In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities.

This is joint work with Bob Fernholz and Ioannis Karatzas

# Past Mathematical Finance Internal Seminar

We give necessary and sufficient conditions for the variance of the partial sums of stationary processes to be regularly varying in terms of the spectral measure associated with the shift operator. In the case of reversible Markov chains, or with normal transition operator we also give necessary and sufficient conditions in terms of the spectral measure of the transition operator. The two spectral measures are then linked through the use of harmonic measure. This is joint work with S. Utev(University of Leicester, UK) and M. Peligrad (University of Cincinnati, USA).

This talk will discuss work-in-progress on the numerical approximation

of reflected diffusions arising from applications in engineering, finance

and network queueing models. Standard numerical treatments with

uniform timesteps lead to 1/2 order strong convergence, and hence

sub-optimal behaviour when using multilevel Monte Carlo (MLMC).

In simple applications, the MLMC variance can be improved by through

a reflection "trick". In more general multi-dimensional applications with

oblique reflections an alternative method uses adaptive timesteps, with

smaller timesteps when near the boundary. In both cases, numerical

results indicate that we obtain the optimal MLMC complexity.

This is based on joint research with Eike Muller, Rob Scheichl and Tony

Shardlow (Bath) and Kavita Ramanan (Brown).

When estimated volatilities are not in perfect agreement with reality, delta hedged option portfolios will incur a non-zero profit-and-loss over time. There is, however, a surprisingly simple formula for the resulting hedge error, which has been known since the late 90s. We call this The Fundamental Theorem of Derivative Trading. This is a survey with twists of that result. We prove a more general version and discuss various extensions (including jumps) and applications (including deriving the Dupire-Gyo ̈ngy-Derman-Kani formula). We also consider its practical consequences both in simulation experiments and on empirical data thus demonstrating the benefits of hedging with implied volatility.

We examine the Foreign Exchange (FX) spot price spreads with and without Last Look on the transaction. We assume that brokers are risk-neutral and they quote spreads so that losses to latency arbitrageurs (LAs) are recovered from other traders in the FX market. These losses are reduced if the broker can reject, ex-post, loss-making trades by enforcing the Last Look option which is a feature of some trading venues in FX markets. For a given rejection threshold the risk-neutral broker quotes a spread to the market so that her expected profits are zero. When there is only one venue, we find that the Last Look option reduces quoted spreads. If there are two venues we show that the market reaches an equilibrium where traders have no incentive to migrate. The equilibrium can be reached with both venues coexisting, or with only one venue surviving. Moreover, when one venue enforces Last Look and the other one does not, counterintuitively, it may be the case that the Last Look venue quotes larger spreads.

a working version of the paper may be found here

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2630662

We provide a characterization in terms of Fatou property for weakly closed monotone sets in the space of P-quasisure bounded random variables, where P is a (eventually non-dominated) class of probability measures. Our results can be applied to obtain a topological deduction of the First Fundamental Theorem of Asset Pricing for discrete time processes, the dual representation of the superhedging price and more in general the robust dual representation for (quasi)convex increasing functionals.

This is a joint paper with T. Meyer-Brandis and G. Svindland.

In finance, the default time of a counterparty is sometimes modeled as the

first passage time of a credit index process below a barrier. It is

therefore relevant to consider the following question:

If we know the distribution of the default time, can we find a unique

barrier which gives this distribution? This is known as the Inverse

First Passage Time (IFPT) problem in the literature.

We consider a more general `smoothed' version of the inverse first

passage time problem in which the first passage time is replaced by

the first instant that the time spent below the barrier exceeds an

independent exponential random variable. We show that any smooth

distribution results from some unique continuously differentiable

barrier. In current work with B. Ettinger and T. K. Wong, we use PDE

methods to show the uniqueness and existence of solutions to a

discontinuous version of the IFPT problem.

Systemic risk in financial markets occurs when activities that are beneficial to an agent in isolation (e.g. reducing microprudential risk) cause unintended consequences due to collective interactions (usually called macroprudential risk). I will discuss three different mechanisms through which this occurs in financial markets. Contagion can propagate due to the market impact of trading among agents with strongly overlapping portfolios, or due to cascading failures from chains of default caused by networks of interlinked counterparty exposures. A proper understanding of these phenomena must take both dynamics and network effects into account. I will discuss four different examples that illustrate these points. The first is a simple model of the market dynamics induced by Basel-style risk management, which from extremely simple assumptions shows that excessive leverage can give rise to a slowly rising price bubble followed by an abrupt crash with a time period of 10 - 15 years. The model gives rise to a chaotic attractor whose time series closely resembles the Great Moderation and subsequent crisis. We show that alternatives to Basel can provide a better compromise between micro and macro prudential risk. The second example is a model of leveraged value investors that yields clustered volatility and fat-tailed returns similar to those in financial markets. The third example is the DebtRank algorithm, which uses a similar method to PageRank to correctly quantify the way risk propagates through networks of counterparty exposures and can be used as the basis of a systemic risk tax. The fourth example will be work in progress to provide an early warning system for financial stress caused by overlapping portfolios. Finally I will discuss an often neglected source of financial risk due to imbalances in market ecologies.