def. Arbitrage Portfolio. Portfolio with value at time is an arbitrage portfolio if:

Alternatively we can define it as:

def. Law of One Price (LOP). Two portfolios with the same future value must have the same value to begin with. if time then:

thm. (Arbitrage is equivalent to LOP) If there is no arbitrage portfolio, then the law of one price holds.

Proof. Contrapositive: If law of one price doesn’t hold, there is an arbitrage portfoilo. Let portfolios such that

Then construct the following new portfolio that:

  • Long position on
  • Short position on Then
  1. At time , value is
  2. At time , value is
    1. The cash investments are risk free, thus this is determined.
  3. Therefore this portfolio is an arbitrage portfolio.

One can similarly prove the other way. Thus LOB is equivalent to No Arbitrage condition.