Paper: arXiv 2409.07477

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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

Cited References (30)

  1. Toward a general theory of stochastic hybrid systems (2006)
  2. Reachability questions in piecewise deterministic M arkov processes (2003)
  3. Processus de M arkov diffusifs par morceaux: outils analytiques et num 'e riques (2007)
  4. Stochastic differential equations and applications (2007)
  5. Option pricing: A simplified approach (1979)
  6. The use of the control variate technique in option pricing (1988)
  7. Options: A monte carlo approach (1977)
  8. Pricing American-style securities using simulation (1997)
and 22 more
  1. Finite difference methods and jump processes arising in the pricing of contingent claims: A synthesis (1978)
  2. Variational inequalities and the pricing of American options (1990)
  3. Chemotaxis in E scherichia C oli analysed by three-dimensional tracking (1972)
  4. Probabilistic and piecewise deterministic models in biology (2017)
  5. Fluid limit theorems for stochastic hybrid systems with application to neuron models (2010)
  6. An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution (2011)
  7. Stochastic hybrid systems for studying biochemical processes (2010)
  8. The regime switching portfolios (2011)
  9. Piecewise-deterministic M arkov processes: A general class of non-diffusion stochastic models (1984)
  10. From piecewise deterministic to piecewise diffusion M arkov processes (1988)
  11. PDE models for American options with counterparty risk and two stochastic factors: Mathematical analysis and numerical solution (2020)
  12. Pricing American options: A comparison of Monte Carlo simulation approaches (2001)
  13. Monte Carlo simulation for American options (2004)
  14. Theory of rational option pricing (1973)
  15. Option pricing when underlying stock returns are discontinuous (1976)
  16. A jump-diffusion model for option pricing (2002)
  17. The fine structure of asset returns: An empirical investigation (2002)
  18. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  19. Numerical valuation of European and American options under Kou’s jump-diffusion model (2008)
  20. Numerical Approximations and Convergence Analysis of Piecewise Diffusion Markov Processes, with Application to Glioma Cell Migration (2024) — link
  21. A stochastic hierarchical model for low grade glioma evolution (2023)
  22. Mod 'e lisation stochastique de systemes biologiques multi- 'e chelles et inhomogenes en espace (2018)

0 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.