Is there a way to retrieve the implied correlation between stock price and zero coupon bonds?
as in the question about average/implied correlation to do this in a straight way (in a perfect world where all this were given) you would need: immplied vol for the stock, implied vol for the bond and implied vol for an option on a portfolio/basket that contains both assets. If you get a quote for the latter then this sounds possible.
I have never seen an exchange traded product with a basket of stocks and bonds - so you would need something OTC. Using such OTC prices I would be careful do derive further conclusions.
As Richard pointed out you will need the implied volatility (iV) for the option on the stock, the iV of an option with the zero bond as underlying and the iV of an option on the stock and zero bond combined (most likely an OTC derivative).
You can then easily derive the implied correlation :
impliedCorrel = (pow(iV[stock],2) + pow(iV[zBond],2) - pow(iV[otcDeriv],2)) / (2*iV[stock]*iV[zBond])
(sorry I am not good at laTeX, and credit to Richard for having first pointed out the implied volatility of the basket option)
Implied correlations can be more easily derived for currencies, where you would just replace the iV of the otc derivative by the iV of the cross currency option contract (most use otc contracts, others futures options on the currencies).
In either way, make sure you pick the implied vol of the at-the-money-forward and matching expiries for a more accurate result.