I have a question considering Financial markets in discrete Time.
One of the main theorems in discrete time is the following. In finite discrete Time with trading times t={1,...,T} the following are equivalent:
The market $(S,\mathbb{F})$ is complete, i.e every $F_T$-measurable random variable $U$ is replicable
andThere is exactly one equivalent martingale measure.
Now I seem to come to a contradiction if I define the following framework (shortly: Black Scholes in discrete time):
Let's assume we are in the Black-Scholes framework but we consider the model as discrete time model with (simplifying heavily) two trading dates $t=0$ and $t=1$. The (discounted) stock price is denoted by $S_t$. $S_0$ is constant and for $T=1$ $$S_T=\exp((\mu-r-1/2\sigma^2)T-\sigma W_t).$$ The Filtration $\mathbb{F}=(F_t)_{t=0,1}$ is the natural one.
It is clear that in this framework there is exactly one equivalent martingale measure, namely the one in which $\mu=r$. Applying the theorem from above, a call-option on that stock S should be replicable (in only one trading day, namely from $t=0$ to $T=1$)
Now this seems to me very doubtful and I don't really know where the Problem is.... ANY HELP IS WELCOME!!!!