Good morning everybody,
i have a doubt concerning volatility and its connection with options payoff, and hope that someone will help me understand. First of all i apologize for my very basic language, i'm not a matemathician.
I come to the point.
Intuitively, the price and the payoff of options ( i refer in particular to equity indexes, as the Dax ) should be positively correlated with volatility of underlying. The greater the volatility, the greater , on average, the price of options, and also the payoff.
so i made the following test, on the Dax index serie from 1980 to today:
i took the realized volatility of the index , at 1 month and at 1 year, and associated with the relative real move of the underlying index, the dax, in absolute value ( simply the percent difference between the index value at end of period and at beginning, in absolute value ). I then calculated the correlation between the two series, and obtained a coefficient close to 0.5 for the 1 month serie, ( as i expected a good positive correlation ), and a coefficient very close to zero for the 1 year serie. Should i conclude that volatility really affects the amplitude of the underlying moves in the short period, but not in the long? does this mean that b&s formula is less efficient in calculating prices for leaps? or is it simply some conceptual mistake in my reasoning?
thks
DARIOS   

Taking the last question first, you could use the BS formula (and your own best volatility estimate) to get a ballpark estimate for a leap price. If it differs from the market price, you're probably best off concluding the market is 'fair' and the model is weak. Ditto for any other listed option price, regardless of maturity.

Re the first question, my suggestion is to first assume the BS assumption of geometric Brownian motion (GBM) is correct and learn how to simulate daily prices under GBM -- say 40 years worth. Now re-do your correlation tests with the simulated data. I suspect you'll find more or less the same thing that you found with the DAX data. I think the right conclusion is that the absolute return (from the beginning to the end of a very long period) is a very poor approximation to the realized volatility computed from, say, daily returns. And that's true regardless of the 'true' process that generates the prices.    

Thks Alan,
i'll follow yr suggestion and make same test on a simulated serie. Will post results. 

First of all, the price of the option and the implied vol is more than correlated, it's bijection and options are often quote in term of implied vols. 

Now coming back to your analysis, imagine you calculate the realised vol and the return over the maximum period, there is no particular reason they should be a link between those two numbers. Now more interesting you should do is take your realised vol versus return - slice by quantile relative to the return and take the average realised: you will a skewed realised vol similar the implied vol skew - you will also see that the steepness is more pronounced for 1m versus 1y. you can also do the correl beween return and realised on those sub-set, might shed some like on what acutally happened. 

EndoftheWorld thks for yr interesting suggestion. But pls i have to ask you a clarification of yr reply, you just demolished one of the pillars of my very basic comprehension and knowledge of option theory! That is: option price is positively correlated with volatility, and option price is discounted option payoff at expiration. So in some way there must be a positive correlation between  volatility and option payoff, and indeed i found that evidence but only for short period expiration ( 1 month). But in yr reply you say: "Now coming back to your analysis, imagine you calculate the realised vol and the return over the maximum period, there is no particular reason they should be a link between those two numbers" so seemengly for you there should not be any correlation between the two....which i don't understand...pls help thks

In a non-mathy way:

OptionValue = FwdValue + VolValue

If you "freeze" all parameters except vol: 

If Vol goes up, then VolValue goes up  ==> if you freeze FwdValue, then OptionValue ALWAYS goes up
If Vol goes dn, then VolValue goes dn ==> if you freeze FwdValue, then OptionValue ALWAYS goes dn

It's just stronger than "correlated with"