# Implied Volatility Surface - log forward moneyness

I'm reading this paper by Fengler (2005) and have came across the below snippet.

context: Implied volatiltiy surface plot has 3 dimensions IV, Strike, Time to Maturity. Author replaced Strike with Moneyness metric.

My questions are:

1. why replace strike price with forward moneyness
2. what is log forward moneyness or some metric of moneyness?
3. For two calls with different maturities, what does "both calls have same forward-moneyness" means. Please refer page 11, proposition 2.1 for this question. I couldn't post the snippet, since i am new to this forum and have less reputations. Apologies.

Thank you in advance. Loving this community. :)

1. The reason is that, as shown in Proposition 2.1 of that paper, in order to exclude static calendar arbitrage, the total variance has to be strictly increasing in forward moneyness. See also the below to links for details on this result.

The intuition is that for European options, only the distribution of the terminal spot price is relevant. Furthermore, $$F_t^T = \mathbb{E}_{\mathbb{Q}} \left[ \left. S_T \right| \mathfrak{F}_t \right]$$ (under the assumptions in the paper). So two options with the same forward moneyness $$\kappa = K_1 / F_t^T$$ are the same relative distance away from their respective forward.

2. I don't understand what your question is here. A metric of moneyness is a measure for how far a given strike is away from some reference level - e.g. the spot or forward.

3. It means that for both of them the ratio $$K_i / F_t^{T_i}$$ is the same. I.e. consider $$K_1 = 110$$, $$F_t^{T_1} = 100$$, and $$F_t^{T_2} = 90$$ (e.g. because there is a dividend between $$T_1$$ and $$T_2$$). Then $$\kappa = K_1 / F_t^{T_1} = 1.1$$. Now you solve $$\kappa = K_2 / F_t^{T_2}$$ for $$K_2$$ to get $$K_2 = 99$$. Both options are 10% out-of-the money relative to their respective forward.