Non-arbitrage in enlarged markets
Anna Aksamit (Université d'Evry, France)
Joint work with Tahir Choulli, Jun Deng and Monique Jeanblanc

Wednesday June 4, 14:30-15:00 | session 5.8 | Trading (Strategies) | room 1+2

Our study addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon or when a random variable satisfying Jacod's hypothesis is incorporated. Precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) condition, which is also known in the literature as the first kind of non-arbitrage. In the general semimartingale setting, we provide a necessary and sufficient condition on the random time for which the non-arbitrage is preserved for any process. Analogous result is formulated for initial enlargement with random variable satisfying Jacod's hypothesis. The crucial intermediate results in enlargement of filtration theory are obtained. For local martingales from the reference filtration we provide special optional semimartingale decomposition up to random time and in initially enlarged filtration under Jacod's hypothesis. An interesting link to absolutely continuous change of measure problem is observed. The importance of thin random times is remarkable for our non-arbitrage considerations. In fact that is our motivation for the analysis of thin random times. We classify random times into thin and strict random times. Taking as a starting point assumption on avoidance of all stopping times from the reference filtration we define a class of thin random times. Then we define a decomposition of a random time into thin and strict parts in analogous way to the stopping time decomposition into accessible and totally inaccessible parts. The notion of dual optional projection plays a crucial role. Furthermore we develop properties of thin random times, namely relationship of thin honest times with a jumping filtration, and entropy of a thin random time.

Modeling capital gains taxes for trading strategies of infinite variation
Björn Ulbricht (Goethe University, Germany)
Joint work with Christoph Kühn

Wednesday June 4, 15:00-15:30 | session 5.8 | Trading (Strategies) | room 1+2

In this article we show that the payment flow of a linear tax on trading gains from a security with a semimartingale price process can be constructed for all c\`agl\`ad and adapted trading strategies. It is characterized as the unique continuous extension of the tax payments for elementary strategies w.r.t. the convergence “uniformly in probability”. In this framework we prove that under quite mild assumptions dividend payoffs have almost surely a negative effect on investor’s after-tax wealth if the riskless interest rate is always positive. (paper available on http://arxiv.org/abs/1309.7368)

VWAP order execution and dynamic trading volume estimation
Christoph Frei (University of Alberta, Canada)
Joint work with Nicholas Westray

Wednesday June 4, 15:30-16:00 | session 5.8 | Trading (Strategies) | room 1+2

We consider the optimal liquidation of a position of stock (long or short) where trading has a temporary market impact on the price. The aim is to minimize both the mean and variance of the order slippage with respect to a benchmark given by the market VWAP (volume weighted average price). In this setting, we introduce a new model for the relative volume curve which allows simultaneously for accurate data fit, economic justification and mathematical tractability. Under the assumption of complete information (observability of relative trading volume), we give explicit formulae for the optimal trading rate and liquidation trajectory by tackling the resulting optimization problem using a stochastic control approach. We then study how this strategy can be implemented using dynamic trading volume estimation and analyze its performance for the stocks in the DJIA. The talk is based on a recent article (accepted in Mathematical Finance) and ongoing work both joint with Nicholas Westray (Deutsche Bank AG).