# Correlation of Asynchronous Brownian Motion

I am trying to use the closing prices of the S&P 500 and the Nikkei Index to see how they are correlated (assuming they are exactly 12 hours apart). In order to test my method, I have generated two Brownian Motions with a known correlation of 1/2. I then take the odd-numbered data points of the first Brownian Motion and the even-numbered data points from the second, and I test my method on these data sets. However, nothing I do seems to work. I have tried the naive approach of shifting one of the BM's over, I have tried to determine the correlation of the "returns" (I know the intervals are i.i.d.), I have tried a naive data-filling method (insert the midpoint of each pair of data points into the BM), and I have tried to create a new path of the midpoints of one of the two BM's and tried finding the correlation of this, and the other BM. All of these methods consistently get values close to zero. Does anyone have any ideas on how to go about getting the correlation from asynchronous data? I understand that the intervals of Brownian Motions are i.i.d., so I am thinking I should use this to my advantage.

• How are you choosing the correlation to simulate the paths to begin with? If you're getting them from best of options on the close price (or some other market implied way) then you already have the numbers you need, so I don't see why you need to do this. – will Oct 16 '16 at 17:06

Bram made a good point about looking at longer-term returns, but that weakens the quality of estimates.

Here are two sources that address the issue directly - a formal approach here and one specifically in financial context here.

Closing prices should be very highly correlated, I assume you care about close to close returns instead.

Given the frequency of the data that you seem to be looking at (ie you don't seem to be looking at correlation of the futures return intraday), I assume this is for some sort of modelling/pricing over a longer horizon.

What I believe most people do here is to look at correlations for 3 or 5 day return periods. The idea here is that in that case common factors driving returns show up again, because as you noticed, on a single day basis there is a de correlation effect due to time zone differences.

Have you tried to simulate both processes together from US close -> JP close -> US close -> JP close and so on? Where the correlation is fixed, but the volatility of each step is proportional to the square root of its length. And then pick US close and JP close points to build your two series.

You will find that the correlation of the asynchronous ones is the input correlation scaled down by the non-overlap fraction. But you don't need to simulate to show that, you can do it analytically.

Which means that you can directly simulate two series with correlation scaled down appropriately.

Brownian motion should be i.i.d, as you noted.

Subsets of two correlated series are not necessarily correlated.

The correlation of a correlated series with one lagged will simply yield the autocorrelation times the initial correlation of the lagged series.