COMMERCIAL AND INDUSTRIAL MATHEMATICS

Marcel Nutz - Columbia University
Optimal Transport and Robust Finance

After a brief introduction to classical optimal transport, we shall focus on the so-called martingale optimal transport and its connection to finance, the problem of robust semi-static hedging. Some differences with the classical transport problem will be highlighted, in particular the failure of duality in the usual sense. We explain how to obtain a complete duality theory using notions related to Knightian uncertainty about pricing models. Based on joint work with Mathias Beiglböck and Nizar Touzi.

Joe Langsam - University of Maryland
Concentration Risk

Most financial disasters are a result of either outright fraud or some form of concentration risk. Concentration risk as it impacts financial firms can be classified into three categories: a firm in a well-diversified market whose capital is exposed to one (or a small number) of risk, a firm (or small number of firms) who have taken one side of a risk while facing a large market on the other side of the risk, a market where a sizeable amount of banking capital is exposed to a small number of highly correlated risks. The first of these represents little risk to the system with the possible exception that the exposed firm is a SIFI. If the risks prove too great for this firm, it will go out of business. The market, being well diversified can absorb this. The second, which can generate a fire sale, can be of systemic concern. The third type of concentration, one that we saw in the recent crisis where so great a part of the system was exposed to real estate risk, poses the greatest systemic risk. How to measure these concentrations remains, to my knowledge, an open question. A bank's risk manager may know the bank's exposures, but is unlikely to have sufficient knowledge of how these risks are distributed in the market place. Now is a good time for complete disclosure: I am not as interested in finding the answer to measuring these risks as I am to finding the answer as to why these risks continue to evolve.

It is far easier, and much less useful, to identify risk concentrations after an event. In this talk, I will explore possible means for measuring concentration risks examining both data and modeling requirements. What I will not do is answer the important questions of why it exists. That is beyond my current knowledge, but an important step for designing regulatory actions that reduce the likelihood and severity of future crisis without seriously reducing the effectiveness of the capital markets.

Sasha Stoikov - Cornell University
Estimating the cost of latency in trading

Starting with a diffusion model for the evolution of the best bid and ask sizes of an asset, we compute the probability that the next price move is upward. This probability is a function of the ratio of the best bid to ask sizes, the correlation between changes in the bid/ask sizes and a hidden liquidity parameter. We then formulate a trade execution problem and solve it using dynamic programming. The objective is to sell a single lot of an asset in a short time horizon, while monitoring the ratio of bid to ask sizes. The optimization problem takes into account the latency of the trading algorithm, which affects the prices at which the asset is traded. Identifying good trading times is equivalent to solving an optimal stopping problem, where the objective is to stop whenever the best bid to ask size ratio is small. The solution divides the state space into a ``trade'' and ``no-trade'' region. We calculate the cost of latency per lot traded and demonstrate that the advantage of observing the limit order book can dissipate quickly as latency increases.

Yaacov Mutnikas - Bank of England
Financial Networks and Systemic Risk

Joe Campolieti - Wilfrid Laurier
Dual Families of Solvable Diffusion Models: Applications to Modelling and Derivative Pricing in Finance

In this talk I will firstly present a framework for the construction and classification of a class of dual families of solvable diffusion models. The dual models have highly nonlinear volatility specifications. One main family is characterized by an affine drift specification, while the other has a nonlinear drift specification. The models with affine drift are useful for modelling asset prices, while those with nonlinear drift specification are useful for modelling other quantities such as interest rates. The nonlinearity features, together with the multiply adjustable parameters in the models, are desirable for model calibration to market data such as option data. In the second part of my talk I will discuss the derivation of closed-form spectral expansions for various transition densities, first hitting time distributions, joint distributions of the process value and its maximum or minimum, as well as distributions and expected values for occupation times of the solvable diffusion processes. As an application of the closed-form spectral expansions, I will present some results for pricing barrier, lookback and occupation-time options. The talk will conclude with a brief discussion of some future extensions and applications of the solvable models.

Terry Rockafellar - University of Washington
Risk, Utility and Regression

Maximizing the expected utility of a random variable representingprofit or gain is a widely used approach to financial decision-making.Alternatively, portfolios can be put together to minimize risk ascaptured by the choice of a measure of risk as a functional applied torandom variables representing costs or losses. Some connections areknown between the two approaches, but there is a deeper linkage, not yetfully appreciated, in which a measure of risk can very broadly be portrayedas coming from trade-off rules with respect to "utility" as afunctional on a space of random variables.

That requires extending beyond just expectations and on the other hand considering utility to be relative to some benchmark. Such extension opens remarkable connections between utility and statistical analysis using generalized regression tuned to particular types of risk.

Paolo Guasoni - Boston University and Dublin City University
Nonlinear Price Impact and Portfolio Choice

In a market with price-impact proportional to a power of the order flow, we derive optimal trading policies and their implied welfare and trading volume, for long-term investors with constant relative risk aversion, who trade one safe asset and one risky asset that follows geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the price-impact exponent. As with linear impact, trading rates are finite, but they are lower near the target portfolio, and higher away from the target. The model nests the square-root impact law and, as extreme cases, linear impact and proportional transaction costs.

Anna Obizhaeva - University of Maryland (coauthors: Torben G. Andersen, Oleg Bondarenko, Albert S. Kyle)
High-Frequency Trading Invariance for Equity-Index Futures

The high-frequency trading patterns of the S&P500 E-mini futures contracts between January 2008 and November 2011 are consistent with the following invariance relationship: the number of transactions is proportional to a product of dollar volume and volatility in 2/3 power. Equivalently, the return variation per transaction is log-linearly related to trade size, with a slope coecient of -2. This factor of proportionality deviates sharply from those associated with prior hypotheses relating volatility to the transactions count or trading volume. High-frequency trading invariance is, a priori, motivated by the notion of market microstructure invariance introduced by Kyle and Obizhaeva (2013), though it does not follow from it directly.

Yuri Lawryshyn - University of Toronto (coauthor: Sebastian Jaimungal, University of Toronto) (Slides)
Incorporating Managerial Cash-Flow Estimates and Risk Aversion to Value Real Options Projects

Real options analysis (ROA) is widely recognized as a superior method for valuing projects with managerial flexibilities, yet, its adoption within industry remains limited due to varied difficulties in its implementation. Models proposed by practitioners often lack financial rigour, while the more rigorous mathematical models are not conducive to practical implementation. In this work, we propose a method that matches managerial cash-flow estimates, consisting of arbitrary distributions that can be integrated within many of the rigorous mathematical model frameworks. We achieve this by introducing an observable, but not traded, market stochastic driver process which drives the cash-flows. We present our methodology first in a practical real options setting, then expand the model to account for managerial risk aversion.

(D. Saunders talk has been postponed)
David Saunders - University of Waterloo
Applications of Quantitative Finance to Private Pension Plan Valuation and Management

Private pensions have undergone a dramatic upheaval in recent years. The needs of providers to avoid the risks associated with defined benefit (DB) structures have led to an exodus from these plans and into defined contribution (DC) arrangements. However, employees under DC plans suffer from greater volatility, and lack the benefits of security and risk pooling associated with traditional DB pensions. These concerns have led to the rise of hybrid designs, which seek to combine features of both DB and DC plans. The methods of quantitative finance have much to say about the valuation and management of these plans, and their many embedded options. In particular, we will discuss two examples: cash balance plans, offered by many private employers in the U.S., and variable payout life annuities, such as the one offered by the University of British Columbia.

Damiano Brigo - Imperial College London.
Nonlinear valuation under credit gap risk, collateral margins, funding costs and multiple curves (Slides)

Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, includin deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.

Paul Glasserman - Columbia University.
Hidden Illiquidity with Multiple Central Counterparties (Slides)

The ongoing transformation of the swaps market from over-the-counter trading to central clearing reduces counterparty risk but may create systemic risk by concentrating risk in central counterparties (CCPs). Margin requirements provide the first line of defense against the failure of a CCP. To reflect liquidation costs in the event of a clearing member’s failure, initial margin should be convex in the size of a position. But convex margin requirements create an incentive for dealers to split positions across multiple CCPs, leading each CCP to underestimate potential liquidation costs. We analyze the problem of correcting for this effect. We show that CCPs may set different yet consistent margin requirements provided they agree on true liquidation costs. Different views on true liquidation costs create a potential race to the bottom in which lower margin requirements drive out higher margin requirements. This is joint work with Ciamac Moallemi and Kai Yuan.

September 24, 2014
at 5 p.m.