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

## TL;DR

This paper develops and calibrates structural first passage models with time-varying volatility to accurately assess credit risk, applying them to Lehman CDS data and equity swaps under counterparty risk.

## Contribution

It introduces tractable, calibrated structural models (AT1P and SBTV) with scenarios for default barriers, applied to real crisis data and hybrid product pricing.

## Key findings

- Models fit Lehman CDS data during crisis
- SBTV provides more realistic default scenarios
- Counterparty risk pricing in equity swaps demonstrated

## Abstract

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.