Rating
1653
Battle Count: 50
Relevance
6/10
Highly relevant for theoretical quant finance, particularly for understanding arbitrage conditions in markets with discrete price ticks (e.g., crypto or specific equity exchanges). It provides a rigorous foundation for models that do not rely on continuous diffusion assumptions, which is crucial for high-frequency trading strategies dealing with discrete order books.
Implementation Complexity
9/10
Extremely high complexity. The paper is a dense mathematical proof involving functional analysis, Hilbert spaces, and stochastic calculus. It does not provide an algorithm or code for direct implementation but rather establishes theoretical equivalences.
Reproducibility
3/5
The paper is purely theoretical. All mathematical definitions, assumptions, and proofs are fully described within the text. No external data or code is required to verify the logical consistency of the proofs, though implementing the numerical implications would require custom development.
Abstract
Motivated by discrete price quotes in continuous time, we seek a fundamental theorem based on no arbitrage rather than a condition on limiting trading opportunities. We consider a vector of discounted prices taking values in a locally finite subset of R^d in continuous time. For a locally bounded càdlàg price, we prove that no arbitrage, no free lunch with vanishing risk, no free lunch, and the existence of an equivalent local martingale measure are equivalent. An elementary finite-dimensional reduction of portfolios yields a common pathwise bound on admissible wealth. This bound reduces closedness to convex averaging in a weighted Hilbert space, after which the Kreps-Yan theorem applies. Trading gains are finite Stieltjes sums, and the proof uses no Itô integration.
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