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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?

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  • $\begingroup$ The call option price (in the Black-Scholes world) is a function of time $t$ and stock price $S_t$ and certainly not constant -- even if $r=0$ and the option is ATM. Recall that a martingale has constant unconditional mean, i.e. $\mathbb{E}[M_t]=\mathbb{E}[M_0]$ for all $t$ but time-varying conditional mean $\mathbb{E}[M_t|\mathcal{F}_s]=M_s$ for $s\leq t$. The (first) FTAP states that without arbitrage asset prices are conditional expectations of their discounted payoffs: $C(t,S_t)=B_t\mathbb{E}^\mathbb{Q}\left[\frac{1}{B_T}\max\{S_T-K,0\}|\mathcal{F}_t\right]$, where $B_t=e^{rt}$. $\endgroup$ – Kevin Apr 24 '20 at 23:12

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