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Hello everyone,Can someone explain why the vega is the average of gammas?can you give quantitative and qualitative reasons please?Thanks a lot for your help.

Quotethe vega is the average of gammasI do not think it is literally true, though perhaps useful as a shorthand or mnemonic device. The actual statement is more like:Assuming interest rates are zero, the Vega (sensitivity of option value to variance of the stock) is equal to the integral from 0 to T of the Dollar Gamma of the option (where dollar gamma is defined as [$] \frac{1}{2} S_t^2 \Gamma[$]). For details see Carr and Wu.

Vega is the expected sum of gamma PnL due to volatility changes. The relationship is given by: Vega = sigma t S^2 GammaQuantitative answer can be found at Tabel (97) or a short paper by Mercurio "Vega-Gamma Relationship".