# Abstracts

**23.06.2017:** Johannes Muhle-Karbe (Michigan)

**Title: **A Risk-Neutral Equilibrium Leading to Uncertain Volatility Pricing

**Abstract:** We study the formation of derivative prices in equilibrium between risk-neutral agents with heterogeneous beliefs about the dynamics of the underlying. Under the condition that the derivative cannot be shorted, we prove the existence of a unique equilibrium price and show that it incorporates the speculative value of possibly reselling the derivative. This value typically leads to a bubble; that is, the price exceeds the autonomous valuation of any given agent. Mathematically, the equilibrium price operator is of the same nonlinear form that is obtained in single-agent settings with strong aversion against model uncertainty. Thus, our equilibrium leads to a novel interpretation of this price.

This is joint work with Marcel Nutz.

**09.06.2017:** Bruno Bouchard (Paris Dauphine)

**Title: **Super-hedging with proportional transaction costs under uncertainty: a randomization approach

**Abstract:** We prove a super-hedging duality for models with proportional transaction costs under model uncertainty, under a robust version of the no-arbitrage of second kind condition. It is based on a randomization approach that allows one to reduce to frictionless markets and apply earlier results of Bouchard and Nutz.

**26.05.2017:** Sebastian Herrmann (Michigan)

**Title: **Robust Pricing and Hedging around the Globe

**Abstract:** We study the martingale optimal transport duality for càdlàg processes with given initial and terminal laws. Strong duality and existence of dual optimizers (optimal robust semi-static superhedging strategies) is proved for a class of reward functions including American, Asian, Bermudan, and European options with intermediate maturity. Our approach sheds light on the structure of primal and dual optimizers. In the case of finitely supported marginal laws, dual optimizers are obtained by solving a semi-infinite linear program.

This talk is based on joint work with Florian Stebegg (Columbia University).

**19.05.2017: **Luciano Campi (LSE)

**Title: **N-player games and mean field games with absorption

**Abstract:** We introduce a simple class of mean field games with absorbing boundary over a finite time horizon. In the corresponding N-player games, the evolution of players' states is described by a system of weakly interacting Itô equations with absorption on first exit from a bounded open set. Once a player exits, her/his contribution is removed from the empirical measure of the system. Players thus interact through a renormalized empirical measure. In the definition of solution to the mean field game, the renormalization appears in form of a conditional law. We justify our definition of solution in the usual way, that is, by showing that a solution of the mean field game induces approximate Nash equilibria for the N-player games with approximation error tending to zero as N tends to infinity. This convergence is established provided the diffusion coefficient is non-degenerate. The degenerate case is more delicate and gives rise to counter-examples.

This talk is based on a joint work with Markus Fischer (Padua University).

**05.05.2017: **Paolo Guasoni (Dublin City)

**Title:** Leveraged Funds: Robust Replication and Performance Evaluation

**Abstract:** Leveraged and inverse exchange-traded funds seek daily returns equal to fixed multiples of indexes' returns. Trading costs implied by frequent adjustments of funds' portfolios create a tension between tracking error, reflecting short-term correlation with the index, and excess return, the long-term deviation from the leveraged index' performance. With proportional costs, the optimal replication policy is robust to the index' dynamics. Overall fund performance is summarized by the implied spread, the product of tracking error and excess return, rescaled for leverage and volatility. The implied spread is insensitive to risk-premia and enables comparisons of funds tracking different factors of an index.

**17.03.2017: **Frank Seifried (Trier)

**Title:** Epstein-Zin Stochastic Differential Utility

**Abstract:** This talk presents some recent results on the foundations of continuous-time recursive utility in the Epstein-Zin parametrization: We establish existence, uniqueness, monotonicity, concavity, and a utility gradient inequality in a fully general semimartingale setting.

**10.03.2017: **Daniel Schwarz (UCL)

**Title:** The existence of densities of BSDEs

**Abstract:** We introduce sufficient conditions for the solution of a multi-dimensional, Markovian BSDE to have a density. We show that a system of BSDEs possesses a density if its corresponding semilinear PDE exhibits certain regularity properties, which we verify in the case of several examples.

**20.01.2017: **Christoph Czichowsky (LSE)

**Title:** Portfolio Optimisation, Transaction Costs, Shadow Prices and Fractional Brownian Motion

**Abstract:** While absence of arbitrage in frictionless financial markets (i.e. without transaction costs) requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this talk, I will present an overview over several results that provide a way how to use non-semimartingale price processes such as the fractional Black-Scholes model in portfolio optimisation under proportional transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs.

The talk is based on joint work with Walter Schachermayer.

**02.12.2016:** Saul Jacka (Warwick)

** Title:** General Controlled Markov Processes and Optimal Stopping.

**Abstract:** This talk will introduce abstract control via the weak (martingale problem) formulation and give two current examples-one involving general Markovian optimal stopping problems.

**25.11.2016: **Frank Kelly (Cambridge)

**Title:** A Markov model of a limit order book: thresholds, recurrence, and trading strategies.

**Abstract:** In this talk we discuss an analytically tractable model of a limit order book where the dynamics are driven by stochastic fluctuations between supply and demand. The model has a natural interpretation for a highly traded market on short time-scales where there is a separation between the time-scale of trading, represented in the model, and a longer time-scale on which fundamentals change. We describe our main result for the model, which is the existence of an explicit limiting distribution for the highest bid, and for the lowest ask, where the limiting distributions are confined between two thresholds. Fluid limits play an important role in establishing the recurrence properties of the model. We use the model to analyze various high-frequency trading strategies (for example market-making, sniping and mixtures of these), and comment on the Nash equilibria that emerge between high-frequency traders when a market in continuous time is replaced by frequent batch auctions.

This is joint work with Elena Yudovina (University of Michigan). https://arxiv.org/abs/1504.00579

**11.11.2016:** Thomas Cayé (ETH Zürich)

** Title:** Trading with small nonlinear price impact.

**Abstract:** We study a portfolio choice problem with nonlinear price impact in a general setting. Using probabilistic techniques, we show that the limiting control problem for small price impact can be reduced to the ergodic control of an OU-type process with nonlinear mean-reversion speed. This problem can be solved explicitly up to a single nonlinear ODE, which identifies the optimal trading speed and the welfare loss due to the trading friction. Previous asymptotic results for proportional and quadratic trading costs are obtained as particular limiting cases.

**04.11.2016:** Tiziano De Angelis (Leeds)

** Title:** The dividend problem with a finite horizon.

**Abstract:** We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at an elastic boundary.

This is joint work with Erik Ekstr\"om (University of Uppsala). http://arxiv.org/abs/1609.01655

**21.10.2016: **Blanka Horvath (Imperial)

**Title:** Aspects of asymptotic expansions in fractional volatility models.

**Abstract:** We revisit small-noise expansions in the spirit of Benarous, Baudoin-Ouyang, Deuschel-Friz-Jacquier-Violante for bivariate diffusions driven by fractional Brownian motions with different Hurst exponents. A particular focus is devoted to rough stochastic volatility models which have recently attracted considerable attention. We derive suitable expansions (small-time, energy, tails) in these fractional stochastic volatility models and infer corresponding expansions for implied volatility. This sheds light (i) on the influence of the Hurst parameter in the time-decay of the smile and (ii) on the asymptotic behaviour of the tail of the smile, including higher orders.