The risk and return profiles of a broad class of dynamic trading strategies, including pairs trading and other statistical arbitrage strategies, may be characterized in terms of excursions of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a Markovian setting.

We introduce the notion of $\delta$-excursion, defined as a path which deviates by $\delta$ from a reference level  before returning to this level. We show that every continuous path has a unique decomposition into such $\delta$-excursions, which turns out to be useful for the scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss, and drawdown.  As $\delta$ is decreased to zero, properties of this decomposition relate to the local time of the path.

When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent $\delta$-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursions match those observed in a data set.

This is based on joint work with Anna Ananova and Rama Cont (Oxford).

The workshop will take place August 25-28 in Berlin. More information will be posted here soon. The lead organiser is Dirk Becherer. 

We study a mean field utility maximization game, where each player manages a stock whose return process depends on a hidden factor, which cannot be observed by the manager. The manager needs to infer the return process and rewrite the dynamics of the stock price based on the information available to her. Moreover, each manager is concerned not only with her own terminal wealth but also with the relative performance of her competitiors. We use the probabilistic approach to consider exponential and power utilities. Due to the mean field interaction and the nature of the game, the FBSDE systems characterizing the equilibria of our problem become coupled mean-field FBSDEs with possibly quadratic growth. We establish the well-posedness result of the mean-field FBSDEs in some suitable BMO space; firstly we work on a short time interval and secondly the local solution is extended to an arbitrary interval by considering the corresponding variational FBSDEs.

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in a path-dependent setting and under weak regularity conditions, that any solution to this dual problem actually allows one to construct explicitly a perfect hedging portfolio. From a pure probabilistic point of view, our approach also allows one to exhibit solutions to a specific class of second order forward backward stochastic differential equations, in the sense of Cheridito et al. (2007). Existence of a solution to the dual optimal control problem is also addressed in particular settings. As a by-product of our arguments, we prove a version of Itô’s Lemma for path-dependent functionals that are only Cˆ{0,1} in the sense of Dupire.