Oxford-Man Institute

This paper models a firm’s rollover risk generated by con.ict of interest between debt and equity holders. When the firm faces losses in rolling over its maturing debt, its equity holders are willing to absorb the losses only if the option value of keeping the firm alive justifies the cost of paying off the maturing debt. Our model shows that both deteriorating market liquidity and shorter debt maturity can exacerbate this externality and cause costly firm bankruptcy at higher fundamental thresholds. Our model provides implications on liquidity- spillover effects, the flight-to-quality phenomenon, and optimal debt maturity structures.

In some options markets (eg. commodities), options are listed with only a single maturity for each underlying.

In others, (eg. equities, currencies),

options are listed with multiple maturities.

In this paper, we assume that the risk-neutral process for the underlying futures price is a pure jump Markov martingale and that European option prices are given at a continuum of strikes and at one or more maturities. We show how to construct a time-homogeneous process which meets a single smile and a piecewise time-homogeneous process, which can meet multiple smiles.

We also show that our construction leads to partial differential difference equations (PDDE's), which permit both explicit calibration and fast numerical valuation

We consider the analysis for a class of random differential equations driven by rough noise and with a trajectory that is influenced by its own law. Having described the mathematical setup with great precision, we will illustrate how such equations arise naturally as the limits of a cloud of interacting particles. Finally, we will provide examples to show the ubiquity of such systems across a range of physical and economic phenomena and hint at possible extensions.

Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying efficiency.

In particular they facilitate precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have only been proved for target measures with a product structure, severely limiting their applicability to real applications. The purpose of this talk is to desribe a research program aimed at identifying diffusion limits for a class of naturally occuring problems, found by finite dimensional approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure.

The diffusion limit to a Hilbert space valued SDE (or SPDE) is proved.

Joint work with Natesh Pillai (Warwick) and Jonathan Mattingly (Duke)

In a quantum network model, unitary matrices are assigned to each edge and node of a graph.  The quantum amplitude for a particle to propagate from node A to node B is the sum over all random walks (Feynman paths) from A to B, each walk being weighted by the ordered product of matrices along the path.  In most cases these models are too difficult to solve analytically, but I shall argue that when the matrices are random elements of SU("), independently drawn from the invariant measure on that group, then averages of these quantum amplitudes are equal to the probability that a certain kind of self-avoiding *classical* random walk reaches B when started at A.  This leads to various conjectures about the generic behaviour of such network models on regular lattices in two and three dimensions.