# Some basics of option pricing

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 $$t, the value of the option is defined to be $$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 ?