Paper: arXiv 2212.01119

Authors: Zbigniew Palmowski, Paweł Stępniak

Abstract

We derive the explicit price of the perpetual American put option cancelled at the last passage time of the underlying above some fixed level. We assume the asset process is governed by a geometric spectrally negative Lévy process. We show that the optimal exercise time is the first epoch when asset price process drops below an optimal threshold. We perform numerical analysis as well considering classical Black-Scholes models and the model where logarithm of the asset price has additional exponential downward shocks. The proof is based on some martingale arguments and fluctuation theory of Lévy processes.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical contribution to option pricing using advanced fluctuation theory of spectrally negative Lévy processes and martingale methods. However, empirical rigor is limited primarily to basic numerical examples rather than comprehensive backtesting against real market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Price Perpetual American Put<br>Cancelled at Last Passage Time] --> B[Inputs: Asset Process as Geometric<br>Spectrally Negative Lévy Process]
    B --> C[Methodology: Apply Martingale Arguments<br>& Fluctuation Theory]
    C --> D[Analytical Derivation:<br>Determine Optimal Exercise Threshold]
    D --> E[Numerical Analysis:<br>Test Black-Scholes & Model<br>with Downward Exponential Shocks]
    E --> F[Key Outcome: Explicit Pricing Formula &<br>Proof that Optimal Stopping is First Epoch<br>Below Threshold]