126 FM12 Abstracts IC1 which happens in applications to barrier option pricing or Optimal Execution in a General One-Sided Limit structural credit risk models. In this talk, I will present Order Book novel adaptive discretization schemes for the simulation of stopped Lvy processes, which are several orders of magni- We construct an optimal execution strategy for the pur- tude faster than the traditional approaches based on uni- chase of a large number of shares of a financial asset over form discretization, and provide an explicit control of the a fixed interval of time. Purchases of the asset have a non- bias. The schemes are based on sharp asymptotic estimates linear impact on price, and this is moderated over time by for the exit probability and work by recursively adding dis- resilience in the limit-order book that determines the price. cretization dates in the parts of the trajectory which are The limit-order book is permitted to have arbitrary shape. close to the boundary, until a specified error tolerance is The form of the optimal execution strategy is to make an met. initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is Peter Tankov set to match the order book resiliency. At the end of this Universit´e Paris-Diderot (Paris 7) period, another lump purchase is made, and following that [email protected] there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase. Any of the IC4 lump purchases could be of size zero. A simple condition is Talk Title TBA - Avellaneda provided that guarantees that the intermediate lump pur- chase is of size zero. This is joint work with Gennady Abstract not available at time of publication. Shaikhet and Silviu Predoiu. Marco Avellaneda Steven E. Shreve Courant Institute Carnegie Mellon University New York University Dept of Mathematical Sciences [email protected] [email protected] IC5 IC2 Optimal Order Placement in Limit Order Books Stable Diffusions With Rank-based Interactions, Abstract not available at time of publication. and Models of Large Equity Markets Xin Guo We introduce and study ergodic diffusion processes inter- University of California, Berkeley acting through their ranks. These interactions give rise to [email protected] invariant measures which are in broad agreement with sta- bility properties observed in large equity markets over long time-periods. The models we develop assign growth rates IC6 and variances that depend on both the name (identity) Quantitative Absence of Arbitrage and Equivalent and the rank (according to capitalization) of each individ- Changes of Measure ual asset. Such models are able realistically to capture critical features of the observed stability of capital distri- It is well known that absence of arbitrage is a highly desir- bution over the past century, all the while being simple able feature in mathematical models of financial markets. enough to allow for rather detailed analytical study. The In its pure form (whether as NFLVR or as the existence of methodologies used in this study touch upon the question a variant of an equivalent martingale measure R), it is qual- of triple points for systems of interacting diffusions; in par- itative and therefore robust towards equivalent changes of ticular, some choices of parameters may permit triple (or the underlying reference probability (the ”real-world” mea- higher-order) collisions to occur. We show, however, that sure P). But what happens if we look at more quantitative such multiple collisions have no effect on any of the sta- versions of absence of arbitrage, where we impose for in- bility properties of the resulting system. This is accom- stance some integrability on the density dR/dP? To which plished through a detailed analysis of collision local times. extent is such a property robust towards changes of P? We The models have connections with the analysis of Queue- discuss these questions and present some recent results. ing Networks in heavy traffic, and with competing par- The talk is based on joint work with Tahir Choulli (Uni- ticle systems in Statistical Mechanics (e.g., Sherrington- versity of Alberta, Edmonton). Kirkpatrick model for spin-glasses). Their hydrodynamic- limit behavior is governed by generalized porous medium Martin Schweizer equations with convection, whereas limits of a different ETH Math kind display phase transitions and are governed by Poisson- Zurich, Switzerland Dirichlet laws. [email protected] Ioannis Karatzas Columbia University SP1 [email protected] AWM-SIAM Sonia Kovalevsky Lecture : The Role of Characteristics in Conservation Laws IC3 Sonya Kovalevsky, in the celebrated Cauchy-Kovalevsky Simulation Schemes for Stopped Lvy Processes theorem, made clear the significance of characteristics in partial differential equations. In the field of hyperbolic Jump processes, and Lvy processes in particular, are noto- conservation laws, characteristic curves (in one space di- riously difficult to simulate. The task becomes even harder mension) and surfaces (in higher dimensions) dominate the if the process is stopped when it crosses a certain boundary, behavior of solutions. Some examples of systems exhibit FM12 Abstracts 127 interesting, one might even say pathological, characteristic scalar Banach algebra. Positive results are obtained for behavior. This talk will focus on ways that characteristics both commutative and noncommutative algebras. in systems of conservation laws give information about the systems being modeled. Ruth Curtain University of Groningen, Netherlands Barbara Lee Keyfitz [email protected] The Ohio State University Department of Mathematics bkeyfi[email protected] SP5 I.E. Block Community Lecture: Creating Reality: the Mathematics Behind Visual Effects SP2 The John Von Neumann Lecture: Liquid Crystals Abstract to follow. for Mathematicians Robert Bridson Liquid crystals form an important class of soft matter sys- University of British Columbia tems with properties intermediate between solid crystals [email protected] and isotropic fluids. They are the working substance of liq- uid crystal displays, which form the basis of a huge multi- national industry. The lecture will describe these fascinat- SP6 ing materials, and what different branches of mathematics, SIAG/FME Junior Scientist Prize: Market-Based such as partial differential equations, the calculus of vari- Aapproach to Modeling Derivatives Prices ations, multiscale analysis, scientific computation, dynam- ical systems, algebra and topology, can say about them. Most of the existing quantitative methods in Finance rely on the assumptions of the underlying mathematical mod- els. The problem of choosing the appropriate model as- John Ball sumptions is one of the cornerstones of modern Financial Oxford Centre for Nonlinear PDE Mathematical Institute Engineering. I am interested in developing modeling frame- Oxford works that facilitate the use of historical observations when [email protected] making the choice of model assumptions. It turns out that, in the markets with a large family of liquid derivative con- tracts, it is rather hard to construct a model that exploits SP3 the information contained in the historical prices of these Past President’s Address: Reflections on SIAM, derivatives. In fact, constructing such models requires the Publishing, and the Opportunities Before Us use of the so-called Market-Based Approach. The idea of this approach is to treat the liquid derivatives as generic Upon taking up the post of president I had, of course, for- financial assets and prescribe the joint evolution of their mulated my priorities for SIAM. This talk provides a good prices in such a way that any future arbitrage-free combi- occasion to revisit some of those. One area turned out nation of prices is possible. In this presentation, I will out- to play a vastly larger role than I would have anticipated, line the main difficulties associated with the construction of namely mathematical publishing and many issues associ- market-based models and will present a general methodol- ated with it, ethical, technological, economic, political, and ogy that bypasses these difficulties. Finally, I will illustrate scientific. The future of scholarly publishing is far from the theory by describing (both mathematically and numer- clear, but one thing seems certain: big changes are needed ically) a family of market-based models for the European and will be coming. We, as mathematicians, are major call options of multiple strikes and maturities. stakeholders. We should also be major agents in guiding these changes. I will present some of my observations and Sergey Nadtochiy thoughts as we confront the opportunities before us. Oxford University Oxford-Man Institute Douglas N. Arnold [email protected] School of Mathematics University of Minnesota [email protected] JP1 Systemic Risk SP4 What is systemic risk, how do we model it, how to we an- W. T. and Idalia Reid Prize Lecture: Large Alge- alyze it, and what are the implications of the analysis? I braic Properties of Riccati Equations. will address these issues both in a larger historical context and within current research mathematical finance. The key In the eighties there was considerable interest in the alge- property of systems subject to systemic risk is their inter- braic properties of the following Riccati equation connectivity and the way individual risk can become over- all, systemic risk when it is diversified by inter-connectivity. A∗X + XA XBB∗X + C∗C =0, (1) I will discuss theoretical issues that come up with mean- − field and other models and will also show results of numer- where A, B, C A, a Banach algebra with identity, and ical simulations. the involution operation∈ . Conditions are sought to ensure that the above equation∗ has a solution in A.Theresults George C. Papanicolaou were disappointing and the problem was forgotten until Stanford University this century when engineers studied the class of spatially Department of Mathematics distributed systems.