Why is volatility an important topic in financial econometrics? 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.
 A: 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.
A: I think the main reason is that many financial time series exhibit high volatility and the standard ARIMA models do not fit well to data with high volatility.  So special time series models that account for this are important to generate better predictions.
The ARIMA models are well established while time series models such as GARCH that model volatility are newer and more open for extensions and theoretical development.  These are reasons why this topic would be appealing to academics.
A: I'm not an economist, but my guesses would be that: 1) there are several options/futures-based techniques for profiting from high volatility, and 2) higher volatility corresponds to greater risk in some sense, or at least investor confidence/nervousness.
A: Modern Portfolio Theory (MPT) and the Efficient Market Hypothesis (EMH) share important assumptions.  Amongst them is the assumption that all investors are at all times profit maximizing, rational and risk-averse.  If this is the case then excess vol and vol clusters are said to violate a strict form of the EMH since this indicates that prices may deviate from fundamentals.  Of course, excess vol/vol clusters are seen at all levels of granularity in financial time series.   
Financial economists seek to explain why excess vol exists and how best to incorporate it in their models.  While the ideas about how to model vol don't necessarily come from financial economists, important ones, like ARCH/GARCH did, and were subsequently incorporated by financial firms in their pricing models and trading strategies.
A: Volatility also has an application that is rather important, which is in options pricing. The famous Black-Scholes model incorporates future volatility in its equation. So being able to forecast volatility with a high degree of accuracy is crucial in everyday life of derivative trade desk professionals. That is why so much attention is being paid in academia to the research and better understanding of volatility.
A: Volatility is a very important concept in finance for the simple reason that it matters how you get there. Let's say you need to get from Chicago to Indianapolis. You have two options to get there, you could drive your nice luxury SUV with air conditioning and cruise control or you could hop into a railcar as a stowaway. Its summer and the railcar is 110 degrees inside and you're so hot after an hour that you feel like jumping out at a stop only halfway there. You get slammed into the side of the railcar and injure your shoulder and now you can't use your right arm and the pain is killing you. You now want to jump out and do anything to help alleviate the pain. You decide to tough it all out and arrive in Indianapolis three hours later, the same amount of time it took to drive. Both options resulted in the same outcome, getting from point A to point B but the railcar option was extremely uncomfortable and you considered not making the whole trip a few times. 
The same goes for stock returns, the degree of discomfort experienced on the trip to year end stock market gains is expressed by standard deviation and volatility. Excessive volatility will make you want to sell out of a stock or portfolio. It can make the ride extremely uncomfortable. A rational investor, even one who claims he's in it for the long term will be tested to sell out at a bottom with too much volatility.  
With volatility and standard deviation, you can calculate one of the most critical metrics of any portfolio which is risk adjusted return. Formulas like the Sharpe and Sortino ratios help quantify the risk adjusted returns of a stock or portfolio. 
Aside from options pricing, this is the best application of volatility and standard deviation to portfolio construction and financial econometrics. 
