Relative Pricing of Options and Defaultable
Bonds under Stochastic Volatility

Yu Xiang

Operations Research and Financial Engineering

“While the mathematical theory is very elegant, I found the practical applications more interesting ...”

My thesis focused on stochastic volatility models of asset price dynamics, which are widely used in the finance industry to price derivative instruments. The famous Black-Scholes model has been used extensively in the past, but modern markets demand more sophisticated models that better capture the ever-more complex behavior of asset prices. One of the most unrealistic assumptions in the Black-Scholes model is that of constant volatility. Volatility is the tendency for the stock price to fluctuate; as the market environment changes, stock prices also would tend to fluctuate in different ways. There are strong empirical observations against the constant volatility assumption, and stochastic volatility models have been introduced to capture this random behavior.

This class of models uses two diffusion processes to model asset prices: one for the price itself, and one for the volatility. The price process itself is modeled just like the classic Black-Scholes model, except now the volatility is a second stochastic process. There is quite a bit of abstract mathematics from stochastic calculus and probability theory that goes into these models. Being a mathematically minded person, I was first drawn into the subject by this more theoretical aspect of derivatives pricing. After taking Professor Erhan Cinlar’s “Probability and Stochastic Systems” and Professor Patrick Cheridito’s “Stochastic Calculus and Finance” courses, I knew I wanted to incorporate what I learned in those two classes in my senior thesis. While the mathematical theory is very elegant, I found the practical applications more interesting; having worked at Citigroup as a summer intern, I saw how the theory has been put into practical use in derivatives pricing, which led me to investigate further applications of mathematical theory in finance.

Stochastic volatility models have been given much attention by both academics and practitioners. One main reason for their popularity is their flexibility in capturing the volatility surface. While increasingly complex models have been introduced to provide better statistical fits to market data, the older models are still studied extensively due to their ease of manipulation, explicit solutions for the prices of vanilla European options, and availability of efficient computational methods for pricing and simulation. While these models largely have been used in the derivatives markets, I had the idea of using them as a link between derivatives and the market for corporate bonds. Under the framework of structural default models, a company defaults when its asset value goes under a certain threshold level. Using the stock price as a proxy for the company’s assets, we can model its dynamics using stochastic volatility models, where the model parameters can be estimated from the market prices of liquid options. In essence, we view a zero-coupon bond as a special kind of derivative: a binary down-and-out option, enabling us to use derivative pricing methods to calculate the fair value of corporate bonds. I wanted to see if we can indeed use stochastic volatility models to relate options and corporate bonds; and if not, I would have liked to further explore the reasons behind this discrepancy—whether it is a problem specific to the models themselves, or whether there is inefficiency in the market that can be systematically exploited by arbitrageurs.

My research was a mix of theoretical, numerical, and empirical work. I began by reviewing the necessary background mathematics, ranging from stochastic calculus to differential equations. My adviser was very helpful in providing me with the best background material to get me up to speed to conduct serious research as fast as possible. Having taken some rigorous proof-based analysis courses certainly helped me grasp the more abstract concepts quickly. After fully understanding the mathematical theories behind these models, I began to study them from a modeling perspective, trying to see why certain processes were chosen and how their properties reflected real market behavior. This is where economic intuition and mathematical rigor have to work together, and often compromises have to be made because reality simply refuses to have any degree of mathematical precision.

The model itself was actually only part of the problem because it only describes the price dynamics; we still had to find methods for parameter fitting and instrument pricing. After I began to tackle these two problems, my research quickly turned from a theoretical one to a more data-driven, empirical, and numerical project. Parameter fitting involves an optimization problem that basically minimizes the errors the model would produce when pricing currently traded liquid options. Because the prices are highly nonlinear in the model parameters, the objective function in our optimization problem was far from convex, rendering it infeasible in practice to get an exact solution. Thus, we had to adopt iterative approximate methods. Also, because the feasibility region is very irregular, many of the popular gradient-based optimization methods would fail to reach a global optimal solution, so I had to look for alternative methods. After much searching through the literature, I discovered the so-called Adaptive Simulated Annealing, or ASA. The idea of simulated annealing is to start at some initial values of the parameters at each iteration, give a random perturbation, and move on to the new values if they are a more optimal solution and, with a small and decreasing probability, accept a less optimal solution. The ASA algorithm refers to the specific distribution of the perturbations and how the probability of accepting a less optimal solution decays at each iteration.

The second problem of pricing also proved to be an interesting research subject. While explicit formulas are found for plain vanilla options, evaluating the formulas can be very time-consuming if we use naïve algorithms. In addition, there is no explicit formula for bond pricing, so I had to devise tree-based and Monte Carlo methods. The difficulty with tree-based methods is that we have to balance between making the tree converge to the underlying diffusion process and keeping the tree recombining and relatively small to reduce computation. My solution to this balancing problem was to use a linear interpolation to “spread” the probability through four adjacent nodes, thus making it possible to force the tree to be recombining with arbitrary node locations. Monte Carlo also has its shortcomings. Because of their probabilistic nature, the computed price will vary from one simulation to another. More importantly, the amount of computation required to achieve a reasonable price estimate can be staggering, making it necessary to implement several variance reduction techniques. Some of these include control variates, importance sampling, and the Girsanov transform. It proved to be quite a task to understand the theories behind all these techniques and to construct a proper implementation.

The whole thesis experience has been quite an intellectual challenge. Not only have I gained more knowledge in the particular subject I was studying, I learned how to properly conduct rigorous academic research. Formulating my ideas, filtering through the vast academic literature for relevant past research, gathering and cleaning data, deriving my own results, analyzing and testing my computations—the senior thesis allowed me to experience every step of the research process. I don’t have much advice for the upcoming seniors, but I do want to say, “Enjoy your thesis.”

Relative Pricing of Options and Defaultable
Bonds under Stochastic Volatility

Yu Xiang

Ramon van Handel

Assistant Professor of Operations Research
and Financial Engineering

“You should have the intellectual curiosity to pursue your topic wherever it may lead you, diving under the surface to learn something interesting and then developing it in a rigorous and thorough fashion.”

The senior thesis is a delightful opportunity to turn the tables on the roles of the student and the professor. The student takes on the role of the professor, educating me about a topic of his own choosing and about which he will know—certainly by the end—much more than I do. Meanwhile, I sit back and listen, do my homework (fortunately it has not yet occurred to any of my senior thesis advisees to assign me weekly problem sets), poke and prod at the appropriate points, offer (un)solicited advice, and generally learn along as the thesis grows and takes shape. It is a thoroughly enjoyable arrangement! But it also is, I believe, a cornerstone of the Princeton education. The thesis allows the student to put his hard-earned knowledge to use on a topic that is of special interest to him (which is highly rewarding in itself). Moreover, the successful pursuit of independent research and its coherent communication in the form of a thesis exercises a set of intellectual skills that is quite distinct from what is encountered in most coursework.

Yu (Evans) Xiang is an unusual student. He had taken several advanced graduate-level classes before his senior year and was therefore extraordinarily well prepared to tackle what is surely a rather technical subject. In the beginning, after Evans described his proposed project to me, we discussed some technical details and I suggested some sources he might want to look at. After this, however, Evans took off largely on his own, ultimately producing a very interesting thesis. Of course, it does not have to work this way, and indeed it usually does not. Graduate coursework is not needed (or even necessarily helpful) in writing a good thesis; once you have found a topic that you are genuinely excited about, we will work together to learn the necessary background. On the other hand, in writing his thesis, Evans encountered many of the idiosyncrasies that all researchers face. For example, the development and comparison of different pricing algorithms, which seemed originally to be a small component of his project, turned out to be interesting in its own right and ultimately became one of the main components of his thesis. This tendency of thesis work to take you in unexpected directions is one of those things that make writing a thesis exciting.

What is the secret to a good thesis? Naturally, it is rather hard to say (otherwise it would not be a secret!). Nonetheless, there are a few ingredients that, at the very least, do not hurt. First, you should find a topic that you are passionate about. Next, you should have the intellectual curiosity to pursue your topic wherever it may lead you, diving under the surface to learn something interesting and then developing it in a rigorous and thorough fashion. Finally, no good thesis is written without the requisite amount of sweat. Such is life—but, I think, you will find it is well worth the effort.