I don't understand why the vega of a call option is not 0 when ATM. Irrespective of the implied volatility the vega of a binary call option when at-the-money is always zero, since you have 50-50 chance of being in the money or out of the money if the volatility increases. Why doesn't this reasoning apply for call options? I'm looking for a "financial" answer.

  • $\begingroup$ Since price = expected payout, you have to weigh payoff profile vs probalities. If the probability of $S_T > K$ is $N(d_2)=N(0)=50%$ ATM at expiry in BS in both cases, the payouts are not the same. In particular, that of a binary is symmetric while that of a call is not. $\endgroup$ – Quantuple Feb 26 '18 at 8:14

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