Ideally you'd be looking for a vol model-independent replication. Seems unlikely --- but then so often in quant finance things I think are unlikely seem to work out, it's one of the fundamental laws that quant finance is v easy!
Yes, and yes it should be relatively easy, but unfortunately on the second point, 95% of the quant finance papers I read nowadays I have no clue what they're talking about and how it can be applied. But I guess that's my problem.
Thinking about the problem below (or above when not logged in) a bit more, I think dynamic replication in the sense of dynamically rebalancing a portfolio of options, is perhaps the way to approach it.
For instance, at T_0 I hold a portfolio of forward start options to replicate the value of
[$] \sqrt{C^2(S_1,T_1,S_1,T_2)/S_1^2} [$]
at [$] T_1 [$].
At [$]T_1[$] I liquidate the portfolio and receive the cash value, and set up the portfolio that replicates the value of
[$] \sqrt{C^2(S_1,T_1,S_1,T_2)/S_1^2 + C^2(S_2,T_2,S_2,T_3)/S_2^2} [$]
at [$] T_2 [$] using a forward start portfolio of option again, where now [$] C^2(S_1,T_1,S_1,T_2)/S_1^2 [$] is just the known constant cash amount, which I have replicated in [$] [0,T_1] [$], and so forth.
I don't know, do we think that this should or could work? I think the strategy is self-financing? If above dynamic replication strategy is correct, then the question is reduced to how to replicate [$] \sqrt{C^2(S_1,T_1,S_1,T_2)/S_1^2} [$] at t=0, since at T=1,2,3, etc it's basically the same type of replication (with some additional cash/constant value under the square root).