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We generalize the asset dynamics assumptions of Leland (1994b) and Leland and Toft (1996) to a state dependent variance with constant elasticity process (CEV) and obtain analytical solutions for corporate debt and equity value. We use the GMM technique to extract the parameters by fitting the empirical data in the equity and credit default swap markets simultaneously. We find that the elasticity parameter is significantly different from zero for most of the firms and that the CEV model performs much better than the model with constant volatility in both in-sample fittings and out-of-sample predictions of CDS spreads.