# No-arbitrage bounds on Implied Volatility under Black-Scholes

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?

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 $$i$$.