Financial Math Seminars 2012-13

The seminar in Financial Mathematics is an integral part of the program and an opportunity to interact with leading academic and industry speakers.

Seminars for the Spring Quarter will be held on Fridays at 3:15pm in Room 370 of Building 01-370, just south of Sloan Mathematics (Math Corner) in the Main Quad.

Abstracts

In this talk, we will discuss the role of market makers in today's security markets which are increasingly dominated by high frequency traders. We consider the fundamental trade-offs faced by market makers, particularly that of balancing profits earned from bid-ask spreads against the risk due to accumulated positions and price fluctuations. We propose a mathematical model of a limit order book market and solve the optimization problem faced by market makers trading in multiple correlated securities. In order to take such a model closer to real markets, one also needs to incorporate hidden variables and unknown model parameters (e.g., volatilities, correlations, order flow volumes, etc.) that may even vary over time. These quantities need to be efficiently inferred and estimated from high frequency data. We introduce a new adaptive filtering technique that is directly applicable to this problem of joint filtering and parameter estimation and demonstrate its utility in the optimal market making context.

In a stochastic pure endowment economy with money and one good market but no financial markets or banks, two types of agents trade one non-durable good using two alternative types of cash constraints, a cash-in-advance or a cash-at end-of-period constraint. Simulations of the corresponding variants are compared to Arrow-Debreu and Autarky equilibriums in order to evaluate the nature of systemic risks and financial innovation associated to the interaction of an outside currency with a market and its payment system design. First, this illustrates how financial innovation or systemic risk may affect market functioning and the specific way they can be respectively encouraged or avoided in a neo-classical model with rational expectations. Second, the price and money partition dynamics generated by the two variants absent any macroeconomic shock, exhibit jumps as well as fat-tails and vary depending on the discount rate, underlining the potential usefulness of cash constraints for macro-modelling. Third, the paper shows why financial development takes time but may benefit from financial crises. Fourth, the various features accompanying such a set-up (fines, direct market interventions, related forbearance, impact of inequality and productivity shocks) complement the understanding of systemic risk generated by the credit risk arising from the creation of inside money. When the second variant is interpreted as a partial equilibrium model for the issuance of long term securities by two market-makers, the impact of overall uncertainty on the discount rate of utility may explain the increasing impairment and volatility of such financial market.

In this talk, we start by going over a range of roles in financial mathematics & engineering, shedding light on the specific skills that are typically employed in each of these roles. We will then cover a few special issues that are relevant to roles at financial firms.

• Sound preparation while in school: Developing computational and econometric modeling skills to augment derivatives modeling skills.
• Moving from theory to practice without compromising on rigor: As examples, we will discuss modeling frictions and evaluating the price of model risk.
• Importance of efficiency in developing and deploying models and technology: Building frameworks and facilitating automation.
• Finance is increasingly about math, stats & computer science, and employers recognize that when hiring (even for sales & trading roles).
• Identifying suitable roles and what to expect during interviews.

We integrate systemic financial instability in an empirical macroeconomic model for the euro area. We find that at times of widespread financial instability the macroeconomy functions fundamentally differently from tranquil times. We employ a richly specified Markov-Switching Vectorautoregression model to capture the dynamic relationships between a set of core macroeconomic variables and a novel indicator of systemic financial stress. Both the variances of the shocks and the parameters that capture the transmission of the shocks through the economy change at times of high stress in the financial system. In particular, the negative output effects of sizeable increases in financial stress are much larger after such a regime change than during tranquil times. Macroprudential and monetary policy makers are well advised to take these nonlinearities into account.

This is joint work with Kirstin Hubrich and Manfred Kremer, also with European Central Bank, and Robert J. Tetlow of the Federal Reserve Board.

High frequency trading has led to widespread efforts to reduce information propagation delays between physically distant exchanges. Using relativistically correct millisecond-resolution tick data, we document a 3-millisecond decrease in one-way communication time between the Chicago and New York areas that has occurred from April 27th, 2010, to August 17th, 2012. We attribute the first segment of this decline to the introduction of a latency-optimized fiber optic connection in late 2010. A second phase of latency decrease can be attributed to line-of-sight microwave networks, operating primarily in the 6-11 GHz region of the spectrum, licensed during 2011 and 2012. Using publicly available information, we estimate these networks' latencies and bandwidths. We estimate the total infrastructure and 5-year operations costs associated with these latency improvements to exceed $500 million.

We consider a N*p dimensional data matrix X where all the rows are i.i.d. samples of mean zero and covariance matrix Sigma. Here the population matrix Sigma is of finite rank perturbation of the identity matrix. This is the "spiked population model" first proposed by Johnstone. As N, p tend to infinity but N/p is kept fixed, for the sample covariance matrix S := XX^T/N, we establish the joint distribution of the largest and the smallest few packs of eigenvalues and apply it in mathematical finance.

Caveats of the complete markets theory of finance have been profusely researched, pointing out that its assumptions do not always hold. Yet, there are few if any, general, meaningful and explanatory theories that replace the Arrow--Debreu framework for complete state preferences or the CCAPM risk pricing framework. Its achievements have, however, blurred the essential premise that financial models and theories are in fact only models of uncertainty, hypotheses never confirmed and always in doubt. In a world where risks have multiple and interactive causes; where risks are both exogenous and endogenous, resulting from what we do; where risks are both associated to micro and macro events often leading to the neglect of one or the other and thus to their mismatch. Risks may also arise due to political agendas or to strategic consideration subjugated to complex financial and regulatory systems, etc. In such an environment, financial pricing models are necessarily only a "work in progress."

The purpose of this presentation is to provide an asset pricing model approach based on an extended and price-sensitive CCAPM. In this framework, the CCAPM model is extended to account for the price effects of additional economic factors such as economic inequalities, debt, debt dependence, etc. For simplicity, we consider only two period problems as well as a number of specific cases while extensions to inter-temporal pricing models, mean field Merton game-type models, as well as specific issues including the effects of regulation and rationing on asset prices, economic inequalities, etc. are discussed. When conditions for complete markets hold, the extended CCAPM model is reduced to the standard CCAPM model with future prices implied by current information. When incomplete markets prevail, the extended CCAPM model is shown to be coherent with a utility rationale that underlies the CCAPM as well as an implied pricing framework.

We introduce a new approach based on random matrix theory in order to study the financial network stability problem. After providing some background and simple intuitions on the method we use, we illustrate it with various financial network models which aim to study credit and/or liquidity contagion phenomena. We then show how this new method can be more broadly used to study the systemic risk problem.

In the first part of the talk, an algorithm for solving continuous-time stochastic optimal control problems is presented. The numerical scheme is based on the stochastic maximum principle (SMP) as an alternative to the widely studied dynamic programming principle (DPP). We show possible performance advantages of the algorithm in the case of feedback control. In the second part, an investment-consumption problem with convex transaction costs and optional stochastic returns is presented. The model is a simplified approach for the investment in a portfolio of real options. We show numerical results that, on one hand, are consistent with the well-known investment-consumption theory and on the other hand, show an investment strategy that may seems counterintuitive.

Market forces in equities trading are driving the need for automation and flow internalization. Managing a large stock inventory in a systematic fashion will soon become a cornerstone of any equity trading business. In this talk we discuss the business drivers and the quantitative components that underlie Systematic Inventory Management and the challenges faced by the brave quant venturing into this complex subject, where systematic trading meets real-time risk management.

In this talk we explore what can be said, from a purely theoretical perspective, about technically-based, model-free trending-following strategies. This is an area of finance that has long been considered "voodoo" by the academic finance community. We describe a specific trend-following strategy, referred to as Simultaneous-Long-Short (SLS), that adheres to the tenets of technical trend-following; namely it is direction independent, lets profits run, and cuts losses short. In particular, we highlight the fact that the SLS strategy is completely model-free (that is, it uses no predictive model of stock prices in its determination of an allocation) and instead relies on simple performance driven feedback loops. Surprisingly, we are able to prove that over the class of stock prices following geometric Brownian motion with rather arbitrary time varying drift and volatility, the SLS trend follower always has a positive expected trading gain. We believe that this remarkable robustness to price dynamics may be responsible for the popularity and longevity of simple trend-following strategies, thus demystifying some of the "voodoo" behind technically-based trend-following trading approaches.

The "Flash Crash" of May 6, 2010, saw some stocks and exchange-traded funds traded at pennies only to rapidly recover in price. We show that the impact of the Flash Crash across stocks is systematically related to prior market fragmentation. Interestingly, fragmentation measured based on quote competition --– reflective of higher frequency activity --– has explanatory power beyond a more standard volume-based definition. Using intraday trade data from January 1994 to September 2011, we find that fragmentation now is at the highest level recorded. We also show divergent intraday behavior of trade and quote fragmentation on the day of the Flash Crash itself. The link to higher frequency quotation activity and the current high levels of fragmentation help explain why a Flash Crash did not occur before and offers a counterpoint to the view that the Flash Crash stemmed from an unlikely confluence of events. Controlling for fragmentation, exchange-traded products were differentially affected reflecting the difficulty in pricing component securities. Market structure reforms enacted since the Flash Crash should help mitigate future such market disruptions, but have not eliminated the possibility that another Flash Crash would occur, albeit with a different catalyst and perhaps in a different asset class.

We review a series of recent results on Mean Field Games, including existence and the construction of approximate Nash equilibria. We also present the analog of the stochastic maximum principle approach to the optimal control of stochastic dynamics of the McKean-Vlasov type. In both cases existence results are proven by solving forward-backward stochastic differential equations of the McKean-Vlasov type. (Joint work with F. Delarue)

In this talk, execution strategies with high frequency signals currently employed by Bank of America Merrill Lynch will be introduced and illustrated with actual orders. In addition, what statistical measures are computed, stored and maintained in DB in order to run performance analysis on strategies, order flows, and clients will be explained. Lastly, market making business will also be covered based on the speaker’s past experience, in particular in FX market.

After a brief review of strengths and weaknesses of the traditional, logistic regression based, approach to modeling default risk in consumer and small business credit card portfolios, we discuss new approaches to credit risk modeling based on machine learning algorithms. We provide a few examples of potential applications of such techniques and outline key challenges from the practitioner’s perspective:

• Ensuring stability of models based on machine learning algorithms under different economic conditions
• Leveraging machine learning algorithms to derive insights useful for analyst-based modeling
• Combining machine learning algorithms with time series aspects of consumer credit risk

This talk gives an overview of a new approach to dynamic panel data in econometrics by using empirical Bayes principles that lead to generalized linear mixed models which are versatile and convenient to implement. Applications to default probabilities and loss given default in retail and wholesale loans are also given. This is joint work with T.L. Lai and Kevin Sun.

We describe few decision problems/models arisen in the credit-card business. They include: segmenting business and personal spending; developing a probability of liquidation model framework; and determining optimal lines for OPEN high-spenders, etc.

We survey recent and ongoing work on large pools of securities, such as credit cards or mortgages. We treat computational tools, including various approximations, and tractable statistical methods for parameter estimation.

Two main causes of the recent financial crisis are excessive risk taking due to the limited liability of fund managers and corporations, which means profits are shared, but not losses, and the inability of valuing housing market fairly, which partly leads to the housing bubble. In this talk we will present two papers related to these. (1) We investigate hedge fund performance fees via behavioral finance. In particular, we show that in most cases it is possible to improve the satisfaction of regulators, fund managers, and fund investors simultaneously by replacing the traditional 20\% performance fee scheme with a new 10-30 first-loss scheme, in which fund managers take 30\% performance fee in return for their 10\% first-loss capital investment. (2) We propose an asset pricing model with spatial interaction for pricing real estate assets. The model connects the capital asset pricing model (CAPM) and arbitrage pricing theory (APT) with spatial statistics. An empirical test using the Case-Shiller U.S. regional real estate indices is also given, which suggests a new housing factor based on a mean-variance efficient portfolio. This is a joint work with Xuedong He, Xianhua Peng and Haowen Zhong.

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy of the approximation. Joint work with Konstantinos Spiliopoulos and Kay Giesecke.