We propose a formal definition of approximate arbitrage which can be used to extend the applicability of theories based on the absence of arbitrage. Our definition is based on the ratio of gain to loss, where gain (loss) is the expectation of the positive (negative) part of the excess payoff. Arbitrage is characterized by infinite gain-loss ratio, and approximate arbitrage by gain-loss ratio close to infinity. Our definition of approximate arbitrage has a useful dual interpretation in terms of pricing kernels. This allows us to compare the pricing kernel restriction implied by a limit on the maximum gain-loss ratio to other pricing restrictions in the literature. We show theoretically that only the gain-loss ratio restriction is consistent with the absence of arbitrage and approximate arbitrage opportunities. We demonstrate the practical differences of these alternative pricing kernel restrictions by examining their implications for the prices of call options on an asset that does not trade.
UCLA Finance Working Paper #18-99
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