Why Continuous Returns Alone Do Not Guarantee an Arbitrage-Free Market

Summary

The document asks whether a one-period market is arbitrage-free whenever the asset’s log return has a continuous distribution over the real line, and whether such a distribution guarantees a martingale measure. The motivating comparison is with a finite-state multinomial model, where the risk-free return must lie between possible asset returns to avoid arbitrage.

The reply says that in a basic market without interest rates, equality between the expected terminal price and the initial price is relevant to the absence of arbitrage, and mentions a butterfly arbitrage otherwise. It cautions that richer market settings can change the conclusion, using differing information between two agents as an example. The exchange does not establish a general theorem or fully specify assumptions, so continuity of returns alone should not be treated as a sufficient arbitrage test based on this discussion.

Key ideas

• The question asks whether continuous log returns over the real line imply a martingale measure.
• The finite-state no-arbitrage condition depends on the risk-free return lying between possible asset returns.
• The reply points to the expected terminal price relative to the initial price in a basic market without interest rates.
• Different information available to market participants can complicate conclusions in richer settings.
• The discussion is informal and does not state a general theorem or all required assumptions.

Tags

# Arbitrage-free market for continuous logreturn distribution?


# 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).

## Answer by Drew (score 1)

https://quant.stackexchange.com/a/15224

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.)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.