Models for time series are very important for the stock market. Fuzzy Takagi – Sugeno models (functional fuzzy systems) are a promising and already common approach, in which different regression dependencies are used for different areas of variation of certain parameters, and soft switching is performed using the fuzzy logic rules. This is the advantage of this approach over conventional stochastic models. Each Takagi-Sugeno model is based on its set of fuzzy rules. These models can be viewed as a generalization of classical econometric models, if one such model corresponds to one fuzzy rule. This paper studies the possibility of using the wavelet transform and fuzzy Takagi – Sugeno model to analyze the dynamics of stock prices for the following Russian companies: Gazprom, Sberbank, Magnit, Yandex and Aeroflot; this approach was previously used to study some foreign stock markets. Wavelet analysis quite often acts as a tool for signal processing, including time series, as it allows for a multi-level approximation. In this paper, the Takagi – Sugeno model is based on untransformed data as well as data transformed using Haar wavelets. Fuzzy clustering is used to construct membership functions. Calculations show that the use of wavelets often improves the predictive characteristics of the model.
This paper examines two Markov chain Monte Carlo methods that have been widely used in econometrics. An introductory exposition of the Metropolis algorithm and the Gibbs sampler is provided. These methods are used to simulate multivariate distributions. Many problems in Bayesian statistics can be solved by simulating the posterior distribution. Invariance condition is of importance, the proofs are given for both methods. We use finite Markov chains to explore and substantiate the methods. Several examples are provided to illustrate the applicability and efficiency of the Markov chain Monte Carlo methods. They include bivariate normal distribution with high correlation, bivariate exponential distribution, mixture of bivariate normals.
There is no accurate answer for the question, which method of modeling of uncertainty is preferable: random or fuzzy. Today both of these approaches are highly popular. Fuzzy and probabilistic approaches are commonly used for modeling of uncertainty. Fuzzy numbers can be used for modeling vagueness of parameters, such as risk-free rate or volatility in option pricing. Under these assumptions, option value depends on believe degree and turns to fuzzy number. In this paper the Black – Scholes formula and it’s modification for American option arbitrage-free value are used. Fuzzy representations of underlying asset price, volatility of asset price and risk-free rate are used as parameters. There is set of papers regarding fuzzy approach for European option pricing. In this paper fuzzy approach is used for arbitrage-free American option pricing for the first time. The fuzzy American call value is compared with fuzzy European option value.
We analyze optimal execution strategies when multiple traders are simultaneously involved in optimal execution. In this case, we obtain new trading strategies that follow from a direct extension of the mean variance approach of Grinold and Kahn, and Almgren and Chriss. However, as we show below, the proposed strategies can be quite different from the standard ones obtained in Grinold and Kahn, and Almgren and Chriss. This is because each trader (assumed to be rational) is trying to minimize her trading cost or "implementation shortfall" and therefore takes into account the price impacts caused by herself and all other traders. We also obtain a close form characterization for the dynamic Nash equilibrium in terms of the system of second-order ODEs, which can be solved explicitly. The resulting equilibrium strategies describe different types of predatory and defensive behavior, though aggregate order flow profile have some properties of standard Almgren, Chriss strategies, e.g. is monotoneous and convex. We show that the traders with smaller holdings are involved in predatory strategies, while traders with larger holdings tend to defend themselves against potential predators by following the delayed trading strategies. We also show that depending on liquidity and volatility parameters, predatory traders may be frontrunners or contrarian traders.