I do not know if it is totally off-topic, but I thought it might be useful to have opinions and an aggregate answer about why volatility is an important topic in financial econometrics.
I think it started with portfolio theory and the need to understand the properties of the underlying second moment of the asset returns. Subsequently the Black-Scholes formula and the popularity of derivatives made this entity very important in Finance.
Past volatility in the price of something is a measure of the inability of the past to predict the present, as otherwise prices would largely change smoothly just reflecting time costs, and so in many (but not all) cases it could be an indicator of how difficult it might be for the present to predict the future.
Hence it becomes an indicator of risk, and affects the values of derivatives: buying an option will tend to be more expensive if both parties believe prices are likely to be volatile in future and the option is more likely to be exercised.
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