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How can exotic and other path dependent, such as asian options be hedged? For example in the case of an asian option, what is the replicating portfolio: what instruments to keep in it and “how much”?

It is known from the standard Black-Scholes model that when we replicate a vanilla European we have to hold $\Delta_{t}$ (the partial derivative of the option PV corresponding to the variable of the stock price) underlying in every $t$, but how is the replication of an asian option (or any other exotic option) maintained in theory/practice?

In general, literature firstly always discuss what the price of an option is as calculating a tipically very tough expectation. It is always good to know what the price is, but the other important question is how to hedge these options, i.e. what strategy to use in order to construct a replicating portfolio. I think this second question is rarely discussed, even though it is probably more important then knowing the price. (Additionally, in my opinion determining the price is also part of the “strategy”, but it is just my opinion.)

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If you've hedged away delta using a replicating portfolio, you become exposed to implied volatility, hence vanilla options are used to hedge exotics.

In regards to your 2nd question, you are thinking of pricing derivatives backwards. Risk-neutral pricing is just an accounting formula and the stock dynamics is the conclusion of that formula.

We construct a portfolio, $\Pi$ that is short an option, $V$ and long $a$ shares.

$$\Pi = - V + aS$$ $$d\Pi = ...$$ Then using the risk-neutral accounting formula and the Feynmann-kac formula, we get the Black-Scholes PDE. The black-scholes SDE is the probabilistic representation of the accounting PDE.

We don't require any assumptions about the dynamics of the stock so there is no "prediction". The Black-Scholes model is a consequence of the accounting formula. That's why the model was awarded a Nobel prize. Previous models assumed some type of asset dynamics to price options, whilst in Black and Scholes' paper, they didn't need to make any assumptions of the underlying’s dynamics.

It's the same with the Heston model. We don't assume the stock's dynamics is like the Heston model - it's a consequence of hedging volatility.

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    $\begingroup$ "in Black and Scholes' paper, they didn't need to make any assumptions of the underlying’s dynamics". Is that so? $\endgroup$
    – nbbo2
    Commented Dec 18, 2023 at 13:50
  • $\begingroup$ @nbbo2 May have gotten ahead of myself with the specifics, but the point still remains. You can construct your P&L and get the BS PDE without assuming stocks are lognormal... $\endgroup$ Commented Dec 19, 2023 at 4:18
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    $\begingroup$ Sorry I can't agree with your last comment. It is true that I didn't read the original paper of Black and Scholes, so in this sense my opinion is irrelevant, however I think I saw the derivation of the Black Scholes PDE (, with the reasoning of Black and Scholes). In that derivation, (in my opinion) it is indeed used that the underlying is some kind of GBM (so some kind of lognormal process), specially when the quadratic variation of the spot dynamic is calculated... $\endgroup$
    – Kapes Mate
    Commented Dec 19, 2023 at 21:56
  • $\begingroup$ ... i.e. : $d\left[S\right]=\sigma^{2}S^{2}ds$. So the $\sigma^{2}S^{2}$ appears in front of the $ds$ term, because the “background source of randomness” is a Wiener-process. If it wasn't a Wiener-process, then I don't think it would be guaranteed that an other $ds$ term appears, so I doubt that the lognormality assumption (i.e. the fact that the underlying is a GBM) isn't used in the derivation. $\endgroup$
    – Kapes Mate
    Commented Dec 19, 2023 at 21:56
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    $\begingroup$ Thanks for the recommendation. But it is just hard to belive, that the $\left\langle \left(\frac{\delta S}{S}\right)^{2}\right\rangle =\hat{\sigma}^{2}\delta t$ equation that you use is always true for any underlying dynamics. As far as I know, the only continuous local martingale that has quadratic variation $t$ is the Wiener-process (see: Levy's theorem) and this is exactly what I wanted to point out in my comments above: anytime when a $dt$ term appears, it is not a bad guess that the driving process is some kind of Wiener-process, because it certanly has quadratic variation $t$. $\endgroup$
    – Kapes Mate
    Commented Dec 20, 2023 at 0:33

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