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I am a mathematician trying to learn finance on my own. Try to avoid financial lingo in your answer when not necessary.

So I am trying to understand (European) option pricing under the no free lunch with vanishing risks (NFLVR) principle. The NFLVR condition is equivalent to the following: Let $S$ be a semimartingale on $(\Omega, \mathcal A,P)$. $S$ satisfies NFLVR iff there exists a measure $Q$ on $(\Omega, \mathcal A)$ such that $Q$ is equivalent to $P$ (they have the same null-sets) and $S$ is a sigma-martingale under $Q$.

In the following, we suppose that our market satisfies the NFLVR condition. My questions will be highlighted in bold.

The Bachelier model:

Let $S_t=S_0+\sigma W_t$ be the price of a stock, $S_0 \in \mathbb R$, $W_t$ Brownian motion on $(\Omega, \mathcal F_t,P)$, $0\leq t\leq T$, $\mathcal F_t$ the natural filtration generated by $W_t$. We are interested in pricing a European option (strike price $K$, maturity time $T$) based on $S_t$. If I understood correctly, this means determining the "value" of the option for times $0\leq t\leq T$. The payoff of a European option is the random variable $X:= (S_T-K)_+$.

1) Why is the "value" of the option at time $T$ defined to be $E[X]$ ? Shouldn't we be interested instead in the whole distribution of X ? Suppose the variance is huge, isn't that worth noting when you "price" options ?

Because we know the law of $S_T$, we can compute $E[X]$. For $t<T$, the value of the option is defined to be $E[X | S_t]$. 2) Again , why ? Notice that $X$ is in $L^1(\Omega, \mathcal F_T,P)$, so we can use the martingale representation theorem: There exists a unique predictable process $H_t$, $t\in [0,T]$ such that

$$X=E[X] + \int_0^T H_sdW_s$$

$$E[X|\mathcal F_t]=E[X]+ \int_0^t H_s dW_s$$

which is equivalent to (because we supposed Bachelier model)

$$X=E[X] + \int_0^T \frac {H_s}{\sigma} dS_s$$

$$E[X|\mathcal F_t]=E[X]+ \int_0^t \frac {H_s}{\sigma} dS_s$$

As $\sigma(S_t) = \mathcal F_t$ (again, Bachelier model), we have

$$E[X|S_t]=E[X]+ \int_0^t \frac {H_s}{\sigma} dS_s$$

So we even can prove the existence of a trading strategy $H$ to replicate $X$ with initial investment $E[X]$.

3) As we know the law of $S_T$, we can compute $E[X]$ and the full distribution of $E[X|S_t]$. We did not use NFLVR. Why ? So we can price option without this hypothesis ? What am I missing ?

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