I just want to add to vonjd's answer some info on the comparison of the 3 methods. This is too big for a comment so I'm posting as a separate answer but please upvote his answer, not mine.
Do the differences in methodologies matter in practice?
To gauge the practical importance of the biases in methods 2 and 3, we calculate the weighted stock correlation for the stocks in the S&P 500 index during the period January 2002 through March 2004. For each month in our sample, we use the daily total returns of each of the S&P 500 constituents to calculate the pair-wise correlations needed in method 1, and the single stock volatilities needed in methods 2 and 3. In addition, we calculate the volatility of the S&P 500 index based on its daily total returns during the month, and we use the start-of-month index weights to obtain the weighted average stock correlation.
Exhibit 2A shows the resulting weighted average cross-stock correlations for each of the 27 months in the sample based on each of the three calculation methods. The choice of method has a modest impact on the average correlation number for a well-diversified portfolio or index such as the S&P 500. Exhibit 2B makes this point even clearer by plotting the differences between the correlations numbers obtained from each of the methods. The absolute difference in correlation fell below 0.05 during the past 2+ years. Exhibit 2B also visualizes the consistent upward bias in method 3 as compared to method 2, but the overestimation is less than 0.01 in absolute value.
Exhibits 3A and 3B analyze the difference between methods 1 and 2 further, by looking at some crude measures associated with the volatility bias in method 2 identified above. They suggest that larger differences tend to happen more often in periods when average stock volatility is higher or when stock volatility has changed by a larger amount.