No-arbitrage pricing under cross-ownership

Abstract

We generalize Merton's asset valuation approach to systems of multiple financial firms where cross-ownership of equities and liabilities is present. The liabilities, which may include debts and derivatives, can be of differing seniority. We derive equations for the prices of equities and recovery claims under no-arbitrage. An existence result and a uniqueness result are proven. Examples and an algorithm for the simultaneous calculation of all no-arbitrage prices are provided. A result on capital structure irrelevance for groups of firms regarding externally held claims is discussed, as well as financial leverage and systemic risk caused by cross-ownership.

Excerpts and ideas from this paper have been presented at the Scientific Conference of the German Association for Actuarial and Financial Mathematics (DGVFM), Bremen, April 30, 2010. Some methods and systems derived from this work have been subject to a provisional (successful) patent filing

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• Systemic Risk: Fire-Walling Financial Systems Using Network-Based Approaches
• Cross-Ownership as a Structural Explanation for Over- and Underestimation of Default Probability
• Valuation Algorithms for Structural Models of Financial Interconnectedness