The fundamental theorem of asset pricing with and without transaction costs

Paper: arXiv 2307.00571

Authors: Christoph Kühn

Abstract

We prove a version of the fundamental theorem of asset pricing (FTAP) in continuous time that is based on the strict no-arbitrage condition and that is applicable to both frictionless markets and markets with proportional transaction costs. We consider a market with a single risky asset whose ask price process is higher than or equal to its bid price process. Neither the concatenation property of the set of wealth processes, that is used in the proof of the frictionless FTAP, nor some boundedness property of the trading volume of admissible strategies usually argued with in models with a nonvanishing bid-ask spread need to be satisfied in our model.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly rigorous mathematical proof unifying the Fundamental Theorem of Asset Pricing across frictionless markets and markets with transaction costs. It relies heavily on advanced continuous-time stochastic analysis and functional analysis without any empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Unified FTAP in Continuous Time<br>Frictionless & Transaction Costs] --> B[Market Inputs: Risky Asset<br>Ask Price ≥ Bid Price]
    B --> C[Methodology: Apply Strict No-Arbitrage Condition]
    C --> D[Analytical Framework: Relax Concatenation &<br>Bounded Trading Volume Assumptions]
    D --> E[Key Outcome: Unified FTAP Valid for Markets<br>With and Without Proportional Transaction Costs]