In this question I asked about the mathematical rationale of using the ATM vol to price quanto options. One of the reasons pointed (as an answer) was, as expected, that the ATM volatility is kind of an average volatility.
What is missing is to prove this mathematically or at least give some intutiton of why this is true through mathematical equations. Could you please provide the mathematical reasoning for the conclusion that the ATM vol is kind of an average volatility?
Thank you
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