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Coming from here: https://quant.stackexchange.com/a/7616/43679 we have that for a European option, and due to the put-call parity, due to the non arbitrage rule, the volatility for a put and a call with the same strike have to be the same.

Does this rule still met for Asian options so that, if the option is Asian, having the premium of two options a call and a put with the same strike and valuating them using an Asian model such as Turnbull-Wakeman, both the volatility for the call and for the put have to be the same?

If you can give me some references/papers as I have searched a lot about this and I didn't find anything

Thank you in advance.

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    $\begingroup$ The put-call parity really only states $(x-K)^+-(K-x)^+=x-K$, where $x$ is the underlying of the option (for vanillas, $x=S_T$). Note that $\mathbb{E}^\mathbb{Q}[S_T]$ does not depend on volatility due to the martingale property. Thus, vanilla options have the same vega. For Asian options, the put-call parity is $$\text{AsianCall}-\text{AsianPut}=e^{-rT}\left(\mathbb{E}^\mathbb{Q}[A_T]-K\right),$$ where $A_T$ denotes the relevant average. In general, the expected average could depend on volatility, I reckon. It is independent of volatility for discrete, arithmetic averages though. $\endgroup$ – Kevin Apr 6 at 12:03

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