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Utility Pricing of Collateralized Debt Obligations

Adviser: K. Ronnie Sircar

Justin A. Sirignano

Operations Research and Financial Engineering

“I cannot overemphasize the importance of having an adviser who is willing to actively supervise your work.”

sirignano-justin

The latter half of my undergraduate career spanned many of the defining moments of the financial crisis of 2008 and 2009. That motivated a multitude of questions about the practices of major investment banks, current financial models, and overall structure and design of the banking sector. There are many unresolved challenges as to how to implement a more stable financial system as well as craft more accurate and realistic asset pricing models. My thesis addressed the topic of one of the most critical failings of Wall Street leading up to the collapse: credit risk modeling. At the center of the financial crisis was the proliferation of structured products such as the asset-backed security (ABS) and the collateralized debt obligation (CDO). During the boom of the mid-2000s, a broad swath of financial institutions—from investment banks to pension funds—steadily amassed extensive portfolios of these structured products. These structured products’ underlying securities were incredibly diverse, ranging from credit card debt to commercial mortgages. Wall Street gobbled up these structured products with a feverish appetite; it is now painfully clear that—almost across the board—the preeminent models of the day had underestimated the potential for credit default of underlying securities.

My senior thesis focused on understanding and modeling the market for a particular structured product at the center of the financial crisis: the CDO. A CDO is essentially an insurance agreement written upon a slice (called a “tranche”) of an underlying credit portfolio. A protection buyer pays a premium in return for compensation in the event of the default of the underlying securities (which could be anything from investment-grade bonds to the now infamous subprime mortgage–backed securities).

Under the guidance of my adviser Professor Ronnie Sircar, my work modeled the arrival of defaults in a portfolio using a “hybrid” of bottom-up and top-down approaches. The credit portfolio was divided into groups, with each group’s defaults arriving according to a counting process whose intensity can be excited by a default in that same group or from another group. This is equivalent to each group having its own individual top-down model. The group model captures the economic and financial linkages between different groups or sectors of firms. The default model was fitted to historical data using a maximum likelihood method and then used to price CDO tranches using utility functions. We developed different techniques to compute the utility, including systems of partial differential equations, semi-analytic methods, and Monte Carlo simulation, and we presented both analytic and numerical solutions. We found that the risk aversion of CDO buyers is significantly lower for senior tranches and argue that this stems from the protection such senior tranches afford against systematic risk. Within a buyer’s wider portfolio, the buyer is willing to enter the contract for a higher price. This demonstrates the subtle differentiation between idiosyncratic risk and systematic risk, where investors are willing to pay a premium for protection against the latter.

I highlight here some aspects of the thesis process. First and foremost, it is essential to choose an adviser who has research interests similar to the area you wish to focus on. Professor Sircar’s guidance was indispensable in recommending relevant literature and potential methods or approaches for particular problems, and also taking my initial, general idea for a thesis topic and helping to narrow it down to a single problem of interest in the research field. In addition, I cannot overemphasize the importance of having an adviser who is willing to actively supervise your work. Although Professor Sircar was the acting department chair for operations research and financial engineering (ORFE), he was always extremely generous with his time when discussing my ongoing thesis work and met with me on a regular basis. Finally, it is important to choose a topic that to some extent can be tested or confirmed through data or experiment. Developing a very complicated model may be commendable for its use of serious mathematics, but it will be of little practical use or real-world impact if it cannot be tested. In the case of a financial model, it is essential to have real-world data with which to fit the model. I have heard that this is in particular a problem in ORFE: Many students will choose a—usually very interesting—topic, but in an area where there are no available data. As the celebrated but unapologetically blunt physicist Wolfgang Pauli once noted of a colleague’s theory that could not be proven or disproven: “Not only is it not right, it’s not even wrong!”

The senior thesis process is a great introduction to research. One learns how to first frame a problem of current relevance to the field (which is actually more difficult than it sounds; it is not so easy to find a nontrivial problem that has never been addressed before) and then to approach the problem in hopes of solving it. During the thesis process, one will read a variety of academic papers, which can be both challenging and rewarding. Oftentimes, these papers draw upon very high-level (even esoteric) mathematics. However, this exposure to journal papers that represent the foremost research in the field is extremely valuable. In hindsight, I probably learned more during the thesis process than in all of my senior-year classes combined. Perhaps most importantly, one can only be truly introduced to the most challenging and current problems by delving into these research papers. Even if many of these papers may only deal peripherally with the topic of the senior thesis, there is much merit in exploring so as to gain a general knowledge of the field as well as motivate future projects.

Personally, the senior thesis process was an integral part of my choice to continue research by entering a Ph.D. program. It both confirmed and spurred my interest in research. It was an excellent opportunity to comprehensively examine a topic of personal interest and broad relevance to the finance field. I would encourage other ORFE students to write a senior thesis. I also would like to thank Professor Sircar for being my adviser—his help and guidance were crucial to the success of the final product.

Utility Pricing of Collateralized Debt Obligations

Justin A. Sirignano

K. Ronnie Sircar

Professor of Operations Research and Financial Engineering

“... having a handful of the smartest kids plant seeds in your mind on topics you have no idea even existed is a continually regenerative reward.”

Almost every operations research and financial engineering major writes a senior thesis, and each faculty member typically has five advisees. It is a lot of work (equivalent, I tell my friends at schools with a higher teaching load, to an extra half class), and yet it is a pleasure! The opportunity to work with bright young minds, especially when, like Justin Sirignano, they are motivated and passionate about their topic, is a quintessentially Princeton perk.

An enormous benefit of senior thesis advising is that Princeton students have an antenna into frontier topics of interest that we, as academics, would only discover a year or two later, perhaps because we do not do industry internships, know the right websites, follow those Twitter feeds, and so on. On many occasions I, and colleagues, have gotten into a research area because a senior brought the problem to our attention and educated us through his or her work to the literature, the state of the art, and practical questions. The payoff may be a number of years down the road, but having a handful of the smartest kids plant seeds in your mind on topics you have no idea even existed is a continually regenerative reward.

Justin wrote his senior thesis on the important issue of quantitatively understanding the complex credit derivatives market that has been a source of much of the recent financial troubles. He already began reading and developing preliminary simulation tools during the summer, and learned a lot of challenging graduate-level mathematics involving Hawkes processes, Feynman-Kac partial differential equations (PDEs), and simulation methods. His novelty was to directly incorporate and estimate measures of investor risk-aversion into models of default risk. Justin came to me with a rather general notion of working in this area for his senior thesis. This phase of the thesis was interesting as he looked through a wealth of literature and open questions in the field. I helped him focus on a particular topic: developing a credit model using utility functions to investigate market participants’ attitudes toward risk.

One of the challenging aspects of Justin’s thesis was the lack of closed-form, analytic solutions. As models become more complex in an effort to accurately match the dynamics and statistical characteristics of financial markets, often they do not yield the “nicest” or most tractable solutions. Within the framework of Justin’s credit default model, there is no analytic solution. Instead, he proposed numerical methods using PDEs and Monte Carlo simulation. In some special cases, he used semi-analytical methods where the PDEs were reduced to ordinary differential equations that were solved by numerical methods. Using these solutions, he fit the credit default model to the recent history of defaults and then computed using utility functions the market-implied risk aversion.

My advice to future seniors is first and foremost to choose a topic of personal interest. It is a yearlong project, so having certain sets of knowledge or tools is not necessarily a prerequisite; with sufficient work, these can all be acquired along the way. It is far more essential for the writer to be able to remain engaged for that length of time; the thesis is not easy and its completion is for many the hardest task of their undergraduate careers. Each year, we have an excellent Senior Thesis Writers’ Group run by our best graduate students. It is a valuable resource that the best seniors use to great advantage (but that many regretfully underutilize). The group helps educate seniors in research skills, such as computational platforms like MATLAB, technical word-processing using LaTex, and data analysis.

I was delighted that Justin won the School of Engineering and Applied Science’s von Jaskowsy Prize at Class Day, and a number of top graduate Ph.D. programs competed for him—he eventually ended up at Stanford University where, among other research, he will turn his senior thesis into an academic paper. More typically, my former students go into high-pressure financial jobs. A typical e-mail from them a year or two after graduation mentions: “Of course, the hours are as bad as they say and the work often isn’t analytical. But it’s rewarding in different ways.” They go on to testify how the thesis was their most valued academic experience at Princeton, and I always feel very proud to have been part of that.