Paper: arXiv 2503.11053

Abstract

This paper develops general approaches for pricing various types of American-style Parisian options (down-in/-out, perpetual/finite-maturity) with general payoff functions based on continuous-time Markov chain (CTMC) approximation under general 1D time-inhomogeneous Markov models. For the down-in types, by conditioning on the Parisian stopping time, we reduce the pricing problem to that of a series of vanilla American options with different maturities and their prices integrated with the distribution function of the Parisian stopping time yield the American Parisian down-in option price. This facilitates an efficient application of CTMC approximation to obtain the approximate option price by calculating the required quantities. For the perpetual down-in cases under time-homogeneous models, significant computational cost can be reduced. The down-out cases are more complicated, for which we use the state augmentation approach to record the excursion duration and then the approximate option price is obtained by solving a series of variational inequalities recursively with the Lemke’s pivoting method. We show the convergence of CTMC approximation for all the types of American Parisian options under general time-inhomogeneous Markov models, and the accuracy and efficiency of our algorithms are confirmed with extensive numerical experiments.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematical techniques including CTMC approximation, variational inequalities, and Lemke’s pivoting method for pricing exotic options under time-inhomogeneous models, which is highly mathematically dense. However, it also includes extensive numerical experiments, convergence proofs, and practical algorithms (like the state augmentation approach) that are implementation-heavy and backtest-ready, placing it in the Holy Grail quadrant.

Research Flowchart

  flowchart TD
  A["Research Goal: Pricing American Parisian Options<br>under Time-Inhomogeneous Markov Models"] --> B["Method: CTMC Approximation<br>State Augmentation"]
  
  B --> C{"Input: 1D Time-Inhomogeneous Markov Model"}
  
  C --> D{"Option Type?"}
  
  D -- Down-In --> E["Method: Conditioning & Integration<br>- Condition on Parisian stopping time<br>- Price series of vanilla American options<br>- Integrate with distribution of stopping time"]
  D -- Down-Out --> F["Method: State Augmentation & VI<br>- Record excursion duration via state expansion<br>- Solve recursive Variational Inequalities<br>- Apply Lemke's pivoting method"]
  
  E --> G["Computational Process:<br>CTMC Approximation to calculate<br>required quantities & option price"]
  F --> G
  
  G --> H["Key Outcomes:<br>- Convergence proof for CTMC under general models<br>- Efficient algorithms for finite & perpetual maturities<br>- Numerical accuracy & efficiency confirmed"]