Estimation tail parameter for Geometric Brownian motion

Right-tailed distributions are very important in many applications. There are many studies estimating the tail index. In this paper, we will estimate the tail parameter using the three (the Direct, Bootstrap and Double Bootstrap) methods. Our aim is to illustrate the best way to estimate the -stable with using simulation and real data for the daily Iraqi financial market dataset.

Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution

Tail risk is an important financial issue today, but directly hedging tail risks with an ad hoc option is still an unresolved problem since it is not easy to specify a suitable and asymmetric pricing kernel. By defining two ad hoc underlying “assets”, this paper designs two novel tail risk options (TROs) for hedging and evaluating short-term tail risks. Under the Fréchet distribution assumption for maximum losses, the closed-form TRO pricing formulas are obtained. Simulation examples demonstrate the accuracy of the pricing formulas. Furthermore, they show that, no matter whether at scale level (symmetric “normal” risk, with greater volatility) or shape level (asymmetric tail risk, with a smaller value in tail index), the greater the risk, the more expensive the TRO calls, and the cheaper the TRO puts. Using calibration, one can obtain the TRO-implied volatility and the TRO-implied tail index. The former is analogous to the Black-Scholes implied volatility, which can measure the overall symmetric market volatility. The latter measures the asymmetry in underlying losses, mirrors market sentiment, and provides financial crisis warnings. Regarding the newly proposed TRO and its implied tail index, economic implications can be offered to investors, portfolio managers, and policy-makers.

An Enhanced Method for Tail Index Estimation under Missingness

Extreme events in earthquakes, wind speed, among others are rare but may lead to catastrophic effects on humans and the environment. The primary parameter in the estimation of such rare events is the tail index which measures the tail heaviness of an underlying distribution. Since extreme events are rare, the presence of missing observations may further lead to flawed. In view of this, there is a growing effort by researchers to address this problem. However, the existing methods of estimating the tail index use only the available nonmissing data. Thus, if the missing observations are influential values, ignoring them could introduce more bias and higher mean square error (MSE) in the tail index estimation and subsequently other extreme event--estimators such as high quantiles and small exceedance probabilities. In this study, we propose imputation of the missing observations before applying some standard estimators (Hill and geometric-type) to estimate the tail index. Through a simulation study, we assess the performance of the standard estimators under the proposed data enhancement method and the existing modified estimators of the tail index. The results show that the enhanced estimators have relatively lower bias and MSE. The estimation method was illustrated with a practical dataset on wind speed with missing values. Therefore, we recommend imputation mechanism as viable for enhancing the performance of tail index estimators in the case where there is missingness.

Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations

This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient ρ=ρn is derived uniformly over stationary values in [0,1), focusing on ρn→1 as sample size n tends to infinity. For tail index α∈(0,4) of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on 1−ρn2, but no condition on the rate of ρn is required. It is shown that, for the tail index α∈(0,2), the LSE is inconsistent, for α=2, logn/(1−ρn2)-consistent, and for α∈(2,4), n1−2/α/(1−ρn2)-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index α∈(0,4); and no restriction on the rate of ρn is necessary.