We present a spectral element method to compute the probability of default of a firm in a Levy driven structural model. The approximate solution converges exponentially fast and the method is general enough for the entire class of exponential Levy processes. Since multidimensional structural models of default are not tractable in practice, we present a new structural framework for multidimensional default risk. We define the time of default as the first time the log-return of the stock price of a firm jumps below a (possibly nonconstant) default level. When stock prices are exponential Levy, this framework is equivalent to a reduced form approach, where the intensity process is parametrized by a Levy measure. The dependence between the default times of firms within a basket of credit securities is the result of the jump dependence of their respective stock prices, making the link between the equity and credit markets. We value a first-to-default basket CDS as an application!
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