Notation interpretation (Non-maths background)

Question

Please Explain the highlighted symbols. Usually X-bar shows average but i am not able to figure out what is it in this context?

Answer 1

This is not really a mathematics question, but more one of quantitative finance. Your text seems to come from "A Guide to FX Options Quoting Conventions, Dimitri Reiswich and Uwe Wystup, The Journal of Derivatives Winter 2010, 18 (2) 58-68". The previous lines to your quotation said

The implied volatility as a function of moneyness for a single time to maturity is generally referred to as the smile. To be more precise, the volatility smile is a mapping,

so $\mapsto$ is being used to indicate a mapping or function.

The other terms are described in your quotation, so $\sigma(\bar X)$ is the implied volatility for a particular value $\bar X$ of the moneyness given a particular value $\bar T$ of the time to maturity for the option being considered.

Later the article suggests that prices of options are not quoted in currency units, but in terms of implied volatilities, and moneyness is expressed in terms of the option’s delta, not as the difference between the option strike and the price of the underlying.

It looks to me as if the detail of the notation in your quotation is not particularly significant here, but is just used as a rhetorical flourish to set up saying the wrong variable to plug into the wrong formula to get the correct price. And that is saying that options away from the money have different (usually higher) implied volatilities than options at the money if you use the Black-Scholes formula to calculate them.