Technical Documentation

System Architecture

The Stochastic Portfolio Engine is built with a modular architecture consisting of:

  1. Data Infrastructure Layer
  2. Hidden Markov Model Engine
  3. Stochastic Optimization Framework
  4. Risk Management System
  5. Backtesting & Analytics Engine

Hidden Markov Model Implementation

Mathematical Foundation

The regime detection system uses a Hidden Markov Model with the following structure:

  • Hidden States: Bull Market, Bear Market, High Volatility, Low Volatility
  • Observable Variables: Returns, Volatility, VIX levels, Yield curve slopes
  • Emission Distributions: Multivariate Gaussian with regime-specific parameters

Algorithm Implementation

# Pseudo-code for HMM regime detection
class HMMRegimeDetector:
    def __init__(self, n_states=4):
        self.model = GaussianHMM(n_components=n_states)
        
    def fit(self, observations):
        # Baum-Welch algorithm for parameter estimation
        self.model.fit(observations)
        
    def predict_regimes(self, data):
        # Viterbi algorithm for state sequence
        return self.model.predict(data)

Portfolio Optimization

Stochastic Differential Equations

The portfolio optimization incorporates multiple stochastic processes:

  • Geometric Brownian Motion: dS = μS dt + σS dW
  • Jump Diffusion: dS = μS dt + σS dW + S dN
  • Mean Reversion: dS = κ(θ - S) dt + σS dW

Optimization Framework

The system solves the following optimization problem:

Maximize: E[R_p] - λ * Var(R_p)
Subject to: Σw_i = 1, w_i ≥ 0

Where regime-conditional parameters are used based on HMM state probabilities.

Risk Management

Value at Risk (VaR)

  • Historical Simulation: Based on empirical return distributions
  • Monte Carlo VaR: Using simulated portfolio paths
  • Regime-Conditional VaR: Incorporating regime probabilities

Expected Shortfall (CVaR)

Implementation of coherent risk measures for tail risk assessment.

Performance Analytics

Attribution Analysis

  • Brinson Attribution: Allocation vs Selection effects
  • Factor Attribution: Fama-French factor exposures
  • Regime Attribution: Performance by market regime