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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?

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  • $\begingroup$ How to find $F$ ? If the stock pays no dividend then $F=S e^{r T}$. $\endgroup$ – Alex C Jan 21 at 0:06
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    $\begingroup$ There might be a typo. It Should be probably have been $ln(F/K)$. Then the upper expression is known as the standardized momeyness $\endgroup$ – Sanjay Jan 21 at 0:11
  • $\begingroup$ Thanks for point out the typo, I've already edited it! $\endgroup$ – Francis Gong Jan 21 at 10:23
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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.

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