Abstract
In this work, we give a generalized formulation of the Black–Scholes model. The novelty resides in considering the Black–Scholes model to be valid on ’average’, but such that the pointwise option price dynamics depends on a measure representing the investors’ ’uncertainty’. We make use of the theory of non-symmetric Dirichlet forms and the abstract theory of partial differential equations to establish well posedness of the problem. A detailed numerical analysis is given in the case of self-similar measures.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Availability of data and materials
Not applicable.
Notes
Courant-Friedrichs-Lewy.
When the option has what is also called an intrinsic value i.e. the real value of the option, that is to say the profit that could be made in the event of immediate exercise. It means that the value is at a favorable strike price relative to the prevailing market price of the underlying asset. Yet, this does not mean that the trader will be making profit, since the expense of buying, and the commission prices have also to be considered.
When the option has what is also called an extrinsic value i.e. a value at a strike price higher than the market price of the underlying asset. In such a case, the Delta, i.e. the Greek which quantifies the risk, is less than 50.
The call is an option on a financial instrument, which consists in a right to buy. Concretely, it consists in a contract which allows the subscriber to get the targeted financial product, at a price fixed in advance - the strike price - at a given date - the expiry one, or maturity of the call.
As for the put it is this time a right to sell - or not - at the maturity date.
References
Achdou Y, Pironneau O (2005) Computational methods for option pricing. Society for Industrial and Applied Mathematics, Philadelphia
David C (2017) Control of the Black–Scholes equation. Comput Math Appl 73(7):1566–1575
Fortune P (1996) Anomalies in option pricing: the Black–Scholes model revisited. N Engl Econ Rev 17–40
Hu J, Lau K-S, Ngai S-M (2006) Laplace operators related to self-similar measures on \({\mathbb{R} }^d\). J Funct Anal 239:542–565
Kigami J (2001) Analysis on fractals. Cambridge University Press, London
Kilpeläinen T (1994) Weighted sobolev spaces and capacity. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 19(1):95–113
Kangro R, Nicolaides R (2001) Far field boundary conditions for Black–Scholes equations. SIAM J Numer Anal 38(4):1357–1368
Lions J-L, Magenes E (1968) Problèmes aux limites non homogènes et applications. Dunod
Lim GC, Martin GM, Martin VL (2005) Parametric pricing of higher order moments in s &p500 options. J Appl Economet 20(3):377–404
Mandelbrot BB (1997) Fractals and scaling in finance. Springer, New York
Ma Z-M, Röckner M (1992) Introduction to the theory of (non-symmetric) Dirichlet forms. Springer, Berlin
Riane N, David C (2019) The finite difference method for the heat equation on sierpiński simplices. Int J Comput Math 96(7):1477–1501
Riane N, David C (2021) An inverse Black–Scholes problem. Optim Eng 22:2183–2204
Strichartz RS (2006) Differential equations on fractals. A tutorial. Princeton University Press, Princeton
Wloka J (1987) Partial differential equations. Cambridge University Press, London
Zeidler E (1990) Nonlinear functional analysis and its applications. II/A: Linear monotone operators. Springer, Berlin
Acknowledgements
The authors would like to thank the anonymous referee for his judicious remarks about the first version of this paper, which contributed to significantly improve it.
Funding
No funding was received for conducting this study.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Ethical approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Riane, N., David, C. Generalized measure Black–Scholes equation: towards option self-similar pricing. Optim Eng (2024). https://doi.org/10.1007/s11081-024-09885-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11081-024-09885-5
Keywords
- Generalized measure Black–Scholes equation
- Self-similar measure
- Non-symmetric Dirichlet form
- Fractal differential equations
- Finite difference method