Gradient-Based Optimization of Utility-Based Shortfall Risk

A new paper on arXiv proposes a gradient-based stochastic optimization method for minimizing utility-based shortfall risk, a risk measure that generalizes value-at-risk and conditional value-at-risk. The authors, from a European research group, show that the approach converges under standard assumptions and can handle large-scale portfolio problems.

The method works by directly optimizing the shortfall risk objective using stochastic gradient descent, avoiding the need for nested optimization or scenario-based approximations. The paper includes convergence proofs and numerical experiments on synthetic and real market data, demonstrating that the algorithm scales to portfolios with hundreds of assets.

Utility-based shortfall risk is defined as the minimum capital needed to make a position acceptable under a given loss function. The new algorithm treats the risk measure as a smooth objective, enabling gradient-based updates that are computationally efficient. The authors prove convergence to a stationary point under convexity and Lipschitz conditions.

Numerical tests compare the method against standard CVaR optimization and scenario-based approaches. The gradient-based method achieves similar risk levels with lower computational cost, especially as the number of assets grows. The paper also shows how to incorporate transaction costs and position limits.

For practitioners, the key result is that shortfall risk optimization can now be done at scale without specialized solvers. The algorithm fits into standard machine learning pipelines and can be updated online as new data arrives. The authors provide code for reproducibility.

The paper is available on arXiv under the identifier 2506.01101. It has not yet been peer-reviewed but has been submitted to a quantitative finance journal.