How to estimate tail distribution

I got asked this question during an interview with a bank for their market risk department:

You have a large set of historical data on stock price, however there are very limited data in the tail(say beyond the 95% percentile). Assuming you don't have the distribution for the data, how do you estimate/predict the tail/extreme events.

I was thinking about maximum likelihood estimation since I was given the data and I just need to calibrate the parameters so that $$L(\theta|x_1, X_2...X_n)$$ is at maximum.

But does it mean I need use non parametric methods?

I assume this is more of a open ended question, so any thoughts will be welcomed :)

• @NormalHuman Here I think your bot is in error ("question" by itself adds no information, but "interview question" adds information about the open-endedness of the question). – Ian Nov 23 '15 at 18:02
• What do you want to know/prove about the tail? If, say, you want to ensure that the distribution has finite mean, then there is actually nothing you can do without postulating a model for the tail. In other words, without postulating a model, basically all you can do is make statements whose truth is random, and assert that they are true with some large probability. – Ian Nov 23 '15 at 18:05
• I guess, since it concerns a bank, that the theme would be risk.. How extreme are extreme events? – fritzenbauer Nov 23 '15 at 18:07
• @fritzenbauer yes, its about risk! – butterbetter Nov 23 '15 at 18:08
• @Ian Still incidental; compared to the important fact that distribution is unknown. I would rather have the latter in the title than the former. – user147263 Nov 23 '15 at 18:08