Backtesting Value-at-Risk: A Duration-Based Approach

Financial risk model evaluation or backtesting is a key partof the internal model's approach to market risk management aslaid out by the Basle Committee on Banking Supervision. However,existing backtesting methods have relatively low power in realisticsmall sample settings. Our contribution is the exploration ofnew tools for backtesting based on the duration of days betweenthe violations of the Value-at-Risk. Our Monte Carlo resultsshow that in realistic situations, the new duration-based testshave considerably better power properties than the previouslysuggested tests.     

Binomial Option Pricing with Skewed Asset Returns

This research presents a method for estimating the parameters of the binomial option pricing model necessary to appropriately price calls on assets with asymmetric end-of-period return distributions. Parameters of the binomial model are shown to be a function of the mean, variance, and skewness of the underlying return distribution. It is also shown that failure to incorporate skewness results in the mispricing of the call.     

The moments of the Gompertz distribution and maximum likelihood estimation of its parameters

The Gompertz distribution is widely used to describe the distribution of adult deaths. Previous works concentrated on formulating approximate relationships to characterise it. However, using the generalised integro-exponential function, exact formulas can be derived for its moment-generating function and central moments. Based on the exact central moments, higher accuracy approximations can be defined for them. In demographic or actuarial applications, maximum likelihood estimation is often used to determine the parameters of the Gompertz distribution. By solving the maximum likelihood estimates analytically, the dimension of the optimisation problem can be reduced to one both in the case of discrete and continuous data. Monte Carlo experiments show that by ML estimation, higher accuracy estimates can be acquired than by the method of moments.     

Do Sovereign Re‐Ratings Destabilize Equity Markets during Financial Crises? New Evidence from Higher Return Moments

This study investigates the effects of S&P's sovereign re‐ratings on the higher moments of equity market returns over recent financial crises. Using a set of intraday stock market index prices and sovereign credit ratings for a sample of 36 countries that experienced sovereign rating changes over the period from 1996 to 2013, we find that the higher moments of stock market returns are significantly more responsive to sovereign re‐ratings during financial crises, but the effects on stock markets are not the same across different financial crises. The effects during crises are, however, magnified for large downgrades and those that are associated with a loss of investment grade status. We find that there are asymmetric effects during financial crises in that downgrades are consistently more significant than upgrades in increasing realized volatility and realized kurtosis. Both upgrades and downgrades affect realized skewness in times of crises in the expected direction.     

A lottery-preference-based explanation of realized kurtosis puzzle in Chinese stock market

High kurtosis corresponds to fat tails on both sides and under risk-aversion assumption investors’ dislike of left-tail loss outweighs their preference for right-tail gain. Therefore, high kurtosis characteristic of stock should predict high expected returns. However, the high-frequency-data-based empirical results on Chinese stock market are just the opposite, which we refer to as the ‘realized kurtosis puzzle’. Using the double sorts and firm-level cross-sectional regression methods, we further demonstrate investors’ preference for lottery-like stocks or lottery preference is key to solve the puzzle. Our further empirical research verifies stocks with higher retail investors’ shareholding proportion and unavailable for short show stronger ‘realized kurtosis puzzle’. In addition, the puzzle is particularly significant in high lottery preference periods while less apparent in low lottery preference times.     

Evaluating effects of excess kurtosis on VaR estimates: Evidence for international stock indices

The calculus of VaR involves dealing with the confidence level, the time horizon and the true underlying conditional distribution function of asset returns. In this paper, we shall examine the effects of using a specific distribution function that fits well the low-tail data of the observed distribution of asset returns on the accuracy of VaR estimates. In our analysis, we consider some distributional forms characterized by capturing the excess kurtosis characteristic of stock return distributions and we compare their performance using some international stock indices. JEL Classification C15 · G10     

Properties of Multinomial Lattices with Cumulants for Option Pricing and Hedging

In this paper, we analyze properties of multinomial lattices that model general stochastic dynamics of the underlying stock by taking into account any given cumulants (or moments). First, we provide a parameterization of multinomial lattices, and demonstrate that mean, variance, skewness, and kurtosis of the underlying may be matched using five branches. Then, we investigate the convergence of the multinomial lattice when the basic time period approaches zero, and prove that the limiting process of the multinomial lattice that matches annualized mean, variance, skewness and kurtosis is given by a compound Poisson process. Finally, we illustrate the effect of higher order moments in the underlying asset process on the price of derivative securities through numerical experiments using the multinomial lattice, and provide a comparison with jump-diffusion models.     

用统计学原理对学生考试质量的直观评价

采用统计学的一些基本统计量，结合统计分析基本原理，对学生考试成绩进行量化分析，力图使得对学生考试成绩分析计算所得到的结果科学、直观、有效。     

Autoregressive Conditional Kurtosis

This article proposes a new model for autoregressive conditionalheteroscedasticity and kurtosis. Via a time-varying degreesof freedom parameter, the conditional variance and conditionalkurtosis are permitted to evolve separately. The model usesonly the standard Student’s t-density and consequentlycan be estimated simply using maximum likelihood. The methodis applied to a set of four daily financial asset return seriescomprising U.S. and U.K. stocks and bonds, and significant evidencein favor of the presence of autoregressive conditional kurtosisis observed. Various extensions to the basic model are proposed,and we show that the response of kurtosis to good and bad newsis not significantly asymmetric.