Regime-Switching Models in Asset Returns: A Bifurcation Analysis

Code repository for our MAT244 final project (Summer 2026). The project models calm/turbulent volatility regimes in US equity markets as the two stable branches of a scalar nonlinear ODE with saddle-node (fold) bifurcations:

dx/dt = kappa (s + beta x - x^3),   sigma = sigma_bar * e^x,
s(t) = a * D(t) - b   (D = 125-day drawdown of the market index)

Folds at s = ±s*, s* = 2(beta/3)^(3/2), give two drawdown thresholds (calm→turbulent at D↑, turbulent→calm at D↓ < D↑) and hysteresis between them.

Repository layout

code/
  model.py                The ODE model, equilibria, fold thresholds (SymPy-verified).
  data_pipeline.py        Builds the aligned daily dataset 1990-2024 from the raw
                          market return series: log returns, index level, 20-day
                          realized volatility (annualized), 125-day drawdown.
  calibration.py          Two-stage anchored calibration (branch anchoring from
                          labeled regimes + fold anchoring from observed switch
                          drawdowns + 1-D least squares for kappa). Writes the
                          calibrated parameters and the simulated volatility path.
  fitting.py              Unconstrained 6-parameter least squares (the degenerate
                          fit discussed in the report's identifiability analysis).
  fitting_constrained.py  Box-constrained variant (also degenerate; see report).
  figures.py              Regenerates every figure in figures/.
data/
  ff_daily.csv            Raw daily market return series (Ken French market factor,
                          Mkt-RF + RF; see the data note in the report).
  dataset.csv             The processed dataset the model is fit on.
  sigma_model_anchored.csv  Calibrated model volatility path + basin labels.
  turbulent_episodes.csv  All model-identified turbulent episodes (>= 20 days).
  theta_anchored.npy      Calibrated parameters (kappa, beta, a, b, sigma_bar, x0).
figures/                  All report figures (PDF + PNG).
report/main_content.tex   LaTeX source of the report section this code supports.

Reproducing everything

Python 3.10+ with numpy, scipy, pandas, matplotlib (sympy for the symbolic checks):

pip install -r requirements.txt
python code/data_pipeline.py     # data/ff_daily.csv -> data/dataset.csv
python code/calibration.py       # calibrated parameters + simulated path + episodes
python code/figures.py           # regenerates figures/

Each script prints the numbers quoted in the report (thresholds, RMSEs, episode dates). fitting.py / fitting_constrained.py reproduce the two degenerate least-squares fits analyzed in the estimation section.

Headline calibrated values (training window 1990-2019)

beta = 0.216, sigma_bar = 17.9%, a = 1.56, b = 0.155, kappa = 0.97/day; fold thresholds D↑ = 12.4%, D↓ = 7.4%. Held-out 2020-2024: RMSE 9.8 volatility points; COVID-2020 and the 2022 bear market are both crossed as fold events out of sample.