Arbitrage-free is a term used in finance to describe a situation where there is no potential for making an arbitrage profit. An arbitrage profit is made when you are able to buy an asset for less than it is worth and then sell it immediately for a higher price. In order for there to be no potential for an arbitrage profit, the prices of the assets must be such that there is no way to make a profit by buying low and selling high.
The importance of being arbitrage-free is that it ensures that prices are correct and reflects the underlying value of the assets. If prices were not arbitrage-free, then there would be opportunities for investors to make easy profits simply by taking advantage of discrepancies in prices. This would lead to market inefficiencies and would ultimately hurt all investors.
Arbitrage-free finance is therefore important for ensuring that markets are efficient and that prices reflect the true value of assets. This benefits all investors by ensuring that they are not being taken advantage of and that they are getting a fair price for the assets they purchase.