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We examine how to approximate a Lévy process by a hyperexponential jump-diffusion (HEJD) process, composed of Brownian motion and of an arbitrary number of sums of compound Poisson processes with double exponentially distributed jumps. This approximation will facilitate the pricing of exotic options since HEJD processes have a degree of tractability that other Lévy processes do not have. The idea behind this approximation has been applied to option pricing by Asmussen et al. (2007) and Jeannin and Pistorius (2008). In this paper we introduce a more systematic methodology for constructing this approximation which allow us to compute the intensity rates, the mean jump sizes and the volatility of the approximating HEJD process (almost) analytically. Our methodology is very easy to implement. We compute vanilla option prices and barrier option prices using the approximating HEJD process and we compare our results to those obtained from other methodologies in the literature. We demonstrate that our methodology gives very accurate option prices and that these prices are more accurate than those obtained from existing methodologies for approximating Lévy processes by HEJD processes.
We study a relation between higher order comoments and dependence structure of equity portfolio in the US and UK by relying on a simple portfolio approach where equity portfolios are sorted on the higher order comoments. We nd that beta and coskewness are positively related with a copula correlation, whereas cokurtosis is negatively related with it. We also nd that beta positively associates with an asymmetric tail dependence whilst coskewness negatively associates with it. Furthermore, two extreme equity portfolios sorted on the higher order comoments are closely correlated and their dependence structure is strongly time-varying and nonlinear. Backtesting results of value-at-risk and expected shortfall demonstrate the importance of dynamic modeling of asymmetric tail dependence in the risk management of extreme events.
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