Asymmetric Loss Functions

Standard regression loss functions, like squared error, treat an overestimate and an underestimate of the same size as equally bad. But in many trading and risk contexts, the two kinds of mistake have genuinely different costs: underestimating a risk model's volatility forecast can leave a book dangerously under-hedged, while overestimating it just costs a little extra in unused hedges. An asymmetric loss function bakes this imbalance directly into training, so the model is pushed to make the cheaper kind of mistake more often than the expensive one.

A simple example is the "linex" loss, which penalizes underprediction exponentially harder than overprediction (or vice versa, depending on a sign choice), instead of the parabola shape of squared error that treats both sides identically. Another common choice is the pinball loss used in quantile regression, which weights over- and under-shoots by different factors, $\tau$ and $1-\tau$, to train a model that targets, say, the 90th percentile of an outcome rather than its average — useful for a risk model that specifically cares about not underestimating a rare large loss.

Concretely, suppose underestimating next-day volatility costs a desk 3 times as much (in realized losses from being under-hedged) as overestimating it costs in wasted hedging. Training with a loss that weights underprediction errors 3x relative to overprediction errors will shift the model's forecasts systematically higher than a squared-error model would produce, deliberately trading a little average accuracy for a large reduction in the costly kind of error.