In this paper we extend the traditional GARCH(1,1) model by including a functional trend term in the conditional volatility of a time series. We derive the main properties of the model and apply it to all agricultural commodities in the Mexican CPI basket, as well as to the international prices of maize, wheat, pork, poultry and beef products for three different time periods that implied changes in price regulations and behavior. The proposed model seems to adequately fit the volatility process and, according to homoscedasticity tests, outperforms the ARCH(1) and GARCH(1,1) models, some of the most popular approaches used in the literature to analyze price volatility.
A credit default swap (CDS) is a financial contract in which the holder of the instrument will be compensated in the event of a loan default. When available, CDS's are used to monitor the credit risk of countries and companies. In this work we develop a closed form procedure to value a CDS in the case in which the so-called "credit rate index" is modelled as a Bessel bridge of arbitrary order. In particular, these processes seem to capture the nature of a defaultable asset in the sense that they remain strictly positive before default, and thus enrich the existing literature in this field.
The main aim of this work is to price defaultable bonds. In order to achieve this goal we link first hitting densities of Brownian motion with functionals of controlled diffusions. From a practical point of view examples of diffusions with this property are: Brownian motion with linear drfit, the 3D Bessel process, the 3D Bessel bridge, and the Brownian bridge, just to mention a few. In turn, these processes are used in finance and economics since they may fall within the category of controlled processes, and/or mean reverting processes.
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