DH 1st floor SR

Min-Max equations, also called Isaacs equations, arise from many applications, eg in game theory or mathematical finance. For their numerical solution, they are often discretised by finite difference

methods, and, in a second step, one is then faced with a non-linear discrete system. We discuss how upper and lower bounds for the solution to the discretised min-max equation can easily be computed.

We find and describe four futures markets where the bid-ask spread is bid down to the fixed price tick size practically all the time, and which match coun- terparties using a pro-rata rule. These four markets’ offered depths at the quotes on average exceed mean market order size by two orders of magnitude, and their order cancellation rates (the probability of any given offered lot being cancelled) are significantly over 96 per cent. We develop a simple theoretical model to explain these facts, where strategic complementarities in the choice of limit order size cause traders to risk overtrading by submitting over-sized limit orders, most of which they expect to cancel.

Joint work with Jonathan Field.

ARKeX is a geophysical exploration company that conducts airborne gravity gradiometer surveys for the oil industry. By measuring the variations in the gravity field it is possible to infer valuable information about the sub-surface geology and help find prospective areas.

A new type of gravity gradiometer instrument is being developed to have higher resolution than the current technology. The basic operating principles are fairly simple - essentially measuring the relative displacement of two proof masses in response to a change in the gravity field. The challenge is to be able to see typical signals from geological features in the presence of large amounts of motional noise due to the aircraft. Fortunately, by making a gradient measurement, a lot of this noise is cancelled by the instrument itself. However, due to engineering tolerances, the instrument is not perfect and residual interference remains in the measurement.

Accelerometers and gyroscopes record the motional disturbances and can be used to mathematically model how the noise appears in the instrument and remove it during a software processing stage. To achieve this, we have employed methods taken from the field of system identification to produce models having typically 12 inputs and a single output. Generally, the models contain linear transfer functions that are optimised during a training stage where controlled accelerations are applied to the instrument in the absence of any anomalous gravity signal. After training, the models can be used to predict and remove the noise from data sets that contain signals of interest.

High levels of accuracy are required in the noise correction schemes to achieve the levels of data quality required for airborne exploration. We are therefore investigating ways to improve on our existing methods, or find alternative techniques. In particular, we believe non-linear and non-stationary models show benefits for this situation.

Given a diffusion in R^n, we prove a small-noise expansion for its density. Our proof relies on the Laplace method on Wiener space and stochastic Taylor expansions in the spirit of Benarous-Bismut. Our result applies (i) to small-time asymptotics and (ii) to the tails of the distribution and (iii) to small volatility of volatility.

We shall study applications of this result to stochastic volatility models, recovering the Berestycki- Busca-Florent formula (using (i)), the Gulisashvili-Stein expansion (from (ii)) and Lewis' expansions (using (iii)).

This is a joint work with J.D. Deuschel (TU Berlin), P. Friz (TU Berlin) and S. Violante (Imperial College London).

In this talk, approximations to utility indifference prices for a contingent claim in the large position size limit are provided. Results are valid for general utility functions and semi-martingale models. It is

shown that as the position size approaches infinity, all utility functions with the same rate of decay for large negative wealths yield the same price. Practically, this means an investor should price like an exponential investor. In a sizeable class of diffusion models, the large position limit is seen to arise naturally in conjunction with the limit of a complete model and hence approximations are most appropriate in this setting.

The geometric Lévy model (GLM) is a natural generalisation of the geometric Brownian motion (GBM) model. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying Lévy process has been specified, the GLM has four parameters: the initial price, the interest rate, the volatility and the risk aversion. The pricing kernel is the product of a discount factor and a risk aversion martingale. For GBM, the risk aversion parameter is the market price of risk. In this talk I show that for a GLM, this interpretation is not valid: the excess rate of return above the interest rate is a nonlinear function of the volatility and the risk aversion such that it is positive, and is increasing with respect to these variables. In the case of foreign exchange, Siegel’s paradox implies that one can construct foreign exchange models for which the excess rate of return is positive for both the exchange rate and the inverse exchange rate. Examples are worked out for a range of Lévy processes. (The talk is based on a recent paper: Brody, Hughston & Mackie, Proceedings of the Royal Society London, to appear in May 2012).  

Absence of arbitrage is a highly desirable feature in mathematical models of financial markets. In its pure form (whether as NFLVR or as the existence of a variant of an equivalent martingale measure R), it is qualitative and therefore robust towards equivalent changes of the underlying reference probability (the "real-world" measure P). But what happens if we look at more quantitative versions of absence of arbitrage, where we impose for instance some integrability on the density dR/dP? To which extent is such a property robust towards changes of P? We discuss these uestions and present some recent results.

The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton).

TALK 1 -- social media for OII:

TITLE: Computational Perspectives on the Structure and Information

Flows in On-Line Networks

ABSTRACT:

With an increasing amount of social interaction taking place in on-line settings, we are accumulating massive amounts of data about phenomena that were once essentially invisible to us: the collective behavior and social interactions of hundreds of millions of people Analyzing this massive data computationally offers enormous potential both to address long-standing scientific questions, and also to harness and inform the design of future social computing applications: What are emerging ideas and trends? How is information being created, how it flows and mutates as it is passed from a node to node like an epidemic?

We discuss how computational perspective can be applied to questions involving structure of online networks and the dynamics of information flows through such networks, including analysis of massive data as well as mathematical models that seek to abstract some of the underlying phenomena.

TALK 2 -- Community detection:

TITLE: Networks, Communities and the Ground-Truth

ABSTRACT: Nodes in complex networks organize into communities of nodes that share a common property, role or function, such as social communities, functionally related proteins, or topically related webpages. Identifying such communities is crucial to the understanding of the structural and functional roles of networks.Current work on overlapping community detection (often implicitly) assumes that community overlaps are less densely connected than non-overlapping parts of communities. This is unnatural as it means that the more communities nodes share, the less likely it is they are linked. We validate this assumption on a diverse set of large networks and find an increasing relationship between the number of shared communities of a pair of nodes and the probability of them being connected by an edge, which means that parts of the network where communities overlap tend to be more densely connected than the non-overlapping parts of communities. Existing community detection methods fail to detect communities with such overlaps. We propose a model-based community detection method that builds on bipartite node-community affiliation networks. Our method successfully detects overlapping, non-overlapping and hierarchically nested communities. We accurately identify relevant communities in networks ranging from biological protein-protein interaction networks to social, collaboration and information networks. Our results show that while networks organize into overlapping communities, globally networks also exhibit a nested core-periphery structure, which arises as a consequence of overlapping parts of communities being more densely connected.