A quasi-metric is just an asymmetrical measure of distance.
Physical distance is measured symmetrically. The distance from Bloomington to Madrid is the same as the distance from Madrid to Bloomington. Obviously! But if you measured investments symmetrically, you would err. In finance, gains are good and losses are bad. Also obvious! But what that means, mathematically, is that there is only one direction: up (OK, there’s also negative up). Asymmetrical distance.
I think of this like the down on a duck’s back, or the hair on your arm. All the hairs are pointing the same way. ”Hairs” = a 1-form. Which assigns to any point in the theoretical financial space a number: your portfolio returns.
OK, that’s a less obvious way to think about it. But the modeling point is robust: any time you’re implementing a financial model, don’t penalize gains! This idea has only recently been incorporated at my brokerage: they now report the forecast likelihood of 10%, 20%, 30% losses to my portfolio — rather than so-called “risk %” which is really standard deviation, squared.