# Why use moneyness as an axis on a volatility surface

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.

1. Why do they use moneyness as an axis instead of strike?
2. Does it make sense to interpolate along the moneyness axis?

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.

• It is not quite clear what exactly is meant by "Delta" moneyness. The Wiki article says that it is equal to the BS delta, but there seems to be inconsistency since one BS is using S while the article uses F in its definition of $d_{+}$. And if we look at Black's delta instead, then there is a discounting factor in front of the CDF, which doesn't appear in the article. Commented Nov 27, 2023 at 2:31