Mathematical modeling of firms and their capital structure

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Financial crises such as the 2007/08 global meltdown lead to insolvency or default of firms, and in turn such firm defaults create further market distress that compounds the crisis. The damage done worldwide can be in the trillions of $. In accounting, a firm's balance sheet records the capital structure as book values of A (assets), D (debt) and E (equity) on a quarterly basis. On the other hand, Mathematical Finance takes a market perspective that sees the firm and all the securities that trade on it in terms of prices observed in the liquid capital markets. Market values reflect but do not equal book values, and are updated continuously not just 4 times a year. The two strands of this proposal are to develop market models of capital structure and default risk at the one-firm and system-wide levels. Traditional firm models focus on the randomness of A while treating D as non-random, and view securities such as credit default swaps as derivatives on A. A new "hybrid" approach takes A and D as joint stochastic processes and treats both credit and equity securities as derivatives on A and D, enabling trading and hedging across credit and equity markets. The first strand of this proposal is to address the central question about hybrid models: "How well can the collective of all observed market securities written on a firm be understood as derivatives on the unobserved processes A and D?" Financial systemic risk is defined as the risk that insolvency of some banks triggers further ``domino-style'' defaults, transmitted through links representing interbank loans that are identifiable in banks' balance sheets. My second strand is to investigate how capital structure and default risk for banks intertwine with random network theory and lead to mathematical answers to the primary question "Which structural aspects of a financial network most affect systemic risk?" In a nutshell, the twin strands of this proposal will extend the range of Mathematical Finance and have the potential to help protect Canada and the world from future crises inflicting incalculable damage.