In Duan, Gauthier and Simonato (1999), an analytical approximate formula for European options in the GARCH framework was developed. The formula is however restricted to the nonlinear asymmetric GARCH model. This paper extends the same approach to two other important GARCH specifications GJR-GARCH and EGARCH. We provide the corresponding formulas and study their numerical performance. keywords: ...

The EGARCH is a popular model for discrete time volatility since it allows for asymmetric effects and naturally ensures positivity even when including exogenous variables. Estimation and inference is usually done via maximum likelihood. Although some progress has been made recently, a complete distribution theory of MLE for EGARCH models is still missing. Furthermore, the estimation procedure i...

Among 235 extended-spectrum beta-lactamase-producing Klebsiella pneumoniae (ESBL) isolates collected from a nationwide surveillance performed in Taiwan, 102 (43.4%) were resistant to amikacin. Ninety-two of these 102 (90.2%) isolates were carrying CTX-M-type beta-lactamases individually or concomitantly with SHV-type or CMY-2 beta-lactamases. The armA and rmtB alleles were individually detected...

We have derived some matrix equations for speedy computation of the conditional covariance kernel of a discrete-time process obtained from irregularly sampling an underlying continuous-time ARMA process. These results are applicable to both stationary and non-stationary ARMA processes. We have also demonstrated that these matrix results can be useful in shedding new insights on the covariance s...

When we were fitting ARMA models to the data, we first looked at the sample autocovariance or autocorrelation function and we then tried to find the ARMA model whose theoretical acf matched with the sample acf. Now the sample autocovariance function is a nonparametric estimate of the theoretical autocovariance function of the process. In other words, we first estimated γ(h) nonparametrically by...

Various e m p i r i d studies have shown that the time-varying volatility of asset returns can be described by GARCH (generalized autoregressive conditional heteroskedasticity) models. The corresponding GARCH option pricing model of Duan (1995) is capable of depicting the "smile-effect" which often can be found in option prices. In some derivative markets, however, the slope of the smile is not...

In this paper the value at risk (VaR) forecasts are compared using three different GARCH models; ARCH(1), GARCH(1,1) and EGARCH(1,1). The implemented method is a one-day ahead out of sample forecast of the VaR. The forecasts are evaluated using the Kupiec test with a five percent significance level. The focus is on three different markets; commodities, equities and exchange rates. The goal of t...

Abstract – Analysis of time series data can involve the inversion of large covariance matrices. For the class of ARMA (p, q) processes there are no exact explicit expressions for these inverses, except for the MA (1) process. In practice, the sample covariance matrix can be very large and inversion can be computationally time consuming and so approximate explicit expressions for the inverse are...