The fundamental theorem of asset pricing for continuous processes under small transaction costs

A version of the fundamental theorem of asset pricing is proved for continuous asset prices with small proportional transaction costs. Equivalence is established between: (a) the absence of arbitrage with general strategies for arbitrarily small transaction costs ${varepsilon > 0}${varepsilon > 0}, (b) the absence of free lunches with bounded risk for arbitrarily small transaction costs ${varepsilon > 0}${varepsilon > 0}, and (c) the existence of e{varepsilon}-consistent price systems—the analogue of martingale measures under transaction costs—for arbitrarily small ${varepsilon > 0}${varepsilon > 0}. The proof proceeds through an explicit construction, as opposed to the usual separation arguments. The paper concludes comparing numéraire-free and numéraire-based notions of admissibility, and the corresponding martingale and local martingale properties for consistent price systems.