Abstract
We show that the absence of arbitrage in a model with both fixed and proportional transaction costs is equivalent to the existence of a family of absolutely continuous single-step probability measures, together with an adapted process with values within the bid-ask intervals that satisfies the martingale property with respect to each of the measures. This extends Harrison and Pliska’s classical Fundamental Theorem of Asset Pricing to the case of combined fixed and proportional transaction costs.
Highlights
The Fundamental Theorem of Asset Pricing (FTAP), characterising the absence of arbitrage opportunities as equivalent to the existence of a risk neutral probability measure, has been studied for a large variety of financial market models
In Theorem 1 we show that the absence of arbitrage in a market with combined costs is equivalent to the existence of a family of single-step probability measures absolutely continuous with respect to the physical probability, along with a martingale with respect to such a family of measures and taking values between the bid and ask prices
In Corollary 2 we provide another equivalent condition for the absence of combined-cost arbitrage, namely the existence of an embedded arbitrage-free model with fixed costs
Summary
The Fundamental Theorem of Asset Pricing (FTAP), characterising the absence of arbitrage opportunities as equivalent to the existence of a risk neutral probability measure, has been studied for a large variety of financial market models. On the other hand, fixed costs imply that the set of solvent portfolios suffers from the lack of convexity While these difficulties have been tackled separately in the context of proportional costs and, respectively, fixed costs only, they require fresh ideas to handle their compounded effect. We refer to the recent work by Lépinette and Tran (2016, 2017), in which arbitrage in a non-convex market model with friction (including the case of simultaneous fixed and proportional costs) has been considered, and the absence of asymptotic arbitrage has been characterised by the existence of a so-called equivalent separating. No link has been made with risk neutral probabilities, by contrast to the present paper
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