Some economic remarks on arbitrage theory

Today's primarily mathematically oriented arbitrage theory does not address some economically important aspects of pricing. These are, first, the implicit conjecture that there is 'the' price of a portfolio, second, the exact formulation of no-arbitrage, price reproduction, and positivity of the pricing rule under short selling constraints, third, the explicit assumption of a nonnegative riskless interest rate, and fourth, the connection between arbitrage theory (that is almost universal pricing theory) and special pricing theories. Our article proposes the following answers to the above issues: The first problem can be solved by introducing the notion of 'physical' no-arbitrage, the second one by formulating the concept of 'actively' traded portfolios (that is non-dominated portfolios) and by requiring that there is a minimum price for actively traded portfolios and therefore for every admissible portfolio, and the third one by combining the 'invisible' asset 'cash' with the idea of actively traded portfolios - a riskless asset with a rate of return less than zero can never be actively traded in the presence of cash. Finally, the connection between arbitrage theory and special pricing theories ('law-of-one-price-oriented' and 'utility-oriented' pricing) consists in the fact that special pricing theories merely concretize arbitrage theory using different assumptions.