# Arbitrage-free market for continuous logreturn distribution?

Is it true, that a one-period market say $(0,t)$ is arbitrage-free if the logreturn for $S_t$ is continuously distributed on $\mathbb{R}$?

I.e., for continuous distributions on $\mathbb{R}$, there always exists a martingale measure?

E.g. for multinomial model the market is arbitrage free if $r_1<r_f<r_m$, such that on $\mathbb{R}$ for a continuous distribution we would have $-\infty<r_f<\infty$ (which is always true).

Lets look at generic markets with a single market filtration, then if $\mathbb{E}[S_t]=S_0$ then the market should be arbitrage free (absence of interest rates.) Otherwise there would be a butterfly arbitrage.

But for more sophisticated markets, not at all. Consider a market where there are only is only one period and there are two agents, one who knows the final price $S_T$ and one who does not. There is a static equilibrium and one guy rips off the other (but a dynamic one might be very hard to establish.)

• I corrected my question, it should be the return of $S_t$ (not $S_t$ itself), as in the usual binomial model with $1+d<1+r<1+u$. Oct 30, 2014 at 21:41
• What sort of market are you looking at?
– Drew
Nov 2, 2014 at 20:32
• Its just $S_0>0,S_T\sim e^\mathbb{R}$. Is that always arbitrage-free? Nov 2, 2014 at 21:21
• Is $\mathbb{E}[S_T] = S_0$
– Drew
Nov 2, 2014 at 21:22
• I think yes, there should always be a change of measure to $Q$ for that. However, I would need something more formal.. Nov 2, 2014 at 21:28