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A simple volatility surface might have X axis = strike, Y axis = expiry and Z axis = implied volatility. But in many papers I see them use moneyness instead of strike. I have two questions.
First off, there are different types of moneyness one can use when constructing a volatility surface. Each have their own advantages.
Absolute-moneyness: using absolute spot-strike comparison as a measure of moneyness. ATM would correspond with S=K. This has a simplistic interpretation when looking at option payoff diagrams at maturity.
Simple-moneyness: using strike-spot ratio as a measure of moneyness. ATM would correspond with moneyness level of S/K=1 (or K/S). Either ratio will determine whether ITM corresponds with a ratio greater than 1. For instance, call simple moneyness with S/K ratio will be ITM for a ratio greater than one (or less than 1 if using K/S). This ratio standardizes moneyness to correspond with a value greater or less than 1 depending on the ratio used as opposed to a random strike value.
Log-simple-moneyness: moneyness is measured in terms of the forward price of the asset: $\log(F/K)$ where ATM would be when moneyness is 0, which occurs when $F=K \implies \log(1)=0$. Another measure of standardization where moneyness centers around the value of 0 and ITM/OTM can be greater or less than 0. According to Wiki, "While the spot is often used by traders, the forward is preferred in theory, as it has better properties." They are referring to derivative pricing models as it simplifies calculations when using ATMF.
Standardized-forward-moneyness: $\frac{\ln(F/K)}{\sigma\sqrt\tau}$ takes into consideration the volatility and time to maturity of the underlying asset in the measurement of moneyness in terms of standard deviation units (number of standard deviations the current forward price is above the strike price).
Delta moneyness: used to construct vol surface instead of strike moneyness since delta is more consistent measure of how close to the money the option is (e.g. 10% OTM by strike can be close for long maturity but far for short maturity while 10% OTM by delta would remain consistent throughout terms of option). Moreover, delta moneyness describes near the money in more detail (ATM near expiry) and delta surface provides more natural view from hedging activities as investors hedge delta and not the spot.
In regards to your question about interpolation, yes, typically modelers will interpolate the moneyness and time axis. The details will depend on the specific models used but once the axes are interpolated, the surface will need remove any calendar and butterfly arbitrage opportunities. Remember, volatility is essentially used a pricing mechanism so hence the surface needs to be free of arbitrage.
Please refer to this Wiki page for more info on moneyness: https://en.wikipedia.org/wiki/Moneyness
If you use constant strike, the moneyness changes as the underlying changes. Out of the money equity options tend to trade at a premium to at the money options (smiles/skew). Therefore, the moneyness is used to take into account the movement of the underlying.
Yes, if you are trying to price an option with a strike whose moneyness is in between the options used to provide the data along your moneyness axis, you would need to do some interpolation to arrive at the volatility used to price this option.
