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Assume it to be known that the volatility of a stock at any point in time is $\sigma(t)$.

My question is, if we have a number of options priced using some implied volatilities $\sigma_1, ..., \sigma_i$ on the same underlying, different maturities, how do we determine whether this is an exploitable situation, e.g. using gamma trading?

How do I compare my knowledge of $\sigma(t)$ to the multiple $\sigma_i$s and determine what to do?

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    $\begingroup$ The assumed knowledge of the actual σ(t) over relevant future horizon would allow you to know which options are currently underpriced (IV<actual vol) and which are currently overpriced (IV > actual vol). You would buy the former and sell the latter, holding to maturity while delta-hedging at the correct volatility. In this way your P&L would be equal to the initial mispricing. Needless to say such perfect knowledge of future vol is not very common. $\endgroup$ – noob2 Mar 26 '17 at 18:36
  • $\begingroup$ Perhaps you may want to modify the title of your question because comparing apriori value (implied volatility) to aposteriori (realised volatility) for the purposes of finding a trade doesn't make sense - if you know realised value then you have a history (situation already happened). Perhaps you can add instead "view/forecast on realised volatility" $\endgroup$ – Nicholas Mar 27 '17 at 21:09

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