The compressibility of graphene

We present a review of the electronic compressibility of monolayer and bilayer graphene. We focus on describing theoretical calculations of the effects of electron--electron interactions and various types of disorder, and also give a summary of current experiments and describe which aspects of theory they support. We also include a full analysis of all commonly-used contributions to the tight-binding Hamiltonian of bilayer graphene and their effects on the compressibility.Comment: 15 preprint-style pages, 5 figure

Credit risk in the pricing and hedging of derivatives

Credit risk - more specifically, default risk - is introduced in various classical models for option pricing. The consequences of this new parameter in terms of model calibration is studied.

A Mathematical Approach to Order Book Modeling

Motivated by the desire to bridge the gap between the microscopic description of price formation (agent-based modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales, we present a mathematical study of the order book as a multidimensional continuous-time Markov chain and derive several mathematical results in the case of independent Poissonian arrival times. In particular, we show that the cancellation structure is an important factor ensuring the existence of a stationary distribution and the exponential convergence towards it. We also prove, by means of the functional central limit theorem (FCLT), that the rescaled-centered price process converges to a Brownian motion. We illustrate the analysis with numerical simulation and comparison against market data

Tick Size Reduction and Price Clustering in a FX Order Book

We investigate the statistical properties of the EBS order book for the EUR/USD and USD/JPY currency pairs and the impact of a ten-fold tick size reduction on its dynamics. A large fraction of limit orders are still placed right at or halfway between the old allowed prices. This generates price barriers where the best quotes lie for much of the time, which causes the emergence of distinct peaks in the average shape of the book at round distances. Furthermore, we argue that this clustering is mainly due to manual traders who remained set to the old price resolution. Automatic traders easily take price priority by submitting limit orders one tick ahead of clusters, as shown by the prominence of buy (sell) limit orders posted with rightmost digit one (nine).Comment: 17 pages, Minor revision

The times change: multivariate subordination, empirical facts

The normality of multi-asset returns in event time is shown empirically. A multivariate subordination mechanism is proposed in order to explain this phenomenon.

A nonlinear partial integro-differential equation from mathematical finance

We study a nonlinear partial integrodifferential equation arising in the calibration of stochastic volatility models to a market of vanilla options.