Paper: arXiv 2412.11264

Abstract

We introduce a simple, efficient and accurate nonnegative preserving numerical scheme for simulating the square-root process. The novel idea is to simulate the integrated square-root process first instead of the square-root process itself. Numerical experiments on realistic parameter sets, applied for the integrated process and the Heston model, display high precision with a very low number of time steps. As a bonus, our scheme yields the exact limiting Inverse Gaussian distributions of the integrated square-root process with only one single time-step in two scenarios: (i) for high mean-reversion and volatility-of-volatility regimes, regardless of maturity; and (ii) for long maturities, independent of the other parameters.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical concepts like stochastic differential equations, affine structures, and explicit simulation schemes, but also includes numerical experiments and validation on realistic parameter sets with precise performance metrics.

Research Flowchart

  flowchart TD
  A["Research Goal: Simulate square-root process efficiently & accurately"] --> B["Key Methodology: Simulate integrated process first"]
  B --> C["Data/Inputs: Heston model parameters<br>realistic numerical setups"]
  C --> D["Computational Process: Apply new scheme<br>to integrated square-root process"]
  D --> E{"Key Outcomes"}
  E --> F["High precision with low time steps"]
  E --> G["Exact limiting Inverse Gaussian distributions<br>obtained in specific regimes"]
  E --> H["Nonnegativity preservation ensured"]

Cited References (34)

  1. Generating random variates using transformations with multiple roots (1976)
  2. Fast-reversion limit of the H eston model (2015)
  3. The large-maturity smile for the H eston model (2011)
  4. On spatially irregular ordinary differential equations and a pathwise volatility modelling framework (2019) — link
  5. Simulation of square-root processes made simple: applications to the H eston model — → 2412.11264
  6. A Theory of the Term Structure of Interest Rates (1985)
  7. Affine processes and applications in finance (2003)
  8. Credit risk: pricing, measurement, and management (2012)
and 26 more
  1. Reconciling rough volatility with jumps — → 2303.07222
  2. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  3. Continuous martingales and B rownian motion (2013)
  4. The volatility surface: A Practitioner’s Guide (2011)
  5. Laplace transforms and suprema of stochastic processes (2002)
  6. Exact simulation of stochastic volatility and other affine jump diffusion processes (2006)
  7. Efficient simulation of the H eston stochastic volatility model (2007)
  8. Gamma expansion of the H eston stochastic volatility model (2011)
  9. Simulation schemes for the H eston model with P oisson conditioning (2024)
  10. Simulating from the H eston model: A gamma approximation scheme (2015)
  11. Low-bias simulation scheme for the H eston model by I nverse G aussian approximation (2013)
  12. High order discretization schemes for the CIR process: application to affine term structure and H eston models (2010)
  13. On the discretization schemes for the CIR (and B essel squared) processes (2005)
  14. Convergence of discretized stochastic (interest rate) processes with stochastic drift term (1998)
  15. Convergence of M onte C arlo simulations involving the mean-reverting square root process (2005)
  16. Fast strong approximation M onte C arlo schemes for stochastic volatility models (2006)
  17. Euler scheme for SDE s with non- L ipschitz diffusion coefficient: strong convergence (2008)
  18. A note on E uler approximations for SDE s with H "o lder continuous diffusion coefficients (2011)
  19. Weak approximation of CKLS and CEV processes by discrete random variables (2020)
  20. First order strong approximations of scalar SDE s defined in a domain (2014)
  21. The little H eston trap (2007)
  22. Efficient Simulation of the Double H eston Model (2011)
  23. Simulating integrated V olterra square-root processes and V olterra H eston models via I nverse G aussian — → 2504.19885
  24. Efficient, almost exact simulation of the H eston stochastic volatility model (2010)
  25. A simple and accurate simulation approach to the H eston model (2011)
  26. Simple and efficient simulation of the H eston stochastic volatility model. (2008)

3 of 34 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.