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Gamma and Vega are not independent. In Black-Scholes world, aka flat implied volatility, we have:
$$V_y= \sigma_y \tau_y S^2 \Gamma_y $$
and similarly for $z$.
In Black-Scholes world, we have:
$$Vega = \sigma \tau S^2 Gamma.$$$$V_y= \sigma_y \tau_y S^2 \Gamma_y $$
$$Vega = \sigma \tau S^2 Gamma.$$
In Black-Scholes world, aka flat implied volatility, we have:
