Credit Calibration with Structural Models: The Lehman Case and Equity Swaps Under Counterparty Risk

21 Pages Posted: 4 Jan 2010

See all articles by Damiano Brigo

Damiano Brigo

Imperial College London - Department of Mathematics

Massimo Morini

Banca IMI; Bocconi University

Marco Tarenghi


Date Written: December 22, 2009


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) and Brigo and Morini (2006). 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 Brigo and Tarenghi (2004) and Brigo and Morini (2006) 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.

Keywords: Credit Default Swaps, Structural Models, Black Cox Model, Calibration, Analytical Tractability, Monte Carlo Simulation, Equity Return Swaps, Counterparty Risk, Barrier Options, Uncertain Credit Quality, Lehman Brothers Default

JEL Classification: G13

Suggested Citation

Brigo, Damiano and Morini, Massimo and Tarenghi, Marco, Credit Calibration with Structural Models: The Lehman Case and Equity Swaps Under Counterparty Risk (December 22, 2009). Available at SSRN: or

Damiano Brigo (Contact Author)

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom


Massimo Morini

Banca IMI ( email )

Corso Matteotti 6
20121 Milano, 20100

Bocconi University ( email )

Via Sarfatti, 25
Milan, MI 20136

Marco Tarenghi

Mediobanca ( email )

Piazzetta Enrico Cuccia, 1
Milano, MI 20121

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