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Suppose the overnight (1-day) at-the-money implied volatility is X% and the two week (14-day) at-the-money implied volatility is also X%.

How would I go about finding the upper and lower no-arbitrage bounds for the 1-week (7-day) implied volatility assuming a term-structure following the Black-Scholes model?

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Simply make sure the forward variances remain non negative: $\Sigma(T_{i+1})^2 T_{i+1} - \Sigma(T_{i})^2 T_{i} \geq 0$ for all $i$.

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  • $\begingroup$ The lower bound will be given by setting the forward variance for days 2-7 at zero and the upper bound will be given by setting the forward variance for days 8-14 at zero. $\endgroup$
    – dm63
    Apr 30 '21 at 9:14
  • $\begingroup$ And if you want to be very pedantic about it, you should work with the ATMF volatility level, not the ATM one, which might be important if you're looking at a single name distributing divs. $\endgroup$
    – Quantuple
    Apr 30 '21 at 9:47

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