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Arbitrage Conditions for Assets That Always Rise in a Discrete Market

Article Quant Q&A · Author: Wolfy
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Summary

The document considers whether a discrete market is arbitrage-free when every asset price rises strictly with probability one. The answer makes the conclusion depend on the interest rate: when rates are zero or negative, an investor can borrow the purchase price, buy an asset, and later sell it to repay the borrowing, retaining a gain under the stated price movement.

With positive interest rates, the same reasoning does not establish arbitrage, because the proceeds from selling the asset may not exceed the loan repayment. The brief response offers no general proof for that case and does not specify other market assumptions, such as trading constraints or asset availability. Its key lesson is that rising prices alone do not settle the arbitrage question; financing costs matter.

Key ideas

  • If rates are zero or negative, borrowing to buy an asset that strictly rises can produce arbitrage under the stated setup.
  • At positive rates, the purchase-and-borrow argument does not by itself guarantee a profit.
  • Whether rising asset prices imply arbitrage depends on financing conditions.
  • The answer does not analyze other market assumptions or prove a general result for positive rates.

Tags

ArbitrageRisk managementMulti-asset
Full text
# Price of every asset in discrete market model strictly increasing


# Price of every asset in discrete market model strictly increasing












> If the price of every asset in a discrete model is strictly increasing, with probability one, then does the market admit arbitrage?

Thoughts: I believe this is true but I am not sure how to give an arbitrage argument to show that there exists some arbitrage strategy. Perhaps I am not understanding the implications of the question.

Any suggestions are greatly appreciated.

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/30825

Off the top of my head, if interest rates are zero or negative, then yes. Just borrow the purchase price and buy any asset. Sell later and pay off the loan. Otherwise no.

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