There is a script at my university (can't post it for copyright reasons) for a course on discrete time financial mathematics. I decided to give it a try and found this problem:
Consider a financial market model with $\Omega=\left\{\omega_1, \ldots, \omega_N\right\},\mathcal{F}_0=\{\emptyset, \Omega\},$ $\mathcal{F}_1:=2^{\Omega}$, $T=1$ and $d$ arbitrary. Suppose that in the market model every European option is hedgeable. Show the existence of $c=c(d)$, i.e. $c$ only depending on $d$, such that $N \leq c$.
First question I have is: what is $c$ supposed to be here? It is not a used anywhere else in the text and I have no idea what value I'm looking for.
Then, I still tried to work that out:
the definition of headgeability given in the script is that an European option is hedgeable iff one can find a starting value $V_0$ and a trading strategy $H$ s.t.
$$ f = V_0 + (H \cdot S)_T $$
where $(H \cdot S)_T$ denotes a martingale transform.
But i don't know how to set up the system(s) of equations given that there is countably many $\omega$s and arbitrary many $d$s. Any help on how to proceed?
