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.
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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.
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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.
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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
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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