Testing Arbitrage in a Stock Market with Trading Costs

Summary

The document considers a two-period market with a risk-free asset and a stock whose future price can move along several scenarios. It asks whether short selling creates an arbitrage when each stock trade incurs a transaction cost proportional to its volume. The answer checks possible trades at each decision time against what is then known about future prices, emphasizing that a strategy must be feasible using available information and must avoid losses across relevant outcomes.

At the initial time, uncertainty about whether the stock rises or falls prevents a guaranteed gain. At the intermediate time, neither branch offers an arbitrage: the higher-price branch still has outcomes in both directions, while the lower-price branch’s expected rise is too small to overcome both the trading charge and the risk-free alternative. The conclusion depends on the stated price tree, interest accumulation, and transaction-cost rate; it is not a general result for all markets or costs.

Key ideas

• An arbitrage strategy must use only information available when each trade is chosen.
• Short selling alone does not ensure arbitrage when future prices remain uncertain.
• Trading costs can eliminate an apparent profit from a predictable price increase.
• The conclusion is specific to the given price scenarios, risk-free asset, and cost assumptions.

Tags

# Is there an arbitrage strategy if short selling of a stock is allowed?


# Is there an arbitrage strategy if short selling of a stock is allowed?












Consider a market with a risk-free asset such that $A(0) = 100, A(1) = 110, A(2) = 121$ dollars and a risky asset, the price of which can follow three possible scenarios

Is there an arbitrage strategy if short selling of a stock is allowed, but transaction costs of 5% of the transaction volume apply whenever stock is traded?

How can I solve this?

I know the the No-Arbitrage Principle would be violated if there was a self-financing predictable strategy with initial value $V(0) = 0$ and final value $0 \neq V(2) \geq 0 $ such that $V(1)<0$ with positive probability

## Answer by Attack68 (score 1, accepted)

https://quant.stackexchange.com/a/42520

At time 0 you do not know if the asset price will rise to 120 or fall to 90 so you cannot be assured of a profit, in which case there is no arbitrage strategy available at time 0.

At time 1 if the price is 120 you, again, cannot be assured of a profit since the price may fall to 96 or risk to 144.

At time 1 if the price is 90 you can be assured that the price will rise to 96 but this is useless since the 5% commission and weaker than risk free accumulation would preclude you from executing this strategy.

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