No-Arbitrage Questions When Assets Become Tradable at Different Times

Summary

The document raises a question about whether an equivalent local martingale measure is enough to rule out arbitrage when assets enter the tradable market at different times. Its example has one asset available throughout the interval and a second asset that becomes available only at a positive stopping time. The concern is that a relationship between terminal payoffs may create an arbitrage if the assets’ prices do not also satisfy the corresponding relationship when the second asset first becomes tradable.

The example suggests that standard no-arbitrage results for assets available from the initial time may need extra conditions in this setting. However, the text presents this as an open question, not a theorem: it does not specify the full price processes, admissible trading strategies, or precise assumptions needed to establish arbitrage or rule it out. It offers no literature review or general characterization. The useful takeaway is the modeling issue itself: delayed availability can make the price at the entry time relevant, and the usual martingale-measure criterion may not apply without qualification.

Key ideas

• The document asks whether an equivalent local martingale measure excludes arbitrage when assets become tradable at different times.
• One asset is available from the start, while another becomes available at a stopping time.
• A deterministic relationship between terminal asset values may require a matching price relationship when trading in the later asset begins.
• The example raises a concern but does not prove a general no-arbitrage condition.
• A complete answer would depend on the price processes and allowable trading strategies.

Tags

# No Arbitrage condition for assets with different time frame


# No Arbitrage condition for assets with different time frame












In the classic literature, one always assumes that the assets in the market are all available from the very beginning ($t=0$). And under such condition the market is arbitrage free iff there exists an equivalent local martingale measure. But what if there is some asset that is not available for trading until a strict positive amount of time (especially when this is a stopping time?)

Specifically, consider the following model with two assets: $S_1$ is able to trade in [0,1], whereas the other asset $S_2$ is able to trade only in $[\tau,1]$ where $\tau$ is a stopping time. Is the existence of an equivalent local martingale measure sufficient to exclude arbitrage? In general I think not. For instance if the two assets are correlated (like $S_2(1) = 2S_1(1)$ a.s.). Then you need extra condition that $S_2(\tau) = 2S_1(\tau)$ to ensure no arbitrage.

I wonder what exactly one needs more to make sure the model is arbitrage-free in general? Is there any literature

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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