Hello,

I read that the when parameterising the IV curve we should use delta on the x axis with IV on the y axis. However, we need to be careful about choosing a call delta or a put delta because the strike of a 50 call delta will be different from the strike of a 50 put delta.

Is this because an ATM call has delta > 0.5 and an ATM put has delta < -0.5? Thus a call will have a delta of 0.5 below the ATM strike and similarly a put will have a delta of -0.5 above the ATM strike.

Is my reasoning correct?

Bump

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Puts have a negative delta, calls have a positive delta. Always. An ATM option will have a delta of 0.50 (or -0.50), an OTM option will have a delta between that and 0, and an ITM option will have a delta between 0.50 and 1 or -0.50 and -1. The strike of a 50 delta call should be the same as that of a 50 delta put (they will be very slightly different because of the underlying lognormal distribution but this should be trivial). Try reading up on the concept of put-call parity and you'll understand why they should be the same strike. A put and a call option, when properly delta-hedged, are the exact same.

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Puts have a negative delta, calls have a positive delta. Always. An ATM option will have a delta of 0.50 (or -0.50), an OTM option will have a delta between that and 0, and an ITM option will have a delta between 0.50 and 1 or -0.50 and -1. The strike of a 50 delta call should be the same as that of a 50 delta put (they will be very slightly different because of the underlying lognormal distribution but this should be trivial). Try reading up on the concept of put-call parity and you'll understand why they should be the same strike. A put and a call option, when properly delta-hedged, are the exact same.

I agree that for European options for put call parity to hold the implied volatilities must be the same. However, I have found the answer to why there's a difference in implied volatilities between ATM calls and puts in real world.

The put call parity must be held theoretically. However, due to bid ask spreads and liquidity issues observable values of European options do not necessarily follow the put call parity making the implied volatilities of the European puts and calls remain out of whack.

In case of American Options there is no put call parity and hence equivalency of IV between American puts and calls are not guaranteed.

Thanks for helping me realise the fault in my original reason.

"The markets are always changing , and they are always the same."

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The only way I can think of an American put and call option having different IV's is if the strike is ITM for one and OTM for the other, not ATM. In this case you clearly have to consider interest rates. But for an ATM strike the IV should be exactly the same.

The 50 delta strike is the ATM by definition. The strikes can be different for calls and puts i guess because of the forward curve shape. If the forward curve is steeply upward sloping calls are pulled into the money, while puts are dropped out of the money.

Think about the ATM(forward) strike always

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BS delta of an ATMF call will not be exactly 0.5. It's just the feature of the model and the assumed distribution. You can roughly approximate it as Delta(C-ATMF) = 0.5+0.2 * sigma * sqrt(T).

This isn't due to bid/ask or anything like this and it will not invalidate put/call parity.

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