Forward Moneyness and Standardized Log Moneyness

Summary

This explanation compares several conventions for option moneyness. Simple moneyness is expressed as the underlying price relative to the strike, while log moneyness takes the logarithm of that ratio. Standardized log moneyness further scales the log ratio by volatility and the square root of time to maturity. These measures describe how far an option’s strike is from the underlying price, with the standardized form accounting for volatility and horizon.

Replacing the spot price with the forward price gives corresponding forward moneyness definitions. The answer says this is often more natural in the context of Black–Scholes because the model is expressed in terms that align with the forward. For an underlying without dividends, the forward can be calculated from spot and the interest rate over the maturity; it can also be inferred by comparing put and call prices. The source notes that moneyness terminology is not fully standardized and mentions that some references reverse the simple ratio. It does not derive the forward relationship or discuss dividend-paying assets in detail.

Key ideas

• Moneyness may be expressed as a spot-to-strike ratio, its logarithm, or standardized log moneyness.
• Standardized log moneyness scales the log price ratio by volatility and the square root of maturity.
• Forward moneyness uses the forward price in place of spot in these definitions.
• Forward-based measures align naturally with the way the Black–Scholes formula is expressed.
• For an underlying with no dividends, the forward is derived from spot and the interest rate; put and call prices can also inform it.
• Definitions vary across sources, including the direction of the simple price ratio.

Tags

# What is forward moneyness and how to calculate it?


# What is forward moneyness and how to calculate it?












I'm now studying the concept "implied volatility", and my teacher gave us a figure about the implied volatility with respect to the moneyness which is expressed by $\frac{ln(\frac{F}{K})}{\sigma\sqrt{T}}$ , where $F$ should be the forward price at maturity of the underlying I think?

Based on my knowledge, the moneyness should be $\frac{S}{K}$

Could anyone tell me the meaning of the upper expression and the differences between these two kinds of moneyness?

## Answer by Alex C (score 7, accepted)

https://quant.stackexchange.com/a/43597

The definition of moneyness is not completely standardized, you can see different definitions in the literature:

- the simple moneyness is $\frac{S}{K}$ (in some cases you will see $\frac{K}{S}$)

- the log moneyness is $\ln \frac{S}{K}$

- the standardized log moneyness$\frac{\ln(S/K)}{\sigma\sqrt T}$

If the forward price $F$ is used in place of the underlying price $S$ you have (three definitions of) the forward moneyness. The forward moneyness is useful because it is more consistent with the way the Black Scholes formula works, it is more natural.

How to find $F$ ? If the stock pays no dividend then $F=S e^{r T}$. You can also find $F$ by comparing the prices of puts and calls.

Shown 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.