147595880001.CrossRefGoogle Scholar
Cen, Z.-D. and Chen, W.-T., “A HODIE finite difference scheme for pricing American options”, Adv. Difference Equ. 2009 (2009) Article ID 67; doi:10.1186/s13662–018–1917–z.Google Scholar
Chen, W.-T. and He, X.-J., “The pricing of credit default swaps under a generalized mixed fractional Brownian motion”, Phys. A 404 (2014) 2633; doi:10.1016/j.physa.2014.02.046.Google Scholar
Chen, W.-T., Yan, B.-Y., Lian, G.-H. and Zhang, Y., “Numerically pricing American options under the generalized mixed fractional Brownian motion model”, Phys. A 451 (2016) 180189; doi:10.1016/j.physa.2015.12.154.CrossRefGoogle Scholar
Cheng, P. and Xu, Z., “Pricing vulnerable options in a mixed fractional Brownian motion with jumps”, Discrete Dyn. Nat. Soc. 2021 (2021) Article ID 4875909; doi:10.1155/2021/4875909.CrossRefGoogle Scholar
Cheridito, P., “Arbitrage in fractional Brownian motion models”, Finance Stoch. 7 (2002) 533553; doi:10.1007/s007800300101.CrossRefGoogle Scholar
Galai, D. and Schneller, M. I., “Pricing of warrants and the value of the firm”, J. Finance 33 (1978) 13331342; doi:10.1111/j.1540–6261.1978.tb03423.x.CrossRefGoogle Scholar
Han, Y., Li, Z. and Liu, C., “Option pricing under the fractional stochastic volatility model”, ANZIAM J. 63 (2021) 123142; doi:10.1017/S1446181121000225.Google Scholar
Handley, J. C., “On the valuation of warrants”, J. Futures Markets 22 (2002) 765782; doi:10.1002/fut.10032.CrossRefGoogle Scholar
Kangro, R. and Nicolaidesn, R., “Far field boundary conditions for Black–Scholes equations”, SIAM J. Numer. Anal. 38 (2000) 13571368; doi:10.1137/S0036142999355921.CrossRefGoogle Scholar
Lim, K. G. and Terry, E., “The valuation of multiple stock warrants”, J. Futures Markets 23 (2003) 517534; doi:10.1002/fut.10079.CrossRefGoogle Scholar
Lin, S. J., “Stochastic analysis of fractional Brownian motion”, Stoch. Stoch. Rep. 55 (1995) 121140; doi:10.1080/17442509508834021.CrossRefGoogle Scholar
Omari, M. E., “Mixtures of higher-order fractional Brownian motions”, Comm. Statist. Theory Methods 52 (2003) 42004215; doi:10.1080/03610926.2021.1986541.CrossRefGoogle Scholar
Schulz, G. U. and Trautmann, S., “Robustness of option-like warrant valuation”, J. Bank. Finance 18 (1994) 841859; doi:10.1016/0378–4266(94)00030–1.CrossRefGoogle Scholar
Suryawan, H. P. and Gunarso, B., “Self-intersection local times of generalized mixed fractional Brownian motion as white noise distributions”, J. Phys. Conf. Ser. 855 (2017) Article ID 012050; doi:10.1088/1742-6596/855/1/012050.CrossRefGoogle Scholar
Thäle, C., “Further remarks on mixed fractional Brownian motion”, Appl. Math. Sci. 3 (2009) 18851901; Scholar
Ukhov, A. D., “Warrant pricing using observable variables”, J. Financ. Res. 27 (2004) 329339; doi:10.1111/j.1475–6803.2004.00100.x.CrossRefGoogle Scholar
Wilmott, P., Dewynne, J. and Howison, S., Option pricing: mathematical models and computation (Oxford Financial Press, Oxford, 1993).Google Scholar
Xu, W., Xiao, W., Zhang, W. and Zhang, X., “The valuation of equity warrants in a fractional Brownian environment”, Phys. A 391 (2012) 17411752; doi:10.1016/j.physa.2011.10.024.Google Scholar