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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:

  1. Why does that strategy work? I understand it on an intuitive basis, but I don't know how to prove it via the math.
  2. 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?
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    $\begingroup$ 2. The replicating portfolio will contain a bank account and as many assets as there are random variables that affect the option (for example Stock and Foreign Exchange), $\endgroup$ – noob2 Feb 28 '17 at 13:38
  • $\begingroup$ Can you point me to some literature that explains this in detail, so that I can determine the portfolio in any given sitaution? Is there a simple technique, or is it a complicated process for each option? $\endgroup$ – Jaood Feb 28 '17 at 13:39
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    $\begingroup$ See discussion in this question and the reference mentioned there. $\endgroup$ – Gordon Feb 28 '17 at 21:39
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In the Black-Scholes world, one assumes that there is only one source of uncertainty in the model (namely the value of the underlying asset). So, to hedge, you only need to invest in a risk free asset and the underlying. In general, if your payoff dépends on other sources of risk, your hedging portfolio would also have to depend on those sources of risk.

One can think of a hedging portfolio as a Taylor approximation of the value of the asset. So, if the value of your asset is dépendent on several variables, your Taylor expansion (i.e. your hedge portfolio) will have to also depend on those variables.

As far as litterature on the subject goes.......I can't think of one in particular. Bjork's book "Arbitrage theory in continuous time" might be a good place to look.

Hope this helps!

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