Mathematical modeling of firms and their capital structure

2007/08年全球金融危机等金融危机导致企业破产或违约，而这些企业违约反过来又造成了进一步的市场困境，加剧了危机。全球范围内造成的损失可能达到数万亿美元。在会计学上，公司的资产负债表按季度将资本结构记录为A（资产）、D（债务）和E（权益）的账面价值。另一方面，数理金融学从市场的角度来看待公司和所有在其上交易的证券，这些证券在流动资本市场上观察到的价格。市场价值反映但不等于账面价值，并且不断更新，而不仅仅是每年4次。这一建议的两个方面是在一个公司和整个系统的层面上建立资本结构和违约风险的市场模型。传统的公司模型关注A的随机性，而将D视为非随机的，并将信用违约互换等证券视为A的衍生品。 一种新的“混合”方法将A和D视为联合随机过程，并将信贷和股票证券视为A和D的衍生品，从而实现信贷和股票市场的交易和对冲。这个提议的第一部分是解决混合模型的核心问题：“在多大程度上可以将所有可观察到的市场证券的集合理解为未观察到的过程A和D的衍生品？“金融系统性风险的定义是，一些银行的破产引发进一步的”多米诺骨牌式“违约的风险，这种风险通过银行资产负债表中可识别的银行间贷款的联系传播。我的第二条线索是调查银行的资本结构和违约风险如何与随机网络理论相结合，并对主要问题“金融网络的哪些结构方面最影响系统性风险？”“简而言之，这一建议的两条线将扩大数学金融的范围，并有可能帮助保护加拿大和世界免受未来危机造成的不可估量的损害。

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.