Paper: arXiv 2212.12860

Authors: Libo Li, Ruyi Liu, Marek Rutkowski

Abstract

We study the upper and lower bounds for prices of European and American style options with the possibility of an external termination, meaning that the contract may be terminated at some random time. Under the assumption that the underlying market model is incomplete and frictionless, we obtain duality results linking the upper price of a vulnerable European option with the price of an American option whose exercise times are constrained to times at which the external termination can happen with a non-zero probability. Similarly, the upper and lower prices for an vulnerable American option are linked to the price of an American option and a game option, respectively. In particular, the minimizer of the game option is only allowed to stop at times which the external termination may occur with a non-zero probability.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly theoretical and mathematically rigorous framework for vulnerable options pricing with optional hazard processes, deriving duality results for European, American, and game options. However, it is purely mathematical without empirical validation, datasets, or backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Price Vulnerable European & American Options<br>(Incomplete Markets with External Termination)"] --> B["Inputs: Market Model & Optional Hazard Process<br>(Random Contract Termination Risk)"]
    B --> C["Duality Theory & Stochastic Optimization"]
    C --> D["Convert Termination Risk to Constrained Stopping Times<br>(Non-Zero Probability Termination Points)"]
    D --> E["Formulate Hedging & Pricing Bounds<br>(Upper/Lower Prices via Optimal Stopping & Game Options)"]
    E --> F["Key Findings: Vulnerable Options Duality Results<br>• Vulnerable European Upper Price = Constrained American Price<br>• Vulnerable American Upper Price = Standard American Price<br>• Vulnerable American Lower Price = Constrained Game Option Price"]