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Monte Carlo Simulation

Statistical modeling for risk assessment

Historical Simulation Results

Monte Carlo simulation analysis across different market regimes, demonstrating the method's strengths in risk assessment rather than directional prediction.

Monte Carlo Price Path Simulation

Monte Carlo Price Path Simulation

Thousands of simulated price paths showing the range of potential outcomes and probability distributions for risk analysis.

Risk Metrics Analysis

Risk Metrics Analysis

VaR and Expected Shortfall calculations derived from simulation results, comparing model predictions with actual historical outcomes.

Model Validation Results

Model Validation Results

Statistical validation of Monte Carlo model assumptions including return distribution tests and prediction accuracy metrics.

Stochastic Process Modeling and Simulation Theory

Monte Carlo simulation uses massive random sampling to model complex stochastic processes like stock price movements.

Named after the Monte Carlo casino, this method computationally simulates thousands of possible future price paths.

The fundamental question: Can we use Monte Carlo simulation to predict stock prices, even directionally?

Traditional finance assumes stock prices follow a Wiener process (continuous-time random walk with drift).

Reality challenges this assumption: markets close overnight, volatility clusters, and fat-tail distributions exist.

Monte Carlo becomes powerful for risk management, scenario analysis, and options pricing rather than pure prediction.

The method excels at quantifying uncertainty and estimating probability distributions of future outcomes.

Mathematical Foundation

1
Geometric Brownian Motion

dSt=μStdt+σStdWtdS_t = \mu S_t dt + \sigma S_t dW_t

Stochastic differential equation modeling stock price evolution with drift μ and volatility σ.

2
Discrete Price Evolution

St+Δt=Stexp⁡((μ−σ22)Δt+σΔtZ)S_{t+\Delta t} = S_t \exp\left((\mu - \frac{\sigma^2}{2})\Delta t + \sigma\sqrt{\Delta t}Z\right)

Discrete time approximation where Z ~ N(0,1) is a random normal variable for each time step.

3
Log-Normal Distribution

ln⁡(ST)∼N(ln⁡(S0)+(μ−σ22)T,σ2T)\ln(S_T) \sim N\left(\ln(S_0) + (\mu - \frac{\sigma^2}{2})T, \sigma^2 T\right)

Future stock price follows log-normal distribution, ensuring prices remain positive.

4
Value at Risk (VaR)

VaRα=S0−Percentile(ST,α×100)VaR_{\alpha} = S_0 - \text{Percentile}(S_T, \alpha \times 100)

Monte Carlo estimation of Value at Risk at confidence level α using simulated price distribution.

5
Expected Shortfall

ESα=E[S0−ST∣ST<S0−VaRα]ES_{\alpha} = E[S_0 - S_T | S_T < S_0 - VaR_{\alpha}]

Expected loss beyond VaR threshold, providing tail risk measurement from simulation results.

Core Algorithm Implementation

The Monte Carlo simulation algorithm generates thousands of price paths to analyze risk scenarios and estimate probability distributions.

Parameter Estimation

python
def estimate_gbm_parameters(prices, lookback_days=252):
    """Estimate drift and volatility parameters from historical data"""
    returns = np.log(prices / prices.shift(1)).dropna()
    
    # Annualized parameters
    mu = returns.mean() * 252  # Drift (annualized)
    sigma = returns.std() * np.sqrt(252)  # Volatility (annualized)
    
    # Statistical tests
    from scipy import stats
    
    # Test for normality of returns
    shapiro_stat, shapiro_p = stats.shapiro(returns[-lookback_days:])
    
    # Test for autocorrelation (random walk assumption)
    ljung_box = sm.stats.diagnostic.acorr_ljungbox(returns[-lookback_days:], lags=10)
    
    params = {
        'mu': mu,
        'sigma': sigma,
        'current_price': prices.iloc[-1],
        'returns_normality_p': shapiro_p,
        'ljung_box_p': ljung_box['lb_pvalue'].iloc[-1],
        'lookback_period': lookback_days
    }
    
    return params

Estimates drift and volatility from historical returns with statistical validation of model assumptions.

Monte Carlo Price Simulation

python
def monte_carlo_simulation(S0, mu, sigma, T, dt, num_simulations=10000):
    """Generate Monte Carlo price paths using Geometric Brownian Motion"""
    
    # Time parameters
    num_steps = int(T / dt)
    times = np.linspace(0, T, num_steps + 1)
    
    # Pre-allocate simulation matrix
    simulations = np.zeros((num_simulations, num_steps + 1))
    simulations[:, 0] = S0
    
    # Generate random shocks
    np.random.seed(42)  # For reproducibility
    randoms = np.random.standard_normal((num_simulations, num_steps))
    
    # Simulate price paths
    for t in range(1, num_steps + 1):
        dt_sqrt = np.sqrt(dt)
        drift = (mu - 0.5 * sigma**2) * dt
        diffusion = sigma * dt_sqrt * randoms[:, t-1]
        
        simulations[:, t] = simulations[:, t-1] * np.exp(drift + diffusion)
    
    # Create results dataframe
    results = pd.DataFrame(
        simulations.T, 
        index=times,
        columns=[f'Path_{i}' for i in range(num_simulations)]
    )
    
    return results

Generates thousands of potential price paths using geometric Brownian motion with vectorized calculations for efficiency.

Risk Analysis and Statistics

python
def analyze_simulation_results(simulations, confidence_levels=[0.05, 0.01]):
    """Comprehensive analysis of Monte Carlo simulation results"""
    final_prices = simulations.iloc[-1]  # Terminal prices
    initial_price = simulations.iloc[0, 0]
    
    # Basic statistics
    stats_dict = {
        'mean_final_price': final_prices.mean(),
        'median_final_price': final_prices.median(),
        'std_final_price': final_prices.std(),
        'min_final_price': final_prices.min(),
        'max_final_price': final_prices.max(),
        'prob_profit': (final_prices > initial_price).mean(),
        'expected_return': (final_prices.mean() / initial_price - 1) * 100
    }
    
    # Value at Risk calculations
    for conf_level in confidence_levels:
        var_level = np.percentile(final_prices, conf_level * 100)
        var_loss = initial_price - var_level
        
        # Expected Shortfall (Conditional VaR)
        shortfall_scenarios = final_prices[final_prices < var_level]
        expected_shortfall = initial_price - shortfall_scenarios.mean() if len(shortfall_scenarios) > 0 else 0
        
        stats_dict[f'var_{int(conf_level*100)}'] = var_loss
        stats_dict[f'es_{int(conf_level*100)}'] = expected_shortfall
    
    # Path-dependent statistics
    max_drawdowns = []
    for col in simulations.columns:
        path = simulations[col]
        running_max = path.expanding().max()
        drawdown = (path / running_max - 1) * 100
        max_drawdowns.append(drawdown.min())
    
    stats_dict['avg_max_drawdown'] = np.mean(max_drawdowns)
    stats_dict['worst_drawdown'] = np.min(max_drawdowns)
    
    return stats_dict

Comprehensive risk analysis including VaR, Expected Shortfall, probability of profit, and path-dependent metrics.

Trading Strategy Integration

python
def monte_carlo_trading_strategy(prices, forecast_days=30, rebalance_freq=5):
    """Integrate Monte Carlo analysis into trading decisions"""
    
    results = []
    
    for i in range(len(prices) - forecast_days):
        # Historical data for parameter estimation
        hist_prices = prices.iloc[max(0, i-252):i+1]
        
        if len(hist_prices) < 30:  # Minimum data requirement
            continue
            
        # Estimate parameters
        params = estimate_gbm_parameters(hist_prices)
        
        # Run simulation
        simulations = monte_carlo_simulation(
            S0=params['current_price'],
            mu=params['mu'], 
            sigma=params['sigma'],
            T=forecast_days/252,  # Convert to years
            dt=1/252,
            num_simulations=1000
        )
        
        # Analyze results
        analysis = analyze_simulation_results(simulations)
        
        # Generate trading signal based on risk-reward profile
        prob_profit = analysis['prob_profit']
        expected_return = analysis['expected_return']
        var_5 = analysis['var_5']
        
        # Risk-adjusted signal generation
        if prob_profit > 0.6 and expected_return > 5 and var_5 < initial_price * 0.1:
            signal = 1  # Bullish
        elif prob_profit < 0.4 and expected_return < -2:
            signal = -1  # Bearish
        else:
            signal = 0  # Neutral
            
        results.append({
            'date': prices.index[i],
            'price': params['current_price'],
            'signal': signal,
            'prob_profit': prob_profit,
            'expected_return': expected_return,
            'var_5': var_5,
            'forecast_period': forecast_days
        })
    
    return pd.DataFrame(results)

Integrates Monte Carlo analysis into systematic trading decisions using risk-reward metrics and probability thresholds.

Implementation Steps

  1. 1Estimate drift (μ) and volatility (σ) parameters from historical price data
  2. 2Validate model assumptions: test for return normality and independence
  3. 3Configure simulation parameters: time horizon, number of paths, time steps
  4. 4Generate thousands of price paths using Geometric Brownian Motion
  5. 5Calculate comprehensive risk metrics: VaR, Expected Shortfall, probability of profit
  6. 6Analyze path-dependent statistics like maximum drawdown scenarios
  7. 7Generate trading signals based on risk-adjusted expected returns
  8. 8Implement position sizing based on simulation-derived risk estimates
  9. 9Monitor model performance and recalibrate parameters periodically

Key Metrics

Simulation convergence (how many paths needed for stable results)
Model validation metrics (Kolmogorov-Smirnov test, backtesting exceptions)
Calibration accuracy (predicted vs realized volatility and returns)
Risk metric reliability (VaR backtesting, Expected Shortfall coverage)
Signal accuracy (probability forecasts vs actual outcomes)
Computational efficiency (simulation time vs forecast accuracy)
Parameter sensitivity (results stability across different estimation windows)

Risk Considerations

Model assumes constant volatility and drift, which rarely holds in real markets
Fat-tail events and volatility clustering are not captured by standard Brownian motion
Over-reliance on historical data may miss regime changes and structural breaks
Simulation results can provide false confidence in quantitative precision
Computational complexity requires significant processing power for real-time applications
Normal distribution assumption fails during market crises and extreme events
Path independence assumption ignores momentum, mean reversion, and other market dynamics

Practice Implementation

Prerequisites

Mathematical Background

  • • Linear regression and OLS estimation
  • • Time series analysis (stationarity, unit roots)
  • • Hypothesis testing and p-values
  • • Basic econometrics (error correction models)

Technical Skills

  • • Python programming (pandas, numpy)
  • • Statistical libraries (statsmodels)
  • • Data visualization (matplotlib)
  • • Financial data handling (yfinance)

Complete Implementation

Access the full Python implementation from the original quantitative trading repository:

bash
# Complete pair trading implementation
git clone https://github.com/je-suis-tm/quant-trading.git
cd quant-trading
python "Pair trading backtest.py"
# Modify tickers and parameters for your own analysis

Learning Checkpoints

1

Understand Cointegration

Can you explain why two assets might be cointegrated and what breaks this relationship?

2

Interpret Statistical Tests

Practice reading ADF test results and understanding when to accept/reject cointegration.

3

Signal Generation

Implement Z-score calculations and understand threshold selection (±1σ vs ±2σ).

4

Risk Management

Understand position sizing, monitoring regime changes, and exit strategies.

Recommended Learning Path

Immediate Actions

  • Download and run the Python script
  • Test with different asset pairs
  • Experiment with threshold parameters

Advanced Studies

  • Learn Johansen cointegration test
  • Study Vector Error Correction Models
  • Explore multiple asset pair trading

Important Disclaimer

This strategy involves significant risk. Historical cointegration relationships can break permanently. Always use proper risk management, position sizing, and never risk more than you can afford to lose. Paper trade extensively before using real capital.

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