In the simple Black-Scholes model, we can replicate an option by investing its $\Delta$ in the underlying, and keeping that portfolio self-financing via the bank account.
I have two questions. I don't necessarily expect answers served on a platter, some references that I can read on my own would be okay too:
- Why does that strategy work? I understand it on an intuitive basis, but I don't know how to prove it via the math.
- What would you do in more general situations where the payoff of the option depends on something other than the domestic stock asset? Say it was a foreign stock asset. If the exchange rate is X (stochastic), what is the corresponding replicating portfolio?