# Approximate Arbitrage

## Antonio E. Bernardo and Olivier Ledoit

### Abstract

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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