American option pricing using generalised stochastic hybrid systems

ArXiv ID: 2409.07477 “View on arXiv”

Authors: Unknown

Abstract

This paper presents a novel approach to pricing American options using piecewise diffusion Markov processes (PDifMPs), a type of generalised stochastic hybrid system that integrates continuous dynamics with discrete jump processes. Standard models often rely on constant drift and volatility assumptions, which limits their ability to accurately capture the complex and erratic nature of financial markets. By incorporating PDifMPs, our method accounts for sudden market fluctuations, providing a more realistic model of asset price dynamics. We benchmark our approach with the Longstaff-Schwartz algorithm, both in its original form and modified to include PDifMP asset price trajectories. Numerical simulations demonstrate that our PDifMP-based method not only provides a more accurate reflection of market behaviour but also offers practical advantages in terms of computational efficiency. The results suggest that PDifMPs can significantly improve the predictive accuracy of American options pricing by more closely aligning with the stochastic volatility and jumps observed in real financial markets.

Keywords: American options, Markov processes, Stochastic hybrid systems, Option pricing, Finite difference methods

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper presents a highly advanced mathematical framework involving piecewise diffusion Markov processes (PDifMPs), stochastic hybrid systems, and extended generators, supported by extensive formulas and SDEs. While it benchmarks against Longstaff-Schwartz and discusses computational efficiency, the empirical section relies on numerical simulations without real-world backtesting, market data, or robust out-of-sample validation, placing it in the high-math, low-rigor category.

  flowchart TD
    A["Research Goal"] --> B["Methodology Setup"]
    B --> C["Data/Inputs"]
    C --> D{"Computational Process"}
    D --> E["Outcomes/Findings"]

    A -->|American option pricing using PDifMPs| A
    B -->|PDifMP Model Construction| B
    B -->|Benchmarking: LSA & Modified LSA| B

    C -->|Market Data<br>Model Parameters<br>Volatility Inputs| C

    D -->|Finite Difference Method<br>Monte Carlo Simulation| D

    E -->|Improved Predictive Accuracy| E
    E -->|Computational Efficiency Gains| E
    E -->|Realistic Market Representation| E