Markowitz (1952) portfolio selection requires an estimator of the covariance matrix of returns. To address this problem, we promote a nonlinear shrinkage estimator that is more flexible than previous linear shrinkage estimators and has just the right number of free parameters (i. e., the Goldilocks principle). This number is the same as the number of assets. Our nonlinear shrinkage estimator is asymptotically optimal for portfolio selection when the number of assets is of the same magnitude as the sample size. In backtests with historical stock return data, it performs better than previous proposals and, in particular, it dominates linear shrinkage.

Plotting daily stock returns against themselves with one day's lag reveals a striking pattern. Evenly spaced lines radiate from the origin; the thickest lines point in the major directions of the compass. This 'compass rose' pattern appears in every stock. It is caused by discreteness. However, counterexamples demonstrate that the existence of exchange-imposed tick sizes (e.g., eighths) is neither necessary nor sufficient for the compass rose. The compass rose cannot be used to make abnormal profits: it is structure without predictability. Among other consequences, the compass rose may bias estimation of ARCH models and tests for chaos.

Plotting daily stock returns against themselves with one day's lag reveals a striking pattern. Evenly spaced lines radiate from the origin; the thickest lines point in the major directions of the compass. This “compass rose” pattern appears in every stock. It is caused by discreteness. However, counter‐examples demonstrate that the existence of exchange‐imposed tick sizes (e.g. eighths) is neither necessary nor sufficient for the compass rose. The compass rose cannot be used to make abnormal profits: it is structure without predictability. Among other consequences, the compass rose may bias estimation of ARCH models, and tests for chaos.

We develop an approach to asset pricing in incomplete markets that bridges the gap between the two fundamental approaches in finance: model‐based pricing and pricing by no arbitrage. We strengthen the absence of arbtrage assumption by precluding investment opportunities whose attractiveness to a benchmark investor exceeds a specified threshold. In our framework, the attractiveness of an investment opportunity is measured by the gain‐loss ratio. We show that a restriction on the maximum gain‐loss ratio is equivalent to a restriction on the ratio of the maximum to minimum values of the pricing kernel. By limiting the maximum gainloss ratio, we can restrict the admissible set of pricing kernels, which in turn allows us to restrict the set of prices that can be assigned to assets. We illustrate our methodology by computing price bounds for call options in a Black‐Scholes economy without intermediate trading. When we vary the maximum permitted gainloss ratio, these bounds can range from the exact prices implied by a model‐based pricing approach to the loose price bounds implied by the no‐arbitrage approach.

Markowitz (1952) portfolio selection requires an estimator of the covariance matrix of returns. To address this problem, we promote a nonlinear shrinkage estimator that is more flexible than previous linear shrinkage estimators and has just the right number of free parameters (i.e., the Goldilocks principle). This number is the same as the number of assets. Our nonlinear shrinkage estimator is asymptotically optimal for portfolio selection when the number of assets is of the same magnitude as the sample size. In backtests with historical stock return data, it performs better than previous proposals and, in particular, it dominates linear shrinkage. Received January 21, 2014; editorial decision January 25, 2017 by Editor Geert Bekaert.

This paper offers a new approach to estimating time-varying covariance matrices in the framework of the diagonal-vech version of the multivariate GARCH(1,1) model. Our method is numerically feasible for large-scale problems, produces positive semidefinite conditional covariance matrices, and does not impose unrealistic a priori restrictions. We provide an empirical application in the context of international stock markets, comparing the new estimator with a number of existing ones.

Multivariate GARCH models do not perform well in large dimensions due to the so-called curse of dimensionality. The recent DCC-NL model of Engle et al. (2019) is able to overcome this curse via nonlinear shrinkage estimation of the unconditional correlation matrix. In this paper, we show how performance can be increased further by using open/high/low/close (OHLC) price data instead of simply using daily returns. A key innovation, for the improved modeling of not only dynamic variances but also of dynamic correlations, is the concept of a regularized return, obtained from a volatility proxy in conjunction with a smoothed sign of the observed return.