Bayesian Nonlinear Pricing Models with Regime-Switching and Adaptive Volatility Filters

Overview

This repository provides a complete Python implementation of a Bayesian framework for estimating nonlinear asset pricing models that incorporate regime-switching and stochastic volatility. The model integrates a Hidden Markov Model (HMM) for discrete regime states, nonlinear stochastic volatility dynamics, and a Particle Markov Chain Monte Carlo (PMCMC) algorithm for joint state-parameter inference. We employ an Unscented Kalman Filter (UKF)-based filtering procedure to adaptively estimate latent volatility under uncertainty in both parameters and regimes.

The provided code offers:

• Nonlinear State-Space Representation:
  Handles complex observation models relating latent volatility and regimes to observed prices.

• Regime-Switching Dynamics (HMM):
  Integrates discrete state transitions to capture shifts in market conditions, sentiment, or liquidity.

• Stochastic Volatility Modeling:
  Employs a mean-reverting stochastic volatility process (Heston-type dynamics) conditioned on regimes.

• Unscented Kalman Filter (UKF):
  Efficient approximation of nonlinear filtering distributions for the continuous state (volatility).

• Particle MCMC (PMCMC) for Parameter Estimation:
  Joint inference over model parameters and latent states via a combination of particle filtering and MCMC sampling steps.

This project is intended for quantitative researchers and practitioners interested in advanced Bayesian methods for financial time series and state-space modeling.

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Mathematical Foundations

1. Model Setup

We consider a discrete-time asset price series $$(S_t)_{t=1}^T $$. To model price dynamics under uncertainty and structural changes, we introduce:

1. Regime Variable $$R_t$$:
   A latent Markov chain representing market regimes:

$$ R_t \in {1,2,\dots,M}, \quad P(R_t=j \mid R_{t-1}=i) = \Pi_{ij}. $$

The regime influences the parameters governing the price and volatility evolution.

3. Stochastic Volatility State $$v_t$$:
   Let $$v_t > 0$$ denote the latent volatility level at time $$t$$. We assume a mean-reverting square-root process (Heston-type):

$$ v_t = v_{t-1} + \kappa_{v,R_t}(\theta_{v,R_t} - v_{t-1}) + \sigma_{v,R_t}\sqrt{\max(v_{t-1}, 0)},\eta_t, \quad \eta_t \sim N(0,1). $$

5. Observation Model (Prices): We consider log-returns:

$$ r_t = \log\left(\frac{S_t}{S_{t-1}}\right). $$ In regime $$R_t = i$$, we have:

$$ r_t \mid v_t, R_t=i \sim N(\mu_i, v_t\sigma_{level,i}^2). $$

Thus:

$$ \log(S_t) = \log(S_{t-1}) + r_t, \quad r_t \sim N(\mu_i, v_t \sigma_{level,i}^2). $$

The parameters $${\mu_i, \sigma_{level,i}, \kappa_{v,i}, \theta_{v,i}, \sigma_{v,i}}$$ differ across regimes $$i$$.

2. Hidden Markov Model (HMM)

The regime process $$(R_t)$$ is a first-order Markov chain with transition matrix $$\Pi$$. Let:

$$ \Pi = [\Pi_{ij}]_{i,j=1}^M, \quad \Pi_{ij} = P(R_t=j \mid R_{t-1}=i). $$

The joint distribution of $$ {R_t} $$ given $$ \Pi $$ is:

$$ p(R_1,\ldots,R_T \mid \Pi) = \pi_{R_1} \prod_{t=2}^T \Pi_{R_{t-1},R_t}, $$

where $$\pi$$ is the initial regime distribution (often stationary).

3. State-Space Formulation

Define the continuous latent state $$\mathbf{x}_t = v_t$$. The joint model is:

1. State Transition:

$$ v_t \mid v_{t-1}, R_t \sim N\left(v_{t-1} + \kappa_{v,R_t}(\theta_{v,R_t} - v_{t-1}), \sigma_{v,R_t}^2 v_{t-1}\right) \quad \text{(with non-negativity enforced)} $$

2. Observation Model:

$$ \log(S_t) \mid v_t, R_t=i \sim N(\log(S_{t-1}) + \mu_i, v_t \sigma_{level,i}^2). $$

This is a nonlinear state-space model with discrete switching (HMM). The nonlinearity arises from the volatility dynamics and lognormal price observation.

4. Bayesian Hierarchical Structure

We place priors on model parameters. For each regime $$ i $$:

• $$\kappa_{v,i}$$ may have a Gamma prior: $$\kappa_{v,i} \sim \Gamma(a_\kappa,b_\kappa)$$.
• $$\theta_{v,i}$$ may have a Gaussian prior: $$\theta_{v,i} \sim N(\mu_\theta,\sigma_\theta^2)$$.
• $$\sigma_{v,i}$$ may have a Half-Cauchy prior to ensure positivity and heavy-tailed flexibility.
• $$\mu_i \sim N(0,\sigma_\mu^2)$$, and $$\sigma_{level,i}$$ also from a Half-Cauchy.

For the regime transition matrix $$\Pi$$, each row can have a Dirichlet prior to ensure valid probability distributions:

$$ \Pi_{i,:} \sim \text{Dirichlet}(\alpha,\alpha,\dots,\alpha). $$

The resulting joint posterior is:

$$ p(\Theta, {v_t}, {R_t} \mid {S_t}_{t=1}^T) \propto p({S_t} \mid {v_t},{R_t},\Theta)p({v_t},{R_t}\mid\Theta)p(\Theta). $$

5. Nonlinear Filtering: The Unscented Kalman Filter (UKF)

Given known parameters $$ \Theta $$, we seek the filtering distribution:

$$ p(v_t, R_t \mid S_{1:t}, \Theta). $$

Since $${R_t}$$ is discrete, we can compute regime probabilities and state distributions simultaneously. The UKF provides a Gaussian approximation for the conditional distribution of $$v_t$$ given $$R_t=i$$. For each regime, we maintain a Gaussian approximation of the volatility state distribution. The overall distribution is a mixture:

$$ p(v_t \mid S_{1:t}, \Theta) = \sum_{i=1}^M p(R_t=i \mid S_{1:t}, \Theta) , p(v_t \mid R_t=i, S_{1:t}, \Theta). $$

UKF Steps per Regime:

1. Sigma Points: Generate a set of sigma points $${\chi^{(j)}_{t-1}}$$ from the Gaussian distribution at time $$t-1$$.

2. Time Update: Propagate each sigma point through the nonlinear state equation:

$$ \chi^{(j)}_t = f_{R_t}(\chi^{(j)}_{t-1}). $$

Compute the predicted mean and covariance.

3. Measurement Update: Propagate sigma points through the observation model:

   $$ y^{(j)}t = h{R_t}(\chi^{(j)}_t), $$

   update state mean and covariance using the observed $$S_t$$.

Since $$R_t$$ is unknown, we form a mixture by weighting each regime's UKF prediction by the HMM probabilities and integrate out the regimes.

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Particle MCMC (PMCMC) for Joint Parameter and State Inference

With uncertain parameters $$\Theta$$, we must integrate over them as well. Direct inference is challenging due to the nonlinear, non-Gaussian nature of the model.

Approach: Particle MCMC (Andrieu et al., 2010)

1. Particle Filter for Latent States: Given a candidate parameter set $$\Theta$$, we run a particle filter to approximate:

$$ \hat{p}(S_{1:T} \mid \Theta) \approx p(S_{1:T} \mid \Theta). $$

The particle filter simulates ( (v_t^{(n)}, R_t^{(n)}) ) for ( n=1,\ldots,N ) particles, resampling at each step. The likelihood estimate is the product of normalized weights:

$$ \hat{p}(S_{1:T} \mid \Theta) = \prod_{t=1}^T \left(\frac{1}{N} \sum_{n=1}^N w_t^{(n)}\right). $$

3. MCMC Step (Metropolis-Hastings):

We have a current parameter draw ( \Theta^{(m)} ). Propose ( \Theta^* ) from a proposal distribution ( q(\Theta^* \mid \Theta^{(m)}) ). Accept or reject based on the posterior ratio:

$$ A = \frac{p(\Theta^_) \hat{p}(S_{1:T} \mid \Theta^_) q(\Theta^{(m)} \mid \Theta^_)}{p(\Theta^{(m)}) \hat{p}(S_{1:T} \mid \Theta^{(m)}) q(\Theta^_ \mid \Theta^{(m)})}. $$

If ( U \sim \text{Uniform}(0,1) ), accept if ( U < \min(1, A) ), otherwise reject and set ( \Theta^{(m+1)} = \Theta^{(m)} ).

After burn-in, the chain ( {\Theta^{(m)}} ) approximates the posterior distribution over parameters.

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Code Structure

The code is organized as follows:

bayesian_nonlinear_pricing/
    __init__.py          # Package initialization
    data_generation.py   # Synthetic data generation
    model_params.py      # Parameter management, priors, structure
    hmm.py               # Hidden Markov Model utilities (forward-backward, sampling)
    statespace.py        # Nonlinear state & observation models
    filters.py           # Regime-Switching UKF implementation
    inference.py         # Particle filter for latent states
    utils.py             # Numeric utilities (sigma points, density evaluation)
    pmcmc.py             # Particle MCMC integration for parameter inference
    main.py              # Example script: simulate data, run PMCMC, estimate states

Workflow

1. Data Generation:
   Use data_generation.py to produce synthetic $$ (S_t, R_t, v_t) $$ from known parameters. This serves as ground truth for testing inference methods.

2. Initialization:
   Draw initial parameters from specified priors in model_params.py.

3. PMCMC Inference:
   pmcmc.py runs a PMCMC algorithm. Each iteration:

   • Propose new parameters $$ \Theta^* $$.
   • Run particle filter to compute $$ \hat{p}(S_{1:T} \mid \Theta^*) $$.
   • Accept/reject the proposal, forming a posterior chain.

4. Filtering with UKF:
   Once parameters are estimated, use filters.py (RegimeSwitchingUKF) to produce smoothed volatility and regime estimates given the posterior means or samples of $$ \Theta $$.

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Running the Code

Requirements: Python 3, numpy, scipy.

cd bayesian_nonlinear_pricing
python3 main.py

The main.py script:

1. Generates synthetic data.
2. Samples initial parameters from priors.
3. Runs PMCMC for a specified number of iterations.
4. Retrieves posterior parameter estimates.
5. Performs UKF-based state filtering with the estimated parameters.

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Interpreting Results

• Parameter Estimates:
  The PMCMC chain provides draws from the posterior of $$ \Theta $$. Summaries (mean, median, credible intervals) characterize the uncertainty around model parameters.

• Volatility Filtering:
  The final UKF step provides an estimated volatility path $$ \hat{v}_t $$ that can be compared to the true simulated volatility for validation.

• Regime Probabilities:
  At each time $$ t $$, the filtered regime probabilities $$ P(R_t=i \mid S_{1:t}) $$ reveal how the algorithm identifies shifts in market states.

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Extensions and Advanced Topics

1. Multi-Factor Volatility Models: Extend $$ \mathbf{x}_t $$ to include multiple state factors (e.g., jump intensities, risk premia). The UKF and particle filters generalize naturally to higher dimensions.

2. Non-Gaussian Observation Models: If returns exhibit heavy tails or skewness, replace Gaussian assumptions with more flexible distributions (e.g., Student-t), requiring appropriate filtering methods (e.g., particle filters only).

3. Time-Varying Regime Dynamics: Consider more complex regime dynamics or Bayesian updating of the HMM parameters over time.

4. Efficient Proposals and Tuning: The current PMCMC uses simplistic random-walk proposals. Advanced strategies (e.g., adaptive MCMC, Hamiltonian Monte Carlo) could improve mixing and efficiency.

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References

• Particle MCMC:

  • Andrieu, C., Doucet, A., & Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. Royal Statistical Society: Series B, 72(3), 269–342.

• Heston Volatility Model:

  • Heston, S. L. (1993). A closed-form solution for options with stochastic volatility. Review of Financial Studies, 6(2), 327–343.

• Unscented Kalman Filter:

  • Julier, S.J., & Uhlmann, J.K. (1997). A new extension of the Kalman filter to nonlinear systems. Proceedings of AeroSense: The 11th Int. Symp. on Aerospace/Defense Sensing, Simulation and Controls.

• Hidden Markov Models in Finance:

  • Hamilton, J.D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2), 357–384.