# Fundamental Theorem of Asset Pricing under fixed and proportional transaction costs

## TL;DR

This paper extends the Fundamental Theorem of Asset Pricing to models with both fixed and proportional transaction costs, establishing conditions for no arbitrage involving a family of probability measures and a martingale process within bid-ask spreads.

## Contribution

It generalizes the classical theorem to include combined fixed and proportional transaction costs, providing a new theoretical framework.

## Key findings

- Equivalence between no arbitrage and existence of suitable probability measures and martingale process.
- Extension of Harrison and Pliska's classical theorem to more complex transaction cost models.
- Theoretical foundation for pricing and hedging in markets with mixed transaction costs.

## Abstract

We show that the lack 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 between the bid-ask spreads 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.