TAILIEUCHUNG - CONVERTIBLE BONDS IN A DEFAULTABLE DIFFUSION MODEL

Market discipline—exerted at the funding node—was, in principle, supposed to control moral hazard at the lending node. However, incentives for doing so by appropriately pricing risk also weakened. First, because for conforming loans, correctly perceived implicit Federal government guarantees extended to the GSEs meant that banks could substantially reduce capital costs by substituting essentially risk free GSE debt for mortgages. Second, because the credit risk of non-conforming loans targeted for securitization was systematically underestimated by credit ratings agencies (CRAs) and investors. Third, because of the subjective, and yet remarkably uniform, low likelihood assigned by market participants to a break in. | CONVERTIBLE BONDS IN A DEFAULTABLE DIFFUSION MODEL Tomasz R. Bielecki Department of Applied Mathematics Illinois Institute of Technology Chicago IL 60616 USA Stephane Crepeyf Departement de Mathematiques Universite d Evry Val d Essonne 91025 Evry Cedex France Monique JeanblanC Departement de Mathematiques Universite d Evry Val d Essonne 91025 Evry Cedex France and Europlace Institute of Finance Marek Rutkowski School of Mathematics University of New South Wales Sydney NSW 2052 Australia and Faculty of Mathematics and Information Science Warsaw University of Technology 00-661 Warszawa Poland First draft June 1 2007 This version February 16 2009 The research of . Bielecki was supported by NSF Grant 0202851 and Moody s Corporation grant 5-55411. The research of S. Crepey benefited from the support of Ito33 the Chaire Risque de credit and the Europlace Institute of Finance. The research of M. Jeanblanc was supported by Ito33 the Chaire Risque de credit and Moody s Corporation grant 5-55411. The research of M. Rutkowski was supported by the ARC Discovery Project DP0881460. Contents 1 Introduction 3 2 Markovian Equity-to-Credit Framework 4 Default Time and Pre-Default Equity Dynamics . 4 Market Model. 5 Risk-Neutral Measures and Model Completeness. 6 Modified Market Model. 7 3 Convertible Securities 9 Arbitrage Valuation of a Convertible Security. 10 Doubly Reflected BSDEs Approach. 10 Super-Hedging Strategies for a Convertible Security. 12 Solutions of the Doubly Reflected BSDE. 14 Variational Inequalities Approach . 14 Pricing and Hedging Through Variational Inequalities . 16 Approximation Schemes for Variational Inequalities . 18 4 Convertible Bonds 18 Reduced Convertible Bonds. 19 Embedded Bond . 20 Embedded Game Exchange Option . 21 Solutions of the Doubly Reflected BSDEs . 22 Variational Inequalities for Post-Protection Prices . 22 Variational Inequalities for Protection .

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