Tomas Tichy

An application of an n-dimensional fuzzy smoothing filter in financial modeling

Data smoothing is an important step within a data processing procedure that allows one to stress the most important pattern of a function relation between a studied object and given variables. Recently, Holčapek and Tichý (2011) suggested a smoothing filter based on fuzzy transform approach of Perfilieva (2004) and compared it to Nadaraya-Watson estimator. However, within the analysis only one independent variable was assumed. By contrast, many real world problems are multidimensional and hence in this paper we generalize the fuzzy smoothing filter into two dimensions and show its application within a common problem of financial engineering and asset pricing, the option implied volatility surface presentation.

Discrete multivariate F-transform of higher degree

In this paper, we reformulate the multivariate fuzzy transform (Fm-transform) of higher degree proposed originally for an approximation of continuous functions to the discrete case. We introduce the Fm-transform components as multivariate polynomials using the matrix algebra, the inverse Fm-transform is established to obtain a continuous function. We provide an analysis of basic properties of Fm-transform.

No-arbitrage condition of option implied volatility and bandwidth selection

Abstract A standard approach to option pricing is based on Black-Scholes type (BS hereafter) models utilizing the no-arbitrage argument of complete markets. However, there are several crucial assumptions, such as that the option underlying log-returns follow normal distribution, there is unique and deterministic riskless rate as well as the volatility of underlying log-returns. Since the assumptions are generally not fulfilled, the BS-type models mostly provide false results. A common market practice is therefore to invert option pricing model and using market prices of highly liquid options to get a so called implied volatility (IV). The BS model at one time moment can be related to the whole set of IVs as given by maturity/moneyness relation of tradable options. One can therefore get IV curve or surface (a so called smirk or smile). Since the moneyness and maturity of IV often do not match the data of valuated options, some sort of estimating and local smoothing is necessary. However, it can lead to arbitrage opportunity, if no-arbitrage conditions on state price density (SPD) are ignored. In this paper, using option data on DAX index, we analyze the behavior of IV and SPD with respect to different choices of bandwidth parameter h and a set of bandwidths, which violates no-arbitrage conditions, are identified. Moreover, it is documented that the change of h implies interesting changes in the violation interval of moneyness. We also show the impact of h on the total area of SPD under zero, which can be seen as a degree of no-arbitrage violation.

Efficiency analysis of classic risk minimizing portfolios

Portfolio selection problem is one of the most important issues within financial risk management and decision making. It concerns both, financial institutions and their regulator/supervisor bodies. A very challenging question in this context is whether there is some impact of alternative dependency/concordance measures on the efficiency of optimal portfolios. Therefore, the alternative ways of portfolio comparisons were developed, among them a stochastic dominance approach is one of the most popular one. In particular, the definition of the second-order stochastic dominance (SSD) relation uses comparisons of either twice cumulative distribution functions or expected utilities. Alternatively, one can define SSD relation using cumulative quantile functions or conditional value at risk. The task of this paper is therefore to examine and analyze the SSD efficiency of min-var portfolios that are selected on the basis of alternative concordance matrices set up on the basis of either Spearman rho or Kendall tau. It is empirically documented that only Pearson measure in Markowitz model identified a portfolio that can be of interest for at least one risk averse investor. Moreover, a portfolio based on Kendall measure is very poor (at least in terms of SSD efficiency).