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It it written in the book by Giles Jewitt: "If a currency pair had a completely flat volatility smile, the risk reversal strikes would be positioned approximately symmetrically around the ATM strike in logspace. Therefore, the topside strike will be further away from the ATM than the downside strike in regular spot space. At short maturities this effect is small but at longer maturities the impact can be significant." Please explain why strikes are symmetrically distributed in log-space?

Please give mathematical or intuitive rational for the below:

  1. If the implied volatility for a given delta is higher on the smile, the strike moves further away from the ATM; think about the increasing chance of ending up in-the-money at higher volatility.
  2. If the implied volatility for a given delta is lower on the smile, the strike moves closer to the ATM; think about the decreasing chance of ending up in-the-money at lower volatility.
  3. At shorter tenors the risk reversal is largely driven by expectations of spot moves and realized spot volatility.
  4. At longer tenors the risk reversal is largely driven by expectations of spot moves and implied volatility changes.
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