We consider the problem of estimating the true Sharpe ratio of an asset selected for having the highest observed in-sample Sharpe ratio among many assets. We discuss estimators based on the polyhedral lemma, James Stein shrinkage, debiasing the expected maximum Sharpe ratio, thresholding and empirical Bayes. We test these estimators in simulations, computing bias and root mean square error across different values of sample size, number of assets, and spread and shape of population Sharpe ratios. We also compute rank correlation of the estimators against the underlying quantity, simulating how these estimators might be used to compare or rank the output of different teams which perform this selection process. We find that the James Stein estimator provides the best performance across many different realistic values of the relevant parameters, followed by the GMLEB estimator of Jiang and Zhang. These results are fairly robust to correlation of asset returns, with some caveats.

We propose a stochastic game modelling the strategic interaction between market makers and traders. From the trader's perspective, the conventional exogenous permanent price impact is replaced by the endogenous quoting strategies of the market makers. Conversely, from the market maker's perspective, order flows are no longer assumed to be exogenous, but are driven endogenously by the strategic traders. Characterizing the Nash equilibria via forward-backward stochastic differential equations (FBSDEs), we establish a local well-posedness result for the general game. For the specific `Almgren-Chriss-Avellaneda-Stoikov' model, the decoupling approach guarantees the global well-posedness of the FBSDEs by reducing it to a backward stochastic Riccati equation with $M_+$-matrix coefficients. Finally, by introducing small diffusion terms into the inventory processes as an approximation to the general game, we establish its global well-posedness. Simulations reveal a negative correlation between quotes and strategic orders, in contrast to the positive correlation observed between quotes and noise orders.