Paper: arXiv 2603.28948

Authors: Yan Dolinsky, Xin Zhang

Abstract

We study scaled trinomial models converging to the Black–Scholes model, and analyze exponential certainty-equivalent prices for path-dependent European options. As the number of trading dates $n$ tends to infinity and the risk aversion is scaled as $nl$ for a fixed constant $l>0$, we derive a nontrivial scaling limit. Our analysis is purely probabilistic. Using a duality argument for the certainty equivalent, together with martingale and weak-convergence techniques, we show that the limiting problem takes the form of a volatility control problem with a specific penalty. For European options with Markovian payoffs, we analyze the optimal control problem and show that the corresponding delta-hedging strategy is asymptotically optimal for the primal problem.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using duality, martingale theory, and weak convergence to derive scaling limits for utility-based option pricing in incomplete markets. However, it is entirely theoretical with no empirical backtesting or real-market data application provided in the text.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Scaling limit of exponential hedging in trinomial models (n → ∞, risk aversion = nl)"] --> B["Inputs:<br>Scaled Trinomial Model & Path-dependent European Options"]
    
    B --> C["Duality Argument:<br>Transform exponential certainty equivalent to dual formulation"]
    
    C --> D["Probabilistic Analysis:<br>Apply martingale & weak-convergence techniques"]
    
    D --> E["Limiting Problem:<br>Formulate as a Volatility Control Problem with penalty"]
    
    E --> F["Markovian Payoff Analysis:<br>Solve optimal control & derive Delta-Hedging Strategy"]
    
    F --> G["Outcome:<br>Delta-hedging strategy is asymptotically optimal for primal problem"]