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The Generalization of λ-fuzzy measures on Application to Options Pricing 

The Generalization of λ-fuzzy measures on Application to Options Pricing . Liyan Han, Wenli Chen, Juan Zhou School of economics & management, Beihang University, Beijing 100083, China Email: cwl-1028@163.com

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The Generalization of λ-fuzzy measures on Application to Options Pricing 

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  1. The Generalization of λ-fuzzy measures on Application to Options Pricing Liyan Han, Wenli Chen, Juan Zhou School of economics & management, Beihang University, Beijing 100083, China Email: cwl-1028@163.com  The research is supported by National Natural Science Foundation of China, titled as “Research on the pricing methods of options based on the fuzzy measures” (70271010).

  2. Abstract The paper presents the definition of λ-fuzzy signed measure and its generalized transform function. And then we discuss the distribution properties of λ-fuzzy measures with the varying parameters in neutral probability distributions. Furthermore decision makers with different attitude towards risk can make their own expected values across the natural states in the future. Therefore this paper reveals non-identical rationality with a family of λ-fuzzy measures and deduces fuzzy price of the options. • Key words λ-fuzzy measures; signed measures; non-identical rationality; fuzzy options.

  3. Related definitions

  4. Construction and distribution properties

  5. Application to option pricing • Theorem (Expression for fuzzy options pricing): The fuzzy value of the option can be expressed as

  6. Conclusion • 1. Nonadditive measures are suitable to represent information of an uncertain variable. Especially for -fuzzy measures, selecting the parameter  appropriately, we can construct belief, plausibility or probability measures as we want. • 2. In this paper, we propose the definition of the fuzzy signed measure and generalize its transform functions, and discuss fuzzy measure distribution properties. • 3. With the changes in classical distribution parameters, the properties of -fuzzy measures distribution functions are similar to those of classical ones associated with intersections and monotonicities. When <0, -fuzzy measures can increase the classical probability by transform, which the behaviors can apply to overreaction for information by selecting the parameter  appropriately. Hence we can show that -fuzzy measures are useful to explain the occurrences of impossible events in financial market.

  7. Conclusion • 4.This paper is an attempt to consider the heterogeneity in asset pricing. Combining a family of λ-fuzzy measures and the Choquet integral, we propose the concept of “fuzzy option” and construct its pricing models primarily. In this frame, the family of λ-fuzzy measures mathematically stands for agents’ different attitudes towards the same set of market information, and particularly with varying the unique parameter , the diversified individual rationality is expressed. Further the fuzzy value of the option is naturally calculated by taking the Choquet expectations, which covered the classical option value by well-known Black-Scholes model.

  8. References • [1] Yager, R.R., “Measuring the information and character of a fuzzy measure,” IEEE July 2001Page(s):1718 - 1722 vol.3. • [2] Grabish, M.Murofushi, T. and Sugeno, Fuzzy measures and integrals: theory and applications, New York: Phusica Verlag, 2000. • [3] Chengli Zheng, Fuzzy options and their application in risk management. Ph.D. Thesis, Beihang university, 2002. • [4] Umberto Cherubini, “Fuzzy measures and asset prices: accounting for information ambiguity,” Applied Mathematical Finance Vol. 4, 1997. • [5] Umberto Cherubini and Giovanni Della Lunoa, “Fuzzy Value-at-risk: Accounting for Market Liquidity,” Blackwell, 2001. • [6] Liyan Han and Chengli Zheng, “Fuzzy options with application to default risk analysis for municipal bonds in China (to be published),” Journal of Nonlinear Analysis, Theory, Methods and Applications. • [7] Wang, Z. and Klir, G.J., Fuzzy Measure Theory, Plenum Press, New York, 1992. • [8] Liu Yingming, Introduction to Fuzzy mathematics, Sichuan education press, Sichuan, China, 1992. • [9]M.Sugeno, Theory of fuzzy integrals and its applications. Ph.D. Thesis, Tokyo Institute of Technology, 1974. • [10] Paul R. Halmos, Measure Theory, Springer-Verlag New York Inc., 1974. • [11] Klir George J., Zhenyuan Wang, and Harmanec, David, “Constructing fuzzy measures in expert systems.” In Fuzzy Sets Syst., Vol. 92, pp. 251-264, Elsevier.

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