Fuzzy Models And Methods For Finance

Portfolio selection theory and option pricing theory are two core contents of mod-ern financial area. They are faced with the same problem of how to deal with uncer-tainty. Classic models and methods regard uncertainty as randomness. However, as the rise of behavioral finance, the fuzziness in the security market gradually attracts more and more attentions, and fuzzy models and methods for financial problems have become a hot area of academic concern. This dissertation establishes fuzzy models for portfolio selection problems as well as proposes fuzzy methods for solving option pricing problems in the framework of credibility theory.Portfolio selection investigates how to measure investment return and risk under uncertain environment. Based on different risk awareness of investors, a variety of optimization models are established and efficient algorithms are designed to solve the proposed models. First, this dissertation defines several risk measures such as absolute deviation, semi-absolute deviation and credibilistic value-at-risk. Based on these risk measures, fuzzy programming is employed to formulate several return-risk models in-cluding mean-absolute deviation model, mean-semi-absolute deviation model and cred-ibilistic value-at-risk model. In addition, this dissertation also proposes a minimization cross-entropy model by minimizing the deviation between future return and priori re-turn. The mathematical properties such as the deterministic equivalence class of the proposed models are analyzed. In the end, by combining fuzzy simulation and ge-netic algorithm, hybrid intelligent algorithms are designed to solve those models which cannot be converted into the deterministic equivalent forms.Option pricing problem considers how to obtain a reasonable price for some option in uncertain environment. This dissertation considers this problem in the continuous-time environment. Fuzzy calculus is employed to deal with fuzziness in security prices, and the binary option and European option pricing formulas are derived. In addition, several trading strategies involving options are proposed under fuzzy environment, and integral expressions of expected payoff are obtained for each strategy.In conclusion, this dissertation contributes to the research area of portfolio selec- tion and option pricing in fuzzy environment in the following aspects:(1) several risk measures are proposed for fuzzy variable including absolute deviation, semi-absolute deviation and credibilistic value-at-risk; (2) the return-risk models are established for fuzzy portfolio selection including mean-absolute deviation, mean-semiabsolute devia-tion and credibilistic value-at-risk models, and minimization cross-entropy models are also presented; (3) binary option and European option pricing formulas are derived for Liu's stock model; (4) the trading strategies involving options are proposed and integral expressions of expected payoff are obtained for each strategy.