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Recently I have been working a lot with implied volatility and volatility surfaces.

The basic idea is easy to follow: 1) Gather market prices of options at different (Strike,Expiry) 2) Calculate implied volatilities 3) Interpolate/Extrapolate as needed to emulate a continuum of option market prices.

This question concerns point 1), namely the selection of prices and what options to use.

Let's say I use European option prices:

Would it be OK to mix put/call prices such that I only ever calculate implied volatility for in-the-money options?

If so, I assume this surface can then immediately be used to calculate a fair value for an in-the-money call option (even though the implied volatility is calculated from the put option at the same (Strike,Expiry)?

Does it make sense to price American options using this volatility surface constructed from European options?

Let's say I use American option prices:

Does this give additional value as opposed to using a surface constructed from European option prices?

Is it again possible to mix put/call prices into the same surface? Or does that not make sense?

Is it at all possible to mix american option prices with european option prices?

As you may guess, I have had a hard time finding articles or papers that discuss how volatility surfaces should be used in actual practice. Do I need one surface for puts, one for calls, and one each for european and americans? Or do I just need a single surface consisting of a mix of european put and calls?

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Would it be OK to mix put/call prices such that I only ever calculate implied volatility for in-the-money options?

No. Use OTM options because they usually have narrower bid-ask spread. Ideally you calculate all IVs, and then use highest bid IV, smallest ask IV.

If so, I assume this surface can then immediately be used to calculate ... Yes, then you can calculate all puts and calls.

Does it make sense to price American options ... Yes.

Now, the last 3 questions - yes, you can mix European-style, and American-style IVs, or use one IV surface to price the other. But I doubt its usefulness from practical point - AFAIK the only options that trade both styles are SPX (European on CBOE and American/futures on CME) and liquidity on both is among the highest in the world. You are unlikely to see any tradable mispricings in one product vs another.

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