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I encountered this question: "A trader observed that the implied vol of OTM calls and puts are higher than that of ATM option, what is the best strategy among: Calendar Spread, Bull Spread, Bear Spread and Butterfly ?"

Is this a trick question? For me, what the trader observed is simply the standard volatility smile shape, which for foreign currency options is given below: Volatility smile foreign currency option

Only if other traders didn't knew this volatility shape and were using the regular Black-Scholes pricing model, could the trader capitalize on this discovery. Then, the calls with high strike price would be under-priced and the puts with low strike price would be also under-priced. The trader then would want to buy lots of calls with high strike price and lots of puts with low strike price. Besides, none of the payoffs above take advantage of this property. (Maybe shorting a butterfly spread?)

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    $\begingroup$ I concur with you, the volatility premium in the wings is not necessarily an arbitrageable feature. you would be long the body short the wings if there was a way to tell that the realized distribution was tighter than the implied distribution. however, you could not know that in the absence of further information $\endgroup$ – ZRH Apr 2 at 5:33
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Firstly, remember that in general vega is positive for all options. Hence, the fact that the implied volatility is higher in the wings (high and low strikes, i.e. deep ITM/OTM) means that these options are over-priced. Thus, you would want to sell/short these options. By the same logic, you would want to buy/long options around the ATM point where implied volatility is low. Therefore, you would want to short the butterfly spread (since you believe realized volatility will increase in the future).

As you mention, it is correct that this mispricing is compared to a Black-Scholes framework. We know that in reality, the assumptions in B-S are not consistent with what we expect to see in the actual markets. Hence, if you want to implement a vol strategy in real life, you cannot simply use the above strategy.

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