From what I understood the fundamental theorem of asset pricing (FTAP) details that discounted asset prices are martingales under the risk neutral mesure.
As an example:
We consider an ATM call option with zero interest rate. Based on the FTAP, we should have that the expectation of this call option is constant through time, given that the discount factor is always one. However, the Black Scholes formula for an ATM call with zero interest rate can be approximated by $0.4S_t\sigma\sqrt{T-t}$, which is strictly dependent on the time to maturity.
Therefore, the BS approximation formula suggests that the call option is not constant through time. How is that possible?