Paper: arXiv 2412.06343

Authors: Sourav Majumdar, Arnab Kumar Laha

Abstract

We develop diffusion models for time-varying correlation using stochastic processes defined on the unit circle. Specifically, we study Brownian motion on the circle and the von Mises diffusion, and propose their use as continuous-time models for correlation dynamics. The von Mises process, introduced by Kent (1975) as a characterization of the von Mises distribution in circular statistics, does not have a known closed-form transition density, which has limited its use in likelihood-based inference. We derive an accurate analytical approximation to the transition density of the von Mises diffusion, enabling practical likelihood-based estimation. We study inference for discretely observed circular diffusions, establish consistency and asymptotic normality of the resulting estimators, and propose a stochastic correlation model for financial applications. The methodology is illustrated through simulation studies and empirical applications to equity-foreign exchange market data.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces an innovative continuous-time stochastic correlation model using circular diffusions (specifically von Mises processes) and derives a novel analytical approximation to its transition density. It combines strong mathematical derivations with simulation and empirical applications to equity-FX data.

Research Flowchart

  flowchart TD
    A[Goal: Model Time-Varying Correlation via Circular Diffusions] --> B[Data: Equity & Foreign Exchange Market Data]
    B --> C[Method: Formulate Brownian Motion & von Mises Processes on the Unit Circle]
    C --> D[Computation: Derive Analytical Approximation for von Mises Transition Density]
    D --> E[Inference: Maximum Likelihood Estimation for Discretely Observed Data]
    E --> F[Outcomes: Consistent Estimators & Practical Financial Correlation Model]

Cited References (65)

  1. Time-reversible diffusions (1978)
  2. Discussion of Professor Mardia’s Paper (1975)
  3. Simulation and inference for stochastic differential equations: with R examples (2008)
  4. Some angular-linear distributions and related regression models (1978)
  5. A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations (1995)
  6. Infinite product expansion of the Fokker–Planck equation with steady-state solution (2015)
  7. Topics in Circular Statistics (2001)
  8. Directional Statistics (2000)
and 57 more
  1. Langevin diffusions on the torus: estimation and applications (2019)
  2. Estimation of non-normalized statistical models by score matching. (2005)
  3. Correlation risk and optimal portfolio choice (2010)
  4. Pricing Foreign Equity Options with Stochastic Correlation and Volatility. (2009)
  5. Stochastic correlation across international stock markets (2000)
  6. Option pricing when correlations are stochastic: an analytical framework (2007)
  7. On the Heston model with stochastic correlation (2016)
  8. Filtering in finance (2003)
  9. FX markets’ reactions to COVID-19: are they different? (2021)
  10. DEoptim: An R package for global optimization by differential evolution (2011)
  11. Modelling correlation as a stochastic process (2006)
  12. The correlation risk premium: International evidence (2022)
  13. The price of correlation risk: Evidence from equity options (2009)
  14. Modelling stochastic correlation (2016)
  15. Quanto pricing in stochastic correlation models (2018)
  16. The pricing of Quanto options under dynamic correlation (2015)
  17. Derivative pricing with Wishart multivariate stochastic volatility (2010)
  18. Multivariate stochastic volatility via Wishart processes (2006)
  19. Multivariate stochastic volatility: a review (2006)
  20. Quanto option pricing in the presence of fat tails and asymmetric dependence (2015)
  21. Multi-asset option pricing using normal tempered stable processes with stochastic correlation (2023)
  22. Circular-Statistics-Based Estimators and Tests for the Index Parameter $\alpha$ of Distributions for High-Volatility Financial Markets (2023)
  23. An Universal, Simple, Circular Statistics-Based Estimator of $\alpha$ for Symmetric Stable Family (2019)
  24. Analytical approximation to the multidimensional Fokker–Planck equation with steady state (2019)
  25. Asymptotics in statistics: some basic concepts (2000)
  26. sdetorus : Statistical Tools for Toroidal Diffusions (2023) — link
  27. Stochastic Integration and Differential Equations (2005)
  28. Score matching estimators for directional distributions (2016) — link
  29. Asymptotic theory for non-linear least squares estimator for diffusion processes (1983)
  30. Estimation for diffusion processes from discrete observation (1992)
  31. Bootstrap confidence cones for directional data (1985)
  32. Bootstrap confidence regions for directional data (1989)
  33. An introduction to the bootstrap (1994)
  34. Quanto Options (2010) — link
  35. Direct search algorithms for optimization calculations (1998)
  36. The NLopt nonlinear-optimization package (2007)
  37. High-frequency covariance estimates with noisy and asynchronous financial data (2010)
  38. Incorporating global industrial classification standard into portfolio allocation: A simple factor-based large covariance matrix estimator with high-frequency data (2016)
  39. A multiresolution method for parameter estimation of diffusion processes (2012)
  40. Likelihood inference for discretely observed nonlinear diffusions (2001)
  41. Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach (2002)
  42. Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion) (2006)
  43. Likelihood-based inference for correlated diffusions (2011)
  44. Statistical inference for stochastic differential equations (2023)
  45. Brownian motions in a Lie group (1950)
  46. Stochastic differential equations: an introduction with applications (2013)
  47. Naive Lie Theory (2008)
  48. Historical Real-Time Data (2024)
  49. A high-accuracy hybrid method for short-term wind power forecasting (2022)
  50. Numerical Solution of Stochastic Differential Equations (1992) — link
  51. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  52. A Theory of the Term Structure of Interest Rates (1985) — link
  53. Stochastic Differential Equations (2003) — link
  54. Statistical Inference for Ergodic Diffusion Processes (2004) — link
  55. Asymptotic Statistics (1998) — link
  56. "U ber die ``Ganzzahligkeit’’ der Atomgewichte und verwandte Fragen (1918)
  57. A family of toroidal diffusions with exact likelihood inference (2025) — link

0 of 65 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.