<p>In this paper, we use the modern setting of functional empirical processes and recent techniques on uniform estimation for non parametric objects to derive consistency bands for the mean excess function in the i.i.d. case. We apply our results for modelling Dow Jones data to see how good the Generalized hyperbolic distribution fits monthly data.</p>
A new parameter estimation--- pattern search algorithm based on maximum likelihood estimation is used to estimate the parameters of generalized hyperbolic distribution, normal inverse Gaussian distribution and hyperbolic distribution, which are used to fit the log-return of Shanghai composite index. The goodness of fit is tested based on Anderson & Darling distance and FOF distance who pay more attention to tail distances of some distribution. Monte Carlo simulation are used to determin the critical values of Anderson & Darling distance and FOF distance of different distributions.Value at risk (VaR) and conditional value at risk (CVaR) are estimated for the fitted generalized hyperbolic distribution, normal inverse Gaussian distribution and hyperbolic distributio.The results show that generalized hyperbolic distribution family is more suitable for risk measure such as VaR and CVaR than normal distribution.
In this thesis, we explore the uncertainty issues in risk modelling arising from the different approaches proposed in the literature and currently being used in the industry. The first type of methods that we discuss assume that the returns of the stocks follows a generalized hyperbolic distribution. Data is calibrated by the Expectation-Maximization (EM) algorithm in order to estimate the parameters in the underlying distribution. Once we have the parameters, we estimate the Value at Risk (VaR) and Expected Shortfall (ES) by using Monte Carlo simulations.
Furthermore, we calibrate data to different copulas, including the Gauss Copula, the
The premise of this paper is providing a theoretical model for a novel way to portfolio optimization using generalized hyperbolic distribution during crisis with risk measures, expected shortfall and standard deviation. Getting good expected returns from investing in portfolio assets like stocks, bonds and currencies during crisis period chosen is harder where the risks cannot be diverted because of disruptive financial jolts i.e. sudden and unprecedented events like subprime mortgage crises in 2008. Multivariate generalized hyperbolic distribution on joint distribution of risk factors from stocks, bonds and currencies is used because it can simplify the risk factors calculation by allowing them to be linearized. The results show the premise is true. The contributions are discovering both the appropriate probability distribution and risk measure will determine whether the portfolio is optimal or not. The practical application will be taking care of the risk to take care of the profit.
Consistency Bands for the Mean Excess Function and Application to Graphical Goodness-of-fit Test for Financial Data
