The present paper focuses on the methods of derivative contract pricing. The basic differential equation of the popular Black-Scholes model for option contract pricing is derived. Furthermore, its less known modifications by Merton and Garman and Kohlhagen are pointed out. The paper refers to the significant drawbacks and limitations of the option pricing models that are based on constricting and unrealistic assumptions that often fail in comparison to the real market data. Attention is paid to the most serious problem, namely the issue of constant volatility, which is considerably disrupted in practice. Models implementing both stochastic and deterministic volatility in the original model are pointed out, their output being a more accurate option contract price.

The decision-making process on investing in financial markets is a very complex and difficult task, mainly due to the chaotic behavior and high uncertainty in the development of the prices of investment instruments. For this reason, financial markets are increasingly using means of artificial intelligence, namely fuzzy logic, which is able to capture the nonlinear behavior.Fuzzy logic provides a way to draw definitive conclusions from vague, ambiguous, or inaccurate information.However, there are some drawbacks associated with type-1 fuzzy logic, so the type-2 fuzzy logic comes forward, which can work with greater uncertainty. Type-2 fuzzy logic works with a new third dimension fuzzy set that provides additional degrees of freedom and allows to model and process numerical and linguistic uncertainties directly. The paper applies type-2 fuzzy logic to the stock market with the aim to create a simple and understandable model for deciding on investing in investment instruments, which is important for investors in this area. The proposed type-2 fuzzy model uses return, risk, dividend and total expense ratio of ETF as input variables. The created system is able to generate aggregated models from a certain number of language rules, which allows the investor to understand the created financial model. Using type-2 fuzzy logic can lead to more realistic and accurate results than type-1 fuzzy logic.

Extensive research results of stock market time series using classical fuzzy sets (type-1) are available in the literature. However, type-1 fuzzy sets cannot fully capture the uncertainty associated with stock market developments due to their limited descriptiveness. This paper fills a scientific gap and focuses on type-2 fuzzy logic applied to stock markets. Type-2 fuzzy sets may include additional uncertainty resulting from unclear, uncertain, or inaccurate financial data through which model inputs are calculated. Here we propose four methods based on type-2 fuzzy logic, which differ in the level of uncertainty contained in fuzzy sets and compared with the type-1 fuzzy model. The case study aims to create a model to support investment decisions in Exchange-Traded Funds (ETFs) listed on international equity markets. The created models of type-2 fuzzy logic are compared with the classic type-1 fuzzy logic model. Based on the results of the comparison, it can be said that type-2 fuzzy logic with dual fuzzy sets is able to better describe data from financial time series and provides more accurate outputs. The results reflect the capability and effectiveness of the approach proposed in this document. However, the performance of type-2 fuzzy logic models decreases with the inclusion of increasing uncertainty in fuzzy sets. For further research, it would be appropriate to examine the different levels of uncertainty in the input parameters themselves and monitor the performance of such a modified model.

Purpose of the article: The paper is focused on the forecast of stock markets of the Central European countries, known as V4, by means of soft computing. The tested model is constructed by a combination of fuzzy logic and artificial neural networks. A total of four SAX, PX, BUX, WIG stock indices differing in their liquidity and efficiency are selected for the forecast. Methodology/methods: The methods of analysis, synthesis and techniques of mathematical neuro-fuzzy modelling were used to achieve this goal. The proposed neuro-fuzzy decisionmaking model consists of 3 input variables, one block of rules (with 21 fuzzy rules) and one output variable predicting the normalized price of stock indexes of the selected countries. The input variables have three attributes (L-large, M-medium, and S-small). Scientific aim: The aim of the paper is to create a suitable model that will be used to forecast stock indices of the Central European countries with a relatively low error. Findings: The developed ANFIS model is a suitable tool for predicting stock indexes. The importance of the neuro-fuzzy model can be seen especially in the fact that it shows a strong predictive capacity of both efficient and less efficient stock markets. Conclusions: The paper discussed the design of the neuro-fuzzy model as a supporting tool for predicting the selected stock indexes listed on the European stock markets. For further research, it would be appropriate to extend the proposed model with other significant fundamental indicators, or to incorporate technical and psychological indicators and to monitor the strength of the revised model also in several stock markets, for example according to the geographical distribution.