The reason I don't know of any banks who would offer this to you (even if you held the investment account with their bank) is that there is no upside to the bank. It is a good idea for you, but what would they have to gain from this arrangement?
The reason banks require a down payment is underwriting quality. If you can afford a significant down payment, they know that there is a significantly lower chance that you will default. However, if you were to provide an investment account as collateral, you would receive all the upside, and any downside would reduce their collateral as a percent of the amount loaned.
This sort of idea could potentially work along the lines of a margin call (ie you have to provide additional capital if your asset value drops), but this would have the effective of leveraging the bank's risk, when their objective is to lower their risk through requiring a down payment. I don't see a reason why the bank would take on the risk that you would need to provide additional capital down the road with no upside for them. Additionally, many banks have backed away from the kinds of zero-down-payment and negative-amortization-ARM loans that got them (or the people they sold them to) in trouble over the last few years in an effort to reduce how much risk they take on.
I think that in theory, you'd have to offer a lot more benefit to the bank, and that in practice it's probably a non-starter right now.