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Fuzzy Fractional Brownian Motion: Review and Extension
Version 1
: Received: 30 May 2024 / Approved: 30 May 2024 / Online: 31 May 2024 (09:13:18 CEST)
A peer-reviewed article of this Preprint also exists.
Urumov, G.; Chountas, P.; Chaussalet, T. Fuzzy Fractional Brownian Motion: Review and Extension. Algorithms 2024, 17, 289. Urumov, G.; Chountas, P.; Chaussalet, T. Fuzzy Fractional Brownian Motion: Review and Extension. Algorithms 2024, 17, 289.
Abstract
In traditional finance, option prices are typically calculated using crisp sets of variables. However, in a proposed new approach, the assumption is made that these parameters have a degree of fuzziness or uncertainty associated with them. This allows participants to estimate option prices based on their risk preferences and beliefs, considering a range of possible values for the parameters. To determine the belief degree associated with a particular option price, an interpolation search algorithm was proposed recently[1]. Interpolation is a mathematical technique used to estimate values between known data points. In this case, the algorithm is applied to determine the belief degree associated with a specific option price within the range of possible prices. By incorporating fuzzy numbers and the belief degree, this approach provides a more flexible framework for practitioners to make their investment decisions. It allows them to consider their risk preferences and beliefs about the uncertain parameters when selecting an option price. In this paper, we will review a unified framework for combining fractional Brownian motion with fuzzy processes and obtain a joint measure space. We start by constructing a product measure space that captures the randomness and fuzziness inherent in the respective processes. We begin with the usual Kolmogorov filtered probability space (Ω, F, ℙ), where ℙ represents the probability measure, Ω is the set of all possible outcomes or states that a random process can take and {Ft, t ∈ [0, T]} is an information filtration. We will review the work of Merton (1976) on option pricing when the underlying stock returns have jumps [2] and expand it to fuzzy fractional processes.
Keywords
fuzzy; fractional; Brownian motion
Subject
Computer Science and Mathematics, Probability and Statistics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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