Book of abstracts

Find here a booklet containing general information about the conference and the book of abstracts (the schedule in this booklet may be not up to date).

Lecture Notes, slides and articles

Please find below the lecture notes, slides or link to articles the invited speakers of the Summer School used or referred to during their talk. We will continue to upload the notes as we receive them, so please kindly check regularly for updates.

Alexander Cox, slides
Martin Klimmek, slides
Johannes Muhle-Karbe, slides
Marcel Nutz, preprints covered in talk Arbitrage and Duality in Nondominated Discrete-Time Models and Utility Maximization under Model Uncertainty in Discrete Time
Harald Oberhauser, slides
Jan Obloj, slides

Mini Courses

Model-independent pricing and hedging of derivatives
David Hobson

The standard approach for the pricing of financial options is to postulate a model and then to calculate the price of a contingent claim as the suitably discounted, risk-neutral expectation of the payoff under that model. In practice we can observe traded option prices, but know little or nothing about the model. Hence the question arises, if we know vanilla option prices, what can we infer about the underlying model? At one extreme, if we know a single call price, then we can calibrate the volatility of the Black-Scholes model. (However, if we know the prices of more than one call then together they will typically be inconsistent with the Black-Scholes assumption of a constant volatility.) At the other extreme, if we know the prices of call options for all strikes and maturities, then we can find a unique martingale diffusion consistent with those prices. If we know call prices of all strikes for a single maturity, then we know the marginal distribution of the asset price, but there may be many martingales with the same marginal at a single fixed time. Any martingale with the given marginal is a candidate price process. A right-continuous martingale with a given distribution at time 1, can be viewed as a Brownian motion with a given distribution at a random time. In this way the problem of finding candidate price processes translates to one about embedding distributions in Brownian motion via a stopping time --- the Skorokhod embedding problem. These talks are about this correspondence, and the idea that extremal solutions of the Skorokhod embedding problem lead to robust, model independent prices and hedges for exotic options.

Leading-order corrections in Mathematical Finance
Jan Kallsen

Many aspects of real markets are often ignored in order to obtain tractable models and equations. Here, we focus on transaction costs, jumps, and stochastic volatility but one may also think of illiquidity, random endowment, time discretisation etc. In these lectures we consider such phenomena as perturbations of a given simpler setup. Being interested in quantities such as option prices, optimal portfolios etc., the goal is to obtain general explicit and robust leading-order corrections, which highlight the main effects of the perturbation under consideration. Specifically, we consider the effect of small transaction costs on portfolio choice, welfare, and turnover. Moreover, we discuss first-order corrections of hedging errors and option prices in the presence of jumps and stochastic volatility.

Invited Speakers

Optimal robust bounds for variance options Alexander Cox

Robust, or model-independent properties of the variance swap are well-known, and date back to Dupire and Neuberger, who showed that, given the price of co-terminal call options, the price of a variance swap was exactly specified under the assumption that the price process is continuous. In Cox & Wang we showed that a lower bound on the price of a variance call could be established using a solution to the Skorokhod embedding problem due to Root. In this talk, we describe a construction, and a proof of optimality of the upper bound, using results of Rost and Chacon, and show how this proof can be used to determine a super-hedging strategy which is model-independent. In addition, we outline how the hedging strategy may be computed numerically. Using these methods, we also show that the Heston-Nandi model is `asymptotically extreme' in the sense that, for large maturities, the Heston-Nandi model gives prices for variance call options which are approximately the lowest values consistent with the same call price data.

Optimal Liquidity Provision in Limit-Order Markets Johannes Muhle-Karbe

In today's electronic markets, investors can choose to trade by either market or limit orders. Market orders guarantee immediate execution, but investors have to pay the bid-ask spread for taking liquidity out of the order book in this way. In contrast, limit orders allow to earn the spread by providing liquidity, but a posted order is only executed when a suitable counterparty arrives. We study the resulting tradeoff between profits from liquidity provision and inventory risk in a general setting, allowing for arbitrary preferences, asset price and cost dynamics, and arrival rates. In the limit for small spreads, the corresponding non-Markovian singular control problem can be solved in closed form, leading to explicit formulas for the optimal policy and welfare. (Joint work with Christoph Kuehn)

On Model Uncertainty in Discrete Time Marcel Nutz

We study the problems of arbitrage, superhedging and utility maximization in a nondominated model of a discrete-time financial market. We show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures, that a superhedging duality holds, and that optimal strategies for robust utility maximization exist (arXiv:1305.6008 and arXiv:1307.3597). Based on joint work with Bruno Bouchard.

Root's and Rost's solution of the Skorokhod embedding problem Harald Oberhauser

An intuitive solution of the Skorokhod embedding problem is due to Root and Rost who showed that one can find a subset of time-space such that its first hitting time by time-space Brownian motion solves the Skorohod embedding problem. More recently and motivated by applications in finance these solutions received again increased interest (work of Dupire, Cox--Wang, Carr--Lee, etc). I discuss some applications, connections to viscosity theory, reflected FBSDEs, numerics etc. (Joint work with Goncalo dos Reis).

Two tales in tractable (robust) portfolio optimisation Jan Obloj

We consider portfolio choice problems in continuous time. Traditionally, and distinctively in the framework of maximisation of expected utility, the optimal strategies intertwine the choice of the underlying model and the investor's risk preferences. We seek to advance formulations which disentangle these influences and make things tractable. Our aim is to develop a robust and practically relevant approach to portfolio optimisation. In particular we provide a solid theoretical footing to fractional Kelly strategies. The Kelly strategy, or the growth optimal strategy, depends on the choice of reference model. In contrast, the fraction of wealth invested in it does not. In the first part of the talk, it encompasses the investor's risk attitudes expressed through drawdown constraints. In the second part of the talk, it measures the investor's confidence in the model. The first part of the talk in based on a joint paper with Constantinos Kardaras and Eckhard Platen and I also present general existence and uniqueness results on portfolio optimisation subject to drawdown constraints. The second part of the talk is based on a joint work with Sigrid Kallblad and Thaleia Zariphopoulou and I also present some general duality results on robust forward performance criteria.

Vienna University, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Tel: +43 1 4277 50722