Nonparametric Sharpe Ratio Function Estimation in Heteroscedastic Regression Models via Convex Optimization

We consider maximum likelihood estimation (MLE) of heteroscedastic regression models based on a new “parametrization” of the likelihood in terms of the Sharpe ratio function, or the ratio of the mean and volatility functions. While with a standard parametrization the MLE problem is not convex and hence hard to solve globally, our parametrization leads to a functional that is jointly convex in the Sharpe ratio and inverse volatility functions. The major difficulty with the resulting infinite-dimensional convex program is the shape constraint on the inverse volatility function. We propose to solve the problem by solving a sequence of finite-dimensional convex programs with increasing dimensions, which can be done globally and efficiently. We demonstrate that, when the goal is to estimate the Sharpe ratio function directly, the finite-sample performance of the proposed estimation method is superior to existing methods that estimate the mean and variance functions separately. When applied to a financial dataset, our method captures a well-known covariate-dependent effect on the Shape ratio.