I am reading through famous sources like Hull, Wilmott, et.c., where they construct the replicating portfolio in a one-step binomial model. This involves setting up a system of equations, where we have two unknowns, ∆ and B (shares of underlying stock, and risk-free money), corresponding to equations. The no-arbitrage principles allows us to equate the option and portfolio future states, assuming they produce identical cash flows.

What I kind of intuitively understand but would want a more formal explanation of is why we assume that ∆ must be the underlying asset as opposed to some other asset? Intuitively, it makes sense that we need some correlation between the stock and the option, we can't just have ANY random stock with no relation to the option. But is there some formal assumption that this system of equations makes use of? I have read something about complete markets, where all derivatives can be replicated using other assets, but I haven't found a definite statement about it having to be the underlying.

So we assume the portfolio cash flows is identical to the option. And this must be a portfolio with the underlying stock. But "WHY"?

Thanks

I am reading through famous sources like Hull, Wilmott, et.c., where they construct the replicating portfolio in a one-step binomial model. This involves setting up a system of equations, where we have two unknowns, ∆ and B (shares of underlying stock, and risk-free money), corresponding to equations. The no-arbitrage principles allows us to equate the option and portfolio future states, assuming they produce identical cash flows.

What I kind of intuitively understand but would want a more formal explanation of is why we assume that ∆ must be the underlying asset as opposed to some other asset? Intuitively, it makes sense that we need some correlation between the stock and the option, we can't just have ANY random stock with no relation to the option. But is there some formal assumption that this system of equations makes use of? 

I have read something about complete markets, where all derivatives can be replicated using other assets, but I haven't found a definite statement about it having to be the underlying.

So we assume the portfolio cash flows is identical to the option. And this must be a portfolio with the underlying stock. But "WHY"? Or is there some assumption that only these three securities exist in our market?

Thanks