A ceremony preceding this year's Cornell commencement recognized seven Ph.D. degree recipients and three other candidates. Those honored are from ORIE, the Center for Applied Mathematics, and the Department of Statistical Sciences.
ORIE Professor David Shmoys defines research as "the act of doing something that was not known to be doable."
At ORIE's 2011 graduate degree ceremony in Schwartz Auditorium, seven Ph.D. degree recipients and three candidates nearing completion of the degree were honored for their achievements in original research—doing something that was not known to be doable—in the fields of Operations Research, Applied Mathematics, and Statistics. Thesis advisors of each of the seven recipients placed a hood on the new graduate, symbolizing completion of the the degree and a transition from the asymmetric relationship of student and advisor to the symmetric relationship of colleague and colleague.
Advisors of ORIE students Xiaofei (Sophia) Liu, Matthew Maxwell, Abhimanyu Mitra, and Yuemeng (Sunny) Sun, as well as of applied math student Josef Broder and statistics students Luis Lopez-Oliveros and Michael Grabchak, placed hoods on the shoulders of their students. ORIE students Fan Zhu, Collin Chan and Jiawei Qian also participated in the ceremony. Zhu has since received his Ph.D. degree; Chan and Qian are expected to do so in the coming months.
Professor Robert Bland, ORIE's Associate Director of Graduate Studies, introduced the students individually, describing their backgrounds, research work, personal qualities, and hobbies. Their research, described below in links from their names above, is applicable to supply chain management, revenue management, ambulance services, network design, and option pricing.
Liu's work, guided by Professor Philip Protter, deals with changes over time in the prices of financial assets (such as stocks) by providing an economically plausible mathematical explanation for observed attributes of such financial asset returns that are not well accounted for in the standard model due to Black, Scholes and Merton. These attributes include the tendency of observed returns to have heavier tails (higher probability of extreme events) than predicted by the normal (Gaussian, or bell curve) distribution employed in the standard model and the tendency for large returns to be followed by large returns (and small by small) over time, which is also not predicted by the standard model.
While the standard model assumes a large number of homogeneous agents trading independently in the market at any given time, Liu's model explicitly captures interactions among traders who may tend to influence each other or be similarly influenced by a shared source of information. In this sense, traders may behave like birds in a flock, the behavior of which has been predicted by a model due to Cucker and Smale which inspired Liu's work. In her thesis, Liu proves theorems that characterize the behavior of the model and validates the model's properties through computer simulation. According to Professor Bland, Liu's model "explains, for example, how speculative bubbles in the financial market can be initiated by communication among individual traders as well as by the presence of an influential information source."
Liu, from Jinan, China, graduated from the University of Toronto Scarborough, where she majored in computer science and mathematics. She is joining the Interest Rate Product trading group at Credit Suisse Securities (USA) in New York City. Professor Protter characterizes Liu as "a dream student. She is smart, diligent, and hard working, and also creative, imaginative and fearless," who embraces "difficult mathematical challenges with enthusiasm."
Maxwell's work continues and enhances prior Cornell work on the best way to control a fleet of ambulances in real time to ensure that all parts of a city can be reached quickly in the event of an emergency call. In his thesis research Maxwell greatly improved an earlier method for deciding which ambulances to relocate (redeploy) and where they should go, as ambulances are pressed into service, become temporarily unavailable for new calls, and later become available for service again.
In an application where speed is of the essence, he was able to speed up computation by factors of 250 and 1000 in two different cities. For one of these cities, he found that the method reduced the number of calls taking longer than a specified (and often contractual) threshold time interval from 29.5% to 24.9% when compared to a previous approach that serves as a benchmark for the performance of any method. Achieving such an improvement there without the benefit of this research would require at least two additional ambulances, at an investment that would be a significant fraction of the ambulance organization's budget.
In his thesis, Maxwell developed a close interaction between optimization methods (dynamic programming) and computer simulation in arriving at good solutions, in real time, for redeploying ambulance resources. He devised an improved simulation approach for the earlier method and also used simulation in a somewhat counterintuitive way to select values for key parameters in the method. Maxwell's work was advised by Professors Shane Henderson and Huseyin Topaloglu, who observed that Maxwell is highly creative and came up with solution methods that they were sure could not possibly work, and yet they did. So his thesis study "consisted mostly of trying to figure out why his proposed solutions worked so well," they said.
Maxwell proved that, under reasonable conditions, computer-generated redeployment recommendations can be couched in a form that is easily communicated to dispatchers and executed by ambulance teams. Finally, he derived bounds on the minimum number of ambulance calls that take longer than a specified threshold, whatever method is used to redeploy resources.
In his final semester, in addition to working on his thesis, finding a job (he will work on revenue management at SAS in Cary, N.C.) and house, and teaching an undergraduate course, Maxwell wrote the core computer programs for a startup company, Pre Play Sports. He is a computer science graduate of Brigham Young University and grew up in St. George, Utah.
Effective management of risk requires understanding the odds of extreme events, whether in insurance, finance, hydrology, engineering or industry in general. Mitra's thesis makes several contributions to the systematic estimation of the risk associated with extreme events represented in the tail of the outcome distribution.
Many standard loss models with several variables, such as the Gaussian dependence model, do not accurately predict combined overall loss. Mitra investigated the probability of large combined loss for a class of multivariable portfolios under circumstances where individual losses are not independent and are less likely than predicted by the often used Pareto model, and two or more large losses are unlikely.
Mitra has established conditions, for a significant class of distributions, under which the tail of the sum of losses distribution is related to the tail of the individual loss distributions in a straightforward way. As an example, he has applied this result to the problem of minimizing the probability that the total loss from a portfolio of financial assets exceeds a large threshold while achieving a specified level of earnings.
While this example (and the results of Mitra that it illustrates) assumes a fixed number of assets and possible losses, he has also investigated a situation where not only the magnitude of the individual losses is random but the number of such losses is random as well. Such a situation arises when a company, such as a car dealer, sells items that are individually covered for several years by warranties, and must establish a reserve in each quarter to finance claims against the warranties in the subsequent quarter without knowing how many warranty claims to expect or how large each will be.
Mitra has devised a method, based on claims in a quarter, for estimating the warranty cost in the following quarter. His method does not require strong assumptions on the pattern of sales or the nature of arrivals of warranty claims. As an example has applied this method to data on the sales of 34,807 cars resulting in 3868 claims over two quarters.
In his thesis research Mitra also extended a technique called hidden regular variation to estimate the probability that a multivariable observation falls in a remote risk region beyond the threshold where observations are typically observed. He applied this technique to assessing the likelihood that unusually large files are transmitted over the Internet at unusually high rates, which produces a condition, called burstiness, that may tax Internet lines.
Mitra's research was supervised by Professor Sidney Resnick, who has observed that "beneath Abhi's composed, serene exterior is a lively intellect and a penetrating and deep mind. He has an amazing ability to wring from a germ of a suggested idea something profound, important and insightful and to create a satisfying explanation that provides wholeness and completeness to a problem solution."
Mitra, who is from Kolkata, India, received Bachelor and Master of Statistics degrees at the Indian Statistical Institute before coming to Cornell. He has joined the Market Risk Management and Analysis group of Goldman Sachs in New York City, working on models of risk and model validation.
Sun's dissertation covers two financial topics, one which arises in managing inventories and the other relating to a new form of financial exchange, the so called "dark pools."
Although inventory management is not traditionally considered a financial subject, any time inventory of a product is held there is a chance that its monetary value changes due to depreciation of the product, a change in demand -- or a change in the value of money. While the latter is not typically taken into account, in times of rapid monetary inflation inventory managers may stock up on inventory out of concern that the price of acquiring replacements will rise.
Sun developed a mathematical model to analyze this possibility, and showed that this practice can lead to over-investment in inventory relative to the use of financial instruments that do a better job at preserving wealth. In that case Sun showed how inventory managers can develop a strategy for maximizing operational profits while avoiding non-financial risk (from, for example, a change in market conditions) and then finance managers can develop a strategy to hedge the residual financial risk, with the combined result that both strategies are optimal overall.
Further, Sun showed that it is possible, under certain conditions to solve this inventory investment problem one product at a time, even when demand for the products is correlated, there are thousands of products, and there are multiple inventory managers. Moreover, she extended this "separation" result to the realistic case involving multiple time periods (i.e. months) where decisions in one time period have an impact on future time periods. Being able to separate the roles of inventory managers and financial managers and to separate the roles of managers for different inventoried products without sacrificing optimality makes it computationally and organizationally feasible to develop optimal inventory hedging policies.
In her research on dark pools, Sun investigated the possibility of price manipulation when a large block of an asset (e.g. a stock) is being sold by splitting it between a conventional public exchange and one of the relatively new trading platforms (or dark pools) for which buy and sell orders and prices are not visible to the public. Typically, transactions using a dark pool trading platform are carried out with prices from the public exchange. Hence an unscrupulous trader selling a large block can drive proceeds up by buying on the public exchange while selling on the dark pool exchange (sometimes known as "pump and dump.")
Sun formulated a model for this form of trading and derived an optimal strategy for the seller to employ, which entails offering shares in both the public exchange and the dark pool but, at a specific time, removing any remaining unsold shares from the dark pool and offering them on the public exchange. By analyzing the implications of this optimal strategy, she developed a mathematical condition on the specifications of the model under which "pump and dump" price manipulation is not beneficial to the seller. In principle, this mathematical condition could be incorporated into a regulatory framework for example by restricting the asset that can be sold via the dark pool, where the fraction is mathematically chosen so as to prevent the practice from being optimal. Sun's analysis also shows that increasing the liquidity in the public exchange can also reduce price manipulation in the dark pool.
Sun, who was born and raised in Beijing and did her undergraduate studies in mathematics at Nanjing University, has joined the Global Quant Group of Bank of America Merrill Lynch. Her work on dark pools was done under Professor Alexander Schied, and her work on inventory management under Professor Peter Jackson, with assistance from Professor Johannes Wissel.
Professor Jackson notes that Sun "is excellent at taking ideas and formalizing them into mathematical models. When we first formulated the multi-product inventory hedging model I thought that it would be too complex to solve. I was delighted when Sunny came back with the separation result. This is a credit both to her modeling skills and her deep analytical capabilities. She has been a delight to work with."
Applied mathematics Ph.D. graduate Josef Broder did his thesis research under ORIE professor Paat Rusmevichientong. The research focuses on a problem faced by any company that has a new product to offer to the market place but does not know how demand for the product will relate to the price at which it is sold. The challenge is to offer the product at a price or prices that will reveal price-dependent demand without unnecessarily reducing the total revenue for the sales. Broder's work on this challenge is a contribution to the field of revenue management.
Broder developed and analyzed a stylized model of the pricing problem in an effort to understand the properties of two techniques to minimize the extent to which revenue is sacrificed through price experiments, versus what the revenue would be were the response of demand to price were fully known in advance. One key issue is the uncertainty inherent in the response of customers to price: at best what can be discerned from data is the probability that a customer will buy at a particular price. Moreover, if the product is in fact purchased at a given price, would the customer still have bought it had the price been higher?
Broder derived bounds on the best that can be done, over time, by any policy for managing the revenue in such cases and then showed how two specific policies relate to the bounds. In one case, sales at various prices in an exploratory way alternate cyclically with sales at the optimal price calculated as the result of the explorations so far. In another, useful under more restrictive conditions on the relationship of demand to price, no separate exploration is required but prices are continually adjusted over time as the demand response is measured, according to a precise calculation. For both policies, Broder conducted numerical experiments to test the policies. He went on to consider the implications of limiting the number of times prices are adjusted, since changing prices too frequently can be costly and may alienate customers.
In addition to this work on prices, Broder also investigated methods for distributing goods among a number of possible selling venues in the face of uncertain demand. He developed a heuristic algorithm to solve the model, provided a lower bound on the effectiveness of the algorithm, and showed through simulation that the algorithm works better than other known approaches. He then tested his ideas by contacting and obtaining data from Ithaca Carshare, a non-profit organization that provides a fleet of communal vehicles members whose use members can schedule on line and pay at a fixed hourly rate for the time used. Here the problem is how to allocate vehicles at locations in and around Ithaca so as to maximize total usage and corresponding cost recovery. Through his analysis, he identified opportunities for the organization to expand its user base without purchasing any additional vehicles.
Professor Rusmevichientong commented that he is "very impressed with Josef's initiative and his development into an independent researcher. He can identify his own research topic, formulate his own research agenda, and follow through." Broder grew up in Athens, Ga., and received his undergraduate degree in mathematics from the University of Georgia. He recently accepted a position at Google.
As noted in the discussion of Xiaofei Liu's work, financial returns observed in markets have a tendency to have heavier tails than predicted by the Gaussian distribution employed in the standard model. This phenomenon, which is more pronounced the greater the time frequency of the data, has lead to a search for a class of probability distributions that correspond more closely to observed data, while having plausible explanations in terms of the underlying processes being modeled and possessing sufficient flexibility to cover a wide variety of circumstances in finance and other domains.
Statistics Ph.D. graduate Grabchak's thesis research explores the properties of a class known as 'tempered stable distributions.' This class results from a specific modification of stable distributions, which are those that represent the limit of a sum of many independent random variables.
Although the Gaussian distribution is stable, its limitations in explaining real data have led to a focus on other stable distributions (e.g. stable Paretian distributions). Unfortunately these distributions tend to have tails that are overweighted with respect to real data, overstating the probability of very large changes just as the Gaussian understates it. The tempered stable distributions studied by Grabchak provide models for which the probability of a small change is very close to that given by a stable, non-Gaussian distribution but the probability of a very large changes is smaller, hence providing a better explanation for several observed properties of asset returns.
In his thesis, Grabchak gives conditions for tempered stable distributions to have these and other desirable properties and provides explanations for why such models work well in applications and so have been used in physics (where they originated), biology, and computer science as well as finance.
Grabchak, who moved with his family from his birthplace of Odessa, Ukraine to East Brunswick, NJ when he was six years old, received his undergraduate degree in mathematics and computer science from nearby Rutgers University. He will be an Assistant Professor in the department of mathematics and statistics at the University of North Carolina at Charlotte.
Professor Samorodnitsky notes that Grabchak is "very hardworking and a great researcher all around." The two were together at the University of Copenhagen during Samorodnitsky's sabbatical year there, and Grabchak taught there as well as at Cornell while working on his thesis.
Heavy tailed distributions characterize phenomena other than financial returns. As noted in the earlier discussion of Mitra's research, they also characterize traffic on the Internet. In particular, the number of bytes transmitted during a network session, the duration of the session, and the average rate at which data are transferred all have historically shown heavy tails. Working under the guidance of ORIE Professor Sidney Resnick, statistics Ph.D. student Lopez-Oliveros carried out both theoretical and empirical analysis of Internet traffic in order to understand the way in which end users cause data to be transferred over networks.
The overall objective of Lopez-Oliveros' work is to enable network planners to simulate the impact of end user behavior on the performance of networks, so as to design them with sufficient capacity to satisfy contractual and other criteria, and to prevent congestion and identify bottlenecks. In his theoretical work, he reconciled an apparent contradiction between the predictions of previously published theory and examples found in practice. There is empirical evidence that the number of bytes or number of data packets per unit time traveling through a network node ((which is not the same as the rate at which data are transferred) is approximately Gaussian, despite existing theory. He showed that this is a consequence of the fact that the Internet carries many streams of traffic that differ from one another.
When collected, data on Internet traffic is captured in huge data sets that require innovative computing and statistical techniques to understand. Lopez-Oliveros extracted information about different types of user sessions, such as web surfing, downloading email, streaming a movie, etc, from such data sets and and developed mathematical models that match properties and predictions with empirical observations. His findings yield an accurate method for generating network sessions for use in simulations.
Professor Resnick noted that Lopez-Oliveros was "extremely careful and diligent in his work and frequently displayed an eerie ability to spot errors in the huge data sets we had received. He figured out what to do, learned what was necessary from the statistics - computer science - mathematics - networking toolbag and progressed to produce careful, tightly reasoned and impressively insightful analyses."
Lopez-Oliveros, from the Mexican State of Veracruz, received his M.S. degrees in probability from the Center for Mathematical Research (CIMAT) in Guanajuato. His B.S. degree in mathematics is jointly from CIMAT and the University of Guanajuato. He recently started work at Murex in New York City.
Zhu's research focuses on the valuation, or pricing, of what are known as exotic options. An exotic option is a financial contract that is more complex than commonly traded ("vanilla") options such as puts (the purchased right to sell an underlying asset such as a stock at an agreed upon "strike" price by an agreed upon date, or "maturity") and calls (the purchased right to buy an asset at an agreed upon strike price by an agreed upon maturity). Exotic options, contracts with more complicated provisions than vanilla options, have been in the news because they have been implicated in (though are not the root cause of) the recent global financial crisis.
While traders routinely use the standard Black Scholes Merton model to price vanilla options and develop hedging strategies for them, this model cannot be directly applied to price exotic options. Typically, analysts use the model indirectly by deducing a quantity, called implied volatility, from the market price of the related vanilla option, and using that quantity to calibrate formulas for the exotic option price. They also use the related vanilla option to hedge the exotic option.
However this strategy has problems in practice, due to the fact that implied volatility varies unpredictably with strike price and maturity. Nor are other popular models for pricing exotic options without defect. Zhu proposed an alternative framework that entails directly modeling the market price of the vanilla option as a function of both strike price and maturity and incorporating the output of this model into the pricing of the exotic option.
To develop his alternative approach Zhu employed and extended a widely used statistical technique to determine (abstract) factors from which vanilla option prices can be predicted as a function of strike and maturity. He combined novel statistical methods and optimization techniques in this work. He then went on to test this powerful methodology on both historical and simulated data. More accurate pricing may avoid some of the difficulties that have been encountered in the use of exotic options.
Professor Martin Wells, Zhu's advisor, observes that Zhu has worked extremely hard to integrate some deep mathematical and statistical theoretical ideas in order to solve an extremely important and practical financial engineering problem. Zhu is from Beijing, China, and did his undergraduate studies in mathematics at Beijing University. He recently completed his PhD and expects to return to China for employment.
Chan's research, supervised by Professors Peter Jackson and Huseyin Topaloglu, deals with a problem introduced to him by Professor Jack Muckstadt. It is motivated by the issues arising in the overhaul of jet engines for civilian and military aircraft. Depending on the type of engine problem requiring overhaul service, repair crews need different sets of parts, but they may not know which ones until the engine is disassembled. Moreover different overhaul jobs may complete for the same set of parts. Since jet engines arrive at overhaul facility essentially at random, effective procurement policies for the very large number of individual parts must respond to a highly uncertain environment that changes over time.
Chan built mathematical models that are able to prioritize the overhaul jobs so as to optimally balance minimizing the cost of investing in the inventory of parts against the cost of delays in overhauling engines (a particular concern, since if any single part is out of stock the entire repair is held up). Because these models explicitly take uncertainty and multiple time periods into account, they require extensive computation in order to arrive at effective policies. Chan found a way to successively break the problem of computing solutions into smaller problems and reassemble the solutions, so as to bring computation time to within reasonable limits.
Professors Jackson and Topaloglu note that Chan was able to take an existing model intended for a different problem—distributing a single product using a multi-level supply chain—and adapt it to be useful in completing service jobs, each requiring a specific bundle of parts to fix, as exemplified by the engine overhaul problem. "He has the ability to see connections between seemingly unrelated problem domains, taking one model that was developed in one application area and tweaking the model so that it becomes useful in a completely different domain," they said. Chan is expected to complete his degree by the end of the year.
Qian's research, which he is in the process of completing under the supervision of Professor David Williamson, tackles an important problem in the design of fiber optic telecommunications networks such as are currently being rolled out around the country.
Determining where to originate the network cables, laying out the routes for the cables, and establishing points at which the cables branch out to terminate at customer homes in the most cost-effective way is in a class of problems (which includes the well-known traveling problem) known to be hard to solve completely even with the most powerful computers, when hundreds or thousands of locations are involved. Hence Qian sought to develop a more readily computed approximation algorithm that could be shown to yield a solution close to the unobtainable minimum cost solution.
Qian developed his algorithm by generalizing and extending previous algorithms to cover more challenging and realistic situations. It covers the case in which the desired termination points are not all known in advance, but become known over time (for example, as orders for service are received during the course of network construction). The algorithm also helps the service provider determine which connections might better be outsourced to other vendors, at a price.
Qian showed that his algorithm produces designs that have as close to minimum cost as are possible to obtain using an approximation algorithm that computes results in reasonable time for problems with a large number of termination locations. According to Professor Williamson, Qian got his results by being "very tenacious - if he doesn't get the result he is after on the first try, or the second try, or the third try, he gets it by the fourth, fifth, or sixth try." Although his work is deeply theoretical, Qian has also done a great deal of computational work, and once kept ten of Amazon's largest computers busy for a week working on an issue related to the traveling salesman problem.
Qian, who was born and raised in Beijing, earned his undergraduate degree with double majors in computer science and in optimization and combinatorics at the University of Waterloo, Canada.
As Professor Shmoys has pointed out, the Ph.D. degree not only recognizes the research that has been accomplished by the recipients, but "the fact that they have mastered the process of advancing the limits of science, and can now embark on careers of doing it all over, again and again." The ten Ph.D. students honored at the graduation ceremony have mastered the process and are on their way to its continued application.
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