Paper: arXiv 2507.03470

Authors: Pavel V. Gapeev, Libo Li

Abstract

We derive closed-form solutions to the optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in the extensions of the Black-Merton-Scholes model with progressively enlarged filtrations. More specifically, the information available to the insider is modelled by Brownian filtrations progressively enlarged with the times of either the global maximum or minimum of the underlying risky asset price over the infinite time interval, which is not a stopping time in the filtration generated by the underlying risky asset. We show that the optimal exercise times are the first times at which the asset price process reaches either lower or upper stochastic boundaries depending on the current values of its running maximum or minimum given the occurrence of times of either the global maximum or minimum, respectively. The proof is based on the reduction of the original problems into the necessarily three-dimensional optimal stopping problems and the equivalent free-boundary problems. We apply either the normal-reflection or the normal-entrance conditions as well as the smooth-fit conditions for the value functions to characterise the candidate boundaries as either the maximal or minimal solutions to the associated first-order nonlinear ordinary differential equations and the transcendental arithmetic equations, respectively.

Complexity vs Empirical Score

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

Why this score: The paper presents highly sophisticated mathematical proofs involving progressive enlargement of filtrations, optimal stopping, and free-boundary problems in financial mathematics. However, it is purely theoretical with zero empirical backtesting or real-world market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Price Perpetual American Standard & Lookback Options for Insiders] --> B[Inputs: BMS Model & Enlarged Filtrations with Global Max/Min Times]
    B --> C[Reduce to 3D Optimal Stopping Problem]
    C --> D[Formulate Equivalent Free-Boundary Problem]
    D --> E[Apply Smooth-Fit, Normal-Reflection & Normal-Entrance Conditions]
    E --> F[Solve ODEs & Transcendental Equations for Stochastic Boundaries]
    F --> G[Outcomes: Closed-Form Solutions & Optimal Exercise Boundaries]

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0 of 61 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.