How to extrapolate the data?
You are never going to be able to fill in the gaps exactly, so the best approach may depend on what you want to use the data for.  

One reasonable approach might be to use an average offset from the 10-year rate, e.g. adding about $0.3$ to it.  If you did that, then you would get the red data on the chart below, which compares reasonably well with the actual blue data for the 15-year rate, though not perfectly.  So you might use the red line as a proxy for the earlier missing blue data.  This might not be such a good idea if the purpose of your analysis was precisely considering the changing gap between the 10-year and 15-year rates