We know from psychology about “prospect theory”, that a human’s reaction to gain and loss is wildly asymmetric, and powerful enough to change our behaviour when the same payoff structure is presented differently. Good news is not equivalent to bad news; so why should asset returns be any different.
Nor is it unreasonable to suppose that a violation of the assumption of costless and frictionless hedging produces a bias towards frequent near-zero and rare extreme outcomes. If you will, there is some “grey noise” on top of the dominant Gaussian white noise.
None of this would be controversial to the vast majority of market participants. But they’re a different crowd to an academic statistician!
However, here’s a simple mechanism for you. You only have to accept the existence of “greed” and “fear” as distinct emotional states that influence human behaviour in markets.
Then take a long-run series of eg equity market returns, as the archetypal risk asset. Run a two-state Hidden Markov Model on these returns. The outputs will suggest one state with, say, a 70% probability of mean +3% a month with a Gaussian stdev of 4%. Versus a rarer 30% frequency of -5% +/-8%. The analogy with “greed” and “fear” or “calm” and “chaos” are hopefully obvious.
These in aggregate will have a mu and sigma of 7% and 19% per annum respectively, in line with the market’s historical record. Both components are perfectly Gaussian. However, the combination of the two will have left-skew and positive kurtosis, exactly like market history.
You simply have to believe there is more than one “regime” that can drive markets at any point in time. Accept that, and the higher moments will naturally follow.