By "running correlation coefficient" I assume you mean you want the correlation computed for each possible length of x and y from 1:1, 1:2, ..., 1:N for some large maximum number N. If N is very large it becomes inefficient to repeatedly compute using 'corrcoef'. Fortunately it is possible to express the correlation in terms of five running sums which can greatly reduce the total computation required. You can see this form in the Wikipedia site
near the end of the first section.
The following code carries out that computation in terms of the Wikipedia formula.
sx = x(1); sy = y(1);
sx2 = x(1)^2; sxy = x(1)*y(1); sy2 = y(1)^2;
c(1) = 1;
for n = 2:N
sx = sx + x(n);
sy = sy + y(n);
sx2 = sx2 + x(n)^2;
sxy = sxy + x(n)*y(n);
sy2 = sy2 + y(n)^2;
c(n) = (n*sxy-sx*sy)/sqrt((n*sx2-sx^2)*(n*sy2-sy^2));
Note that for each step in the loop there are only about twenty operations performed for updating the correlation value c(n), whereas with a large value for N, each step would involve some multiple of n of such operations.
You should not necessarily take the above code literally, but its method may serve to give you a procedure for performing the computation more efficiently for your "running" type situation.