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Applications of Fuzzy Set Theory, Fuzzy Measure Theory and Fuzzy Differential Calculus

Assi, J. A. (2003). Applications of Fuzzy Set Theory, Fuzzy Measure Theory and Fuzzy Differential Calculus. (Unpublished Doctoral thesis, City, University of London)


This research tackles the issue of uncertainty due to lack of information, alternatively known as Knightian Uncertainty, and its impact on option pricing. In the presence of such uncertainty, Probability Theory becomes restrictive and alternative tools are called for. In this research, we consider tools of Fuzzy Theory. We introduce three Option Pricing Models the first of which is a fuzzy binomial model based on the standard CRR binomial model. The model performs option pricing in a fuzzy world characterized by blurred prices. In such a world, it is no longer possible to price by replication. So we introduce a fuzzy pricing approach that employs Sugeno integration and fuzzy measures, and generates bounds on the possible option price. The second model is a fuzzy Black-Scholes model, which prices options in the presence of uncertain or fuzzy volatility. We model such volatility by establishing bounds on the corresponding fuzzy values thereby generating fuzzy bounds on the possible option price. Finally, the third model is an extension on an existing one period fuzzy binomial model that prices options when the underlying price is characterized by opacity. The option price returned by this model is dependent on a market parameter that summarizes its completeness. However, it is possible to defuzzify the last two models to obtain one crisp price that summarizes market information. The last two models outperform their standard counterparts.

Publication Type: Thesis (Doctoral)
Subjects: H Social Sciences > HG Finance
Q Science > QA Mathematics
Departments: Bayes Business School > Bayes Business School Doctoral Theses
Bayes Business School > Finance
Doctoral Theses
[thumbnail of Assi thesis 2003 PDF-A.pdf]
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