Itô's lemma, in plain English
Background mathsPrices move with a random component. Itô's lemma is the rule for how a function of such a random price changes over time. The extra ½σ² term is the surprise: randomness itself changes the average path. This is the maths behind option pricing and volatility models, and quantitative software often reports it next to statistics such as skewness, kurtosis, the Hurst exponent and ACF(1). It is background maths that describes how uncertainty behaves. It does not predict where a price will go.