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Note that total implied variance defined as $$ V(T,K) = T\Sigma(T,K)^2 $$ should be an increasing function of $T$. Otherwise you have a calendar arbitrage (sell the call with shorter expiry and buy the cheap longer one). If you interpolate linearly your implied volatility is $$ \Sigma(T,K) = w\Sigma(T_i,K) + (1-w)\Sigma(T_{i+1},K) $$ with weight $w = ...


It implies negative forward variance. I have the book, and went through the section following your quote. In math terms, he is making a proof by contradiction. He first assumes that you can interpolate Iinearly, and comes to the conclusion that it is not a good assumption. The argument does involve some calculus. I don't think I have a better explanation, so ...

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