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We formulate a bivariate stochastic volatility jump-diffusion model with correlated jumps and volatilities. An MCMC Metropolis-Hastings sampling algorithm is proposed to estimate the model’s parameters and latent state variables (jumps and stochastic volatilities) given observed returns. The methodology is successfully tested on several artificially generated bivariate time series and then on the two most important Czech domestic financial market time series of the FX (CZK/EUR) and stock (PX index) returns. Four bivariate models with and without jumps and/or stochastic volatility are compared using the deviance information criterion (DIC) confirming importance of incorporation of jumps and stochastic volatility into the model.
Analysis and Comparison of Different Value at Risk Models for Nonlinear Portfolio. Diploma Thesis, MFF UK. The thesis describes Value-at-Risk (VaR) and Expected Shortfall (ES) models for measuring market risk. Parametric method, Monte Carlo simulation, and Historical simulation (HS) are presented. The second part of the thesis analyzes Extreme Value Theory (EVT). The fundamental theory behind EVT is built, and peaks-over-threshold (POT) method is introduced. The POT method is then used for modelling the tail of the distribution of losses with Generalized Pareto Distribution (GPD), and is simultaneously illustrated on VaR and ES calculations for PX Index. Practical issues such as multiple day horizon, conditional volatility of returns, and backtesting are also discussed. Subsequently, the application of parametric method, HS and EVT is demonstrated on a sample nonlinear portfolio designed in Mathematica and the results are discussed.
Estimations of Market and Credit Value at Risk. Bachelor thesis, MFF UK. This present work studies statistical estimations of market and credit risk by measure of risk called Value at Risk. This work describes the defintion of Value at Risk, estimations of Value at Risk for market risk by the variance and covariance method, by historical simu- lation method, by Monte Carlo simulation method and estimations of Value at Risk for credit risk by the most widely known methods CreditMetrics, CreditRisk+ and KMV. This part ends by historical development and cal- culation of capital adequacy. The analytical part of the work analyses main advantages and disadvantages of Value at Risk on the example of portfolio compact of exchange-traded funds. The aim of this work is to describe Value at Risk as a whole, describe its advantages and analyse disadvantages.