A general framework for pricing and hedging under local viability

In this paper, a new approach for solving the problems of pricing and hedging derivatives is introduced in a general frictionless market setting. The method is applicable even in cases where an equivalent local martingale measure fails to exist. Our main results include a new superhedging duality for American options when wealth processes can be negative and trading strategies are subject to a cone constraint. This answers one of the questions raised by Fernholz, Karatzas and Kardaras.

💡 Research Summary

The paper introduces a novel framework for pricing and hedging derivative securities in markets where an equivalent local martingale measure (ELMM) may not exist. Building on the classical Fundamental Theorem of Asset Pricing (FTAP), the authors replace the global No‑Free‑Lunch‑with‑Vanishing‑Risk (NFLVR) condition with a local version that holds only up to a sequence of stopping times. This “local viability” approach allows the analysis of models that exhibit arbitrage on a global scale—such as those driven by a three‑dimensional Bessel process—while still preserving enough structure to develop a coherent pricing theory.

The market consists of d risky assets modeled by a locally bounded ℝ^d‑valued semimartingale S and a risk‑free asset normalized to one. Trading constraints are encoded by a closed convex polyhedral cone C⊂ℝ^d (e.g., no‑short‑sale constraints). An x‑admissible strategy H starts from zero, stays inside C, and guarantees that the wealth process H·S never falls below –x. The set of all such strategies is denoted A_x, and A=∪_{x>0}A_x.

A localizing sequence {T_k} of increasing stopping times with P(T_k=∞)→1 is introduced. For each k, the stopped price process S^{T_k}=S_{·∧T_k} is bounded, and the corresponding cone‑generated set C_k is defined. The local NFLVR condition requires C_k‖∞∩L^∞_+= {0} for every k. Under this condition, a probability measure Q∼P exists such that S is a local Q‑martingale on each interval