Forward Moneyness and Calendar Arbitrage in Implied Volatility Surfaces
Summary
The document explains why option volatility surfaces may use forward moneyness instead of raw strike. Moneyness measures a strike’s distance from a reference price; forward moneyness compares the strike with the forward price for that option’s maturity. This makes options with different maturities comparable by their relative distance from their respective forwards, even when those forwards differ.
The answer relates forward moneyness to a result about avoiding static calendar arbitrage: under the paper’s assumptions, total variance must increase strictly with forward moneyness. It illustrates the comparison with two calls whose strikes and forward prices differ but whose strike-to-forward ratios match. Both are then equally out of the money relative to their own forwards. The explanation is specific to European options and the cited framework; the document does not develop the arbitrage proof or discuss broader surface construction methods.
Key ideas
- Moneyness describes a strike’s distance from a reference price such as spot or forward.
- Forward moneyness compares an option’s strike with the forward for its maturity.
- Options with the same strike-to-forward ratio have the same relative moneyness.
- The cited result links increasing total variance in forward moneyness to excluding static calendar arbitrage.
- The explanation relies on European options and the assumptions of the referenced framework.
Tags
Full text
# Implied Volatility Surface - log forward moneyness
# 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:
- why replace strike price with forward moneyness
- what is log forward moneyness or some metric of moneyness?
- 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. :)
## Answer by LocalVolatility (score 7, accepted)
https://quant.stackexchange.com/a/33805
- 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.
- 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.
- 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.
See also the following related questions and answers:
- How to exploit calendar arbitrage?
- Calendar Arbitrage in a Vol SurfaceShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.