Perpetual American Standard and Lookback Options in Insider Models with Progressively Enlarged Filtrations

2025-07-04

AI Summary

This paper addresses the optimal stopping and pricing problem for perpetual American vanilla and lookback options under progressive enlargement of filtration in an insider trading model. The key feature is that insider information is characterized by the global extremum (rather than a stopping time) of the underlying asset price path. To tackle this, we formulate— for the first time—a three-dimensional free-boundary problem and develop a novel analytical method integrating reflection/entrance boundary conditions with smooth-fit principles. Using stochastic analysis, optimal stopping theory, and ODE techniques, we derive closed-form solutions for perpetual American call and put options. The optimal exercise boundary is shown to be a stochastic surface dependent on the running extremum; we characterize it via a system of nonlinear differential-transcendental equations and analyze its analytic properties. The core contribution lies in incorporating non-stopping-time global extrema into the filtration enlargement framework and achieving explicit resolution of a high-dimensional free boundary.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Derive closed-form solutions for perpetual American options pricing
• Model insider information using enlarged Brownian filtrations
• Determine optimal exercise times via stochastic boundaries

Innovation

• Closed-form solutions for optimal stopping problems
• Progressively enlarged Brownian filtrations modeling
• First-order nonlinear ODEs for boundary solutions