Paper: arXiv 2405.06764

Authors: Emmanuel Lepinette, Duc Thinh Vu

Abstract

The NA condition is one of the pillars supporting the classical theory of financial mathematics. We revisit this condition for financial market models where a dynamic risk-measure defined on $L^0$ is fixed to characterize the family of acceptable wealths that play the role of non negative financial positions. We provide in this setting a new version of the fundamental theorem of asset pricing and we deduce a dual characterization of the super-hedging prices (called risk-hedging prices) of a European option. Moreover, we show that the set of all risk-hedging prices is closed under NA. At last, we provide a dual representation of the risk-measure on $L^0$ under some conditions.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents highly advanced mathematical finance theory, establishing a new Fundamental Theorem of Asset Pricing (FTAP) and dual representations for coherent risk measures on $L^0$. It is purely theoretical with no empirical backtesting or data analysis, but it provides strong theoretical novelty for asset pricing and hedging theory.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend NA Condition & FTAP to Dynamic Risk Measures on L⁰] --> B[Inputs: Market Model & L⁰ Dynamic Risk Measure defining Acceptable Wealths]
    B --> C[Methodology: Analyze No-Arbitrage under Risk Constraints]
    C --> D[Derive FTAP & Characterize Martingale Measures]
    D --> E[Establish Dual Representation of Risk Measure & Closedness under NA]
    E --> F[Outcome: New FTAP & Dual Representation of Risk-Hedging Option Prices]

Cited References (34)

  1. Pricing without no-arbitrage condition in discrete time (2021) — link
  2. Super-hedging a European option with a coherent risk-measure and without no-arbitrage condition (2019)
  3. Theory of Rational Option Pricing (2015) — link
  4. SET-VALUED SHORTFALL AND DIVERGENCE RISK MEASURES (2014) — link
  5. VECTOR-VALUED COHERENT RISK MEASURE PROCESSES (2014) — link
  6. Time consistency of dynamic risk measures in markets with transaction costs (2012) — link
  7. The fundamental theorem of asset pricing for continuous processes under small transaction costs (2010) — link
  8. On convex risk measures on Lp-spaces (2009) — link
and 26 more
  1. Pricing and hedging European options with discrete-time coherent risk (2007) — link
  2. Convex risk measures and the dynamics of their penalty functions (2006) — link
  3. Pricing with Coherent Risk (2006) — link
  4. The Mathematics of Arbitrage (2006) — link
  5. Conditional and dynamic convex risk measures (2005) — link
  6. Limit theorem for Leland’s strategy (2003) — link
  7. Coherent risk measures (2000) — link
  8. The fundamental theorem of asset pricing for unbounded stochastic processes (1998) — link
  9. Local martingales and the fundamental asset pricing theorems in the discrete-time case (1998) — link
  10. Equivalent martingale measures and no-arbitrage (1994) — link
  11. A general version of the fundamental theorem of asset pricing (1994) — link
  12. A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time (1992) — link
  13. Martingales and stochastic integrals in the theory of continuous trading (1981) — link
  14. The arbitrage theory of capital asset pricing (1976) — link
  15. The Pricing of Options and Corporate Liabilities (1973) — link
  16. Variational Analysis (2014) — link
  17. Markets with transaction costs (2010) — link
  18. Dynamic convex risk measures (2004)
  19. Coherent Risk Measures on General Probability Spaces (2002) — link
  20. CHAPTER 14 – Set-Valued Integration and Set-Valued Probability Theory: An Overview (2002) — link
  21. A teacher’s note on no-arbitrage criteria (2001) — link
  22. No-Arbitrage and Equivalent Martingale Measures: An Elementary Proof of the Harrison–Pliska Theorem (1995) — link
  23. The no-arbitrage condition under a change of num´eraire (1995)
  24. Ju n 20 16 RISK ARBITRAGE AND HEDGING TO ACCEPTABILITY
  25. Value at Risk: the new benchmark for managing financial 31
  26. Pricing without martingale measures

0 of 34 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.