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If the discounted price of any asset is a martingale under risk neutral measure, why is $E^Q[e^{-rT} (S_T-K)_+ | F_t]$, not merely $e^{-rt} (S_t-K)_+$?

This is something I wanted to clarify, since that's the definition of a martingale. Instead we use the lognormal distribution of the stock price and solve the expectation completely to get the black Scholes call price.

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    $\begingroup$ Because the function $f(x) = (x-a)_+$ is not linear in $x$, it is a convex function of $x$. Also the discount factor remains $e^{-rT}$, it doesn't become $e^{-rt}$ after taking expectations. $\endgroup$
    – Frido
    May 5, 2023 at 16:15

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The process $Y_t:=(S_t-K)^+$ cannot be the price of a traded asset because of Jensen's inequality. Instead, it is the price of the option which is a martingale.

In the Black-Scholes model, the primitive market model has only two assets: the stock with price $S_t$ and the money market account (MMA) with price $B_t:=e^{rt}$. Within this market, Black and Scholes prove that it is possible to replicate a European (call) option with payoff $(S_T-K)^+$ at some future expiry $T>t$. The value $V$ of this claim is given by its risk-neutral expectation: $$V_t=B_tE^{\mathbb{Q}}\left(\left.\frac{(S_T-K)^+}{B_T}\right|\mathscr{F}_t\right)\tag{1}$$ where $B_t/B_T=e^{-r(T-t)}$.

Given the option can be replicated, it can be viewed as an asset. One can then consider an "augmented" market model with the stock, the MMA and the option. Per risk-neutral theory, it is the discounted price of an asset which is a martingale, that is if $P$ is the price of an asset then the process $$\frac{P_t}{B_t}$$ is a martingale. The price of the option is $V$ therefore letting $s<t$: \begin{align} B_sE^\mathbb{Q}\left(\left.\frac{V_t}{B_t}\right|\mathscr{F}_s\right) &=B_sE^\mathbb{Q}\left(\left.\frac{B_tE^{\mathbb{Q}}\left(\left.\frac{(S_T-K)^+}{B_T}\right|\mathscr{F}_t\right)}{B_t}\right|\mathscr{F}_s\right) \\ &=B_sE^\mathbb{Q}\left(\left.E^{\mathbb{Q}}\left(\left.\frac{(S_T-K)^+}{B_T}\right|\mathscr{F}_t\right)\right|\mathscr{F}_s\right) \\[7pt] &\overbrace{=}^{\text{LIE}}B_sE^\mathbb{Q}\left(\left.\frac{(S_T-K)^+}{B_T}\right|\mathscr{F}_s\right) \\[3pt] &\overbrace{=}^{\text{(1)}}V_s \end{align} where we have used the Law of Iterated Expectations (LIE) and the definition $(1)$. Dividing by $B_s$: \begin{align} E^\mathbb{Q}\left(\left.\frac{V_t}{B_t}\right|\mathscr{F}_s\right) &=\frac{V_s}{B_s}, \end{align} hence the discounted price of the option is indeed a martingale.

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