  In this paper we develop structural first passage models (AT1P and SBTV) with
time-varying volatility and characterized by high tractability, moving from the
original work of Brigo and Tarenghi (2004, 2005) [19] [20] and Brigo and Morini
(2006)[15]. The models can be calibrated exactly to credit spreads using
efficient closed-form formulas for default probabilities. Default events are
caused by the value of the firm assets hitting a safety threshold, which
depends on the financial situation of the company and on market conditions. In
AT1P this default barrier is deterministic. Instead SBTV assumes two possible
scenarios for the initial level of the default barrier, for taking into account
uncertainty on balance sheet information. While in [19] and [15] the models are
analyzed across Parmalat's history, here we apply the models to exact
calibration of Lehman Credit Default Swap (CDS) data during the months
preceding default, as the crisis unfolds. The results we obtain with AT1P and
SBTV have reasonable economic interpretation, and are particularly realistic
when SBTV is considered. The pricing of counterparty risk in an Equity Return
Swap is a convenient application we consider, also to illustrate the
interaction of our credit models with equity models in hybrid products context.
