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In this paper we develop structural first passage models (AT1P and SBTV) withtime-varying volatility and characterized by high tractability, moving from theoriginal work of Brigo and Tarenghi (2004, 2005) [19] [20] and Brigo and Morini(2006)[15]. The models can be calibrated exactly to credit spreads usingefficient closed-form formulas for default probabilities. Default events arecaused by the value of the firm assets hitting a safety threshold, whichdepends on the financial situation of the company and on market conditions. InAT1P this default barrier is deterministic. Instead SBTV assumes two possiblescenarios for the initial level of the default barrier, for taking into accountuncertainty on balance sheet information. While in [19] and [15] the models areanalyzed across Parmalat's history, here we apply the models to exactcalibration of Lehman Credit Default Swap (CDS) data during the monthspreceding default, as the crisis unfolds. The results we obtain with AT1P andSBTV have reasonable economic interpretation, and are particularly realisticwhen SBTV is considered. The pricing of counterparty risk in an Equity ReturnSwap is a convenient application we consider, also to illustrate theinteraction of our credit models with equity models in hybrid products context.

Credit Calibration with Structural Models: The Lehman case and Equity Swaps under Counterparty Risk