Advanced Mathematical Frameworks

Stochastic MEV Opportunity Model

Modeling MEV opportunity size as a stochastic process:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
from scipy.optimize import minimize

class StochasticMEVModel:
    def __init__(self, dt=1/365, T=1.0):
        self.dt = dt  # Time step (daily)
        self.T = T    # Total time horizon
        self.n_steps = int(T / dt)
    
    def simulate_mev_opportunity(self, initial_size, drift, volatility, n_sims=1000):
        """
        Simulate MEV opportunity evolution using Euler-Maruyama scheme
        
        dX(t) = μ(X,t)dt + σ(X,t)dW(t)
        """
        X = np.zeros((n_sims, self.n_steps + 1))
        X[:, 0] = initial_size
        
        for i in range(self.n_steps):
            # Generate random increments
            dW = np.random.normal(0, np.sqrt(self.dt), n_sims)
            
            # Apply drift and volatility
            mu = drift(X[:, i])
            sigma = volatility(X[:, i])
            
            # Euler-Maruyama discretization
            X[:, i + 1] = X[:, i] + mu * self.dt + sigma * dW
        
        return X
    
    def estimate_parameters(self, observed_data):
        """Estimate drift and volatility parameters from data"""
        n = len(observed_data)
        
        # Calculate log returns
        returns = np.diff(np.log(observed_data))
        
        # Estimate drift using maximum likelihood
        drift_est = np.mean(returns) / self.dt
        
        # Estimate volatility
        vol_est = np.std(returns) / np.sqrt(self.dt)
        
        return {
            'drift': drift_est,
            'volatility': vol_est
        }
    
    def optimal_exercise_boundary(self, X, K, r=0.05):
        """
        Find optimal exercise boundary for MEV opportunity
        Problem: When to execute vs wait for better opportunity
        """
        # Value function approximation using PDE
        def value_function(x, t):
            # Simplified boundary conditions
            intrinsic = np.maximum(x - K, 0)
            continuation = 0.95 * np.mean(x) * np.exp(-r * (self.T - t))
            return np.maximum(intrinsic, continuation)
        
        # Find stopping boundary
        boundary = []
        for t_idx in range(self.n_steps + 1):
            t = t_idx * self.dt
            values = [value_function(x, t) for x in X]
            stopping_boundary = np.mean([x for x, v in zip(X, values) if v == np.maximum(x - K, 0)])
            boundary.append(stopping_boundary)
        
        return boundary

# Example usage
model = StochasticMEVModel()

# Define drift and volatility functions
def drift_function(x):
    """Mean-reverting drift"""
    return 0.02 * (1.0 - x)  # Pull towards equilibrium at 1.0

def volatility_function(x):
    """Volatility increases with opportunity size"""
    return 0.3 * np.sqrt(x)

# Simulate MEV opportunities
X = model.simulate_mev_opportunity(
    initial_size=1.0,
    drift=drift_function,
    volatility=volatility_function,
    n_sims=1000
)

# Plot results
plt.figure(figsize=(12, 6))
time_grid = np.linspace(0, model.T, model.n_steps + 1)

# Plot sample paths
for i in range(10):
    plt.plot(time_grid, X[i, :], alpha=0.7)

plt.xlabel('Time (years)')
plt.ylabel('MEV Opportunity Size')
plt.title('Simulated MEV Opportunity Evolution')
plt.grid(True, alpha=0.3)
plt.show()

# Calculate optimal exercise boundary
boundary = model.optimal_exercise_boundary(X, K=0.8)
plt.figure(figsize=(10, 6))
plt.plot(time_grid, boundary, 'r-', linewidth=2, label='Exercise Boundary')
plt.xlabel('Time')
plt.ylabel('Exercise Threshold')
plt.title('Optimal Exercise Boundary for MEV Opportunity')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()

print(f"Simulated {X.shape[0]} scenarios over {X.shape[1]} time steps")
print(f"Final MEV opportunity stats:")
print(f"  Mean: {np.mean(X[:, -1]):.3f}")
print(f"  Std:  {np.std(X[:, -1]):.3f}")
print(f"  Min:  {np.min(X[:, -1]):.3f}")
print(f"  Max:  {np.max(X[:, -1]):.3f}")

Game Theory Validator Auction Model

Modeling strategic interactions in validator auctions:

import numpy as np
from scipy.optimize import minimize
import matplotlib.pyplot as plt

class ValidatorAuctionGame:
    def __init__(self, n_players, block_value, bid_cost=0):
        self.n_players = n_players
        self.block_value = block_value
        self.bid_cost = bid_cost
        
    def nash_equilibrium(self, cost_functions, valuations):
        """
        Find Nash equilibrium in validator auction game
        Players choose bids simultaneously
        """
        def game_objective(bids):
            """Calculate payoffs for all players given bids"""
            payoffs = np.zeros(self.n_players)
            total_bids = np.sum(bids)
            
            if total_bids == 0:
                return payoffs
            
            # Winner takes all
            winner = np.argmax(bids)
            effective_payoff = self.block_value - bids[winner] - self.bid_cost
            
            # Calculate each player's payoff
            for i in range(self.n_players):
                if i == winner:
                    payoffs[i] = effective_payoff
                else:
                    payoffs[i] = -self.bid_cost  # Lost bid cost
            
            return payoffs
        
        # Find equilibrium using best response dynamics
        bids = np.random.uniform(0, self.block_value, self.n_players)
        
        for iteration in range(1000):  # Max iterations
            new_bids = bids.copy()
            total_improvement = 0
            
            for i in range(self.n_players):
                # Find best response for player i
                def objective(bid):
                    test_bids = bids.copy()
                    test_bids[i] = bid
                    return -game_objective(test_bids)[i]  # Minimize negative payoff
                
                # Optimize bid for player i
                result = minimize(objective, bids[i], 
                                bounds=[(0, self.block_value)],
                                method='L-BFGS-B')
                
                if result.success:
                    improvement = abs(bids[i] - result.x[0])
                    total_improvement += improvement
                    new_bids[i] = result.x[0]
            
            bids = new_bids
            
            # Check convergence
            if total_improvement < 1e-6:
                print(f"Converged after {iteration} iterations")
                break
        
        return bids
    
    def repeated_game_analysis(self, n_rounds=100, discount_factor=0.9):
        """Analyze repeated validator auctions"""
        results = {
            'average_bids': [],
            'cooperation_level': [],
            'total_payoffs': np.zeros(self.n_players)
        }
        
        # Tit-for-tat strategy simulation
        for round_num in range(n_rounds):
            # Calculate bid based on history
            bids = np.zeros(self.n_players)
            
            # First round: low bids (cooperative)
            if round_num == 0:
                bids = np.full(self.n_players, 0.3 * self.block_value)
            else:
                # Tit-for-tat: match previous cooperation level
                cooperation = results['cooperation_level'][-1] if results['cooperation_level'] else 0.3
                bids = np.full(self.n_players, cooperation * self.block_value)
            
            # Add some noise to prevent perfect coordination
            bids += np.random.normal(0, 0.05 * self.block_value, self.n_players)
            bids = np.clip(bids, 0, self.block_value)
            
            # Calculate payoffs
            payoffs = self.payoff_function(bids)
            results['total_payoffs'] += (discount_factor ** round_num) * payoffs
            
            # Update cooperation level
            avg_bid = np.mean(bids)
            cooperation_level = avg_bid / self.block_value
            results['average_bids'].append(avg_bid)
            results['cooperation_level'].append(cooperation_level)
        
        return results
    
    def payoff_function(self, bids):
        """Calculate payoffs for given bids"""
        payoffs = np.zeros(self.n_players)
        
        if np.sum(bids) == 0:
            return payoffs
        
        # Winner determination
        winner = np.argmax(bids)
        
        # First-price auction: winner pays their bid
        payoffs[winner] = self.block_value - bids[winner] - self.bid_cost
        
        # Losers pay only bid costs
        for i in range(self.n_players):
            if i != winner:
                payoffs[i] = -self.bid_cost
        
        return payoffs
    
    def analyze_competition_intensity(self, player_types):
        """
        Analyze how different player types affect competition
        player_types: list of 'aggressive', 'conservative', 'random'
        """
        strategies = {
            'aggressive': lambda x: np.random.uniform(0.6, 0.9) * self.block_value,
            'conservative': lambda x: np.random.uniform(0.1, 0.4) * self.block_value,
            'random': lambda x: np.random.uniform(0, self.block_value)
        }
        
        n_scenarios = 100
        competition_metrics = []
        
        for _ in range(n_scenarios):
            # Generate bids based on player types
            bids = np.array([strategies[p_type]() for p_type in player_types])
            
            # Calculate competition metrics
            bid_spread = np.max(bids) - np.min(bids)
            total_value = np.sum(bids)
            concentration = np.max(bids) / total_value if total_value > 0 else 0
            
            competition_metrics.append({
                'bid_spread': bid_spread,
                'total_bids': total_value,
                'concentration': concentration,
                'highest_bid': np.max(bids)
            })
        
        # Aggregate results
        avg_metrics = {
            'avg_spread': np.mean([m['bid_spread'] for m in competition_metrics]),
            'avg_concentration': np.mean([m['concentration'] for m in competition_metrics]),
            'competition_intensity': np.mean([m['highest_bid'] for m in competition_metrics]) / self.block_value
        }
        
        return avg_metrics, competition_metrics

# Example: Analyze validator competition
game = ValidatorAuctionGame(n_players=4, block_value=2.0)

# Compare different player type combinations
scenarios = [
    ['aggressive', 'aggressive', 'aggressive', 'aggressive'],
    ['conservative', 'conservative', 'conservative', 'conservative'],
    ['aggressive', 'conservative', 'random', 'random'],
    ['aggressive', 'aggressive', 'conservative', 'conservative']
]

print("Validator Auction Game Analysis")
print("=" * 40)

for i, scenario in enumerate(scenarios):
    print(f"\nScenario {i+1}: {' vs '.join(scenario)}")
    metrics, _ = game.analyze_competition_intensity(scenario)
    
    print(f"  Competition Intensity: {metrics['competition_intensity']:.3f}")
    print(f"  Average Spread: ${metrics['avg_spread']:.3f}")
    print(f"  Market Concentration: {metrics['avg_concentration']:.3f}")

# Analyze repeated game dynamics
repeated_results = game.repeated_game_analysis(n_rounds=50)
print(f"\nRepeated Game Analysis:")
print(f"  Final cooperation level: {repeated_results['cooperation_level'][-1]:.3f}")
print(f"  Total player payoffs: {repeated_results['total_payoffs']}")

# Plot cooperation dynamics
plt.figure(figsize=(10, 6))
rounds = range(len(repeated_results['cooperation_level']))
plt.plot(rounds, repeated_results['cooperation_level'], 'b-', linewidth=2)
plt.xlabel('Round')
plt.ylabel('Cooperation Level')
plt.title('Validator Auction Cooperation Dynamics')
plt.grid(True, alpha=0.3)
plt.show()

Multi-Factor Risk Model for MEV Portfolio

import numpy as np
import pandas as pd
from scipy import stats
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler
import matplotlib.pyplot as plt

class MEVRiskModel:
    def __init__(self, confidence_level=0.95):
        self.confidence_level = confidence_level
        self.factors = {}
        self.loadings = None
        self.specific_variance = None
        
    def identify_factors(self, returns_data):
        """
        Identify systematic risk factors using PCA and economic intuition
        """
        # Normalize returns
        scaler = StandardScaler()
        normalized_returns = scaler.fit_transform(returns_data)
        
        # Apply PCA to identify principal components
        pca = PCA(n_components=min(10, returns_data.shape[1]))
        principal_components = pca.fit_transform(normalized_returns)
        
        # Get factor loadings
        loadings = pca.components_.T * np.sqrt(pca.explained_variance_)
        
        # Create economic factor interpretation
        factor_interpretation = self._interpret_factors(loadings, returns_data.columns)
        
        self.loadings = pd.DataFrame(
            loadings,
            index=returns_data.columns,
            columns=[f'Factor_{i+1}' for i in range(loadings.shape[1])]
        )
        
        # Calculate specific (idiosyncratic) variance
        reconstructed = np.dot(principal_components, loadings.T)
        residuals = normalized_returns - reconstructed
        self.specific_variance = np.var(residuals, axis=0)
        
        return {
            'loadings': self.loadings,
            'explained_variance': pca.explained_variance_ratio_,
            'factor_interpretation': factor_interpretation,
            'specific_risk': self.specific_variance
        }
    
    def _interpret_factors(self, loadings, strategy_names):
        """Provide economic interpretation of identified factors"""
        interpretations = {}
        
        for i, factor_col in enumerate(loadings.T):
            # Find strategies with highest loadings
            top_strategies = strategy_names[np.argsort(np.abs(factor_col))[-3:]]
            
            interpretations[f'Factor_{i+1}'] = {
                'top_strategies': top_strategies,
                'interpretation': self._factor_economic_meaning(factor_col, strategy_names),
                'variance_explained': np.sum(factor_col**2) / len(factor_col)
            }
        
        return interpretations
    
    def _factor_economic_meaning(self, loadings, strategy_names):
        """Provide economic meaning based on strategy characteristics"""
        # This would typically use domain knowledge to classify strategies
        high_loadings = strategy_names[np.abs(loadings) > 0.3]
        
        if any('arbitrage' in s.lower() for s in high_loadings):
            return 'Arbitrage/Relative Value Factor'
        elif any('liquidation' in s.lower() for s in high_loadings):
            return 'Liquidation/Default Risk Factor'
        elif any('cross-chain' in s.lower() for s in high_loadings):
            return 'Cross-Chain Infrastructure Factor'
        elif any('mev' in s.lower() for s in high_loadings):
            return 'General MEV Market Factor'
        else:
            return 'Systematic Risk Factor'
    
    def calculate_portfolio_var(self, weights, returns, time_horizon=1):
        """
        Calculate Value at Risk for MEV portfolio
        """
        if self.loadings is None:
            raise ValueError("Must identify factors first")
        
        # Normalize returns
        scaler = StandardScaler()
        normalized_returns = scaler.fit_transform(returns)
        
        # Calculate factor returns
        factor_returns = normalized_returns @ self.loadings.values
        
        # Portfolio factor exposure
        portfolio_factor_exposure = weights @ self.loadings.values
        
        # Portfolio factor variance
        factor_covariance = np.cov(factor_returns.T)
        portfolio_factor_variance = portfolio_factor_exposure @ factor_covariance @ portfolio_factor_exposure.T
        
        # Specific risk
        specific_variance = np.sum(weights**2 * self.specific_variance)
        
        # Total portfolio variance
        total_variance = portfolio_factor_variance + specific_variance
        
        # VaR calculation
        std_dev = np.sqrt(total_variance * time_horizon)
        confidence_level = self.confidence_level
        
        if confidence_level > 0.5:  # Right tail
            z_score = stats.norm.ppf(confidence_level)
        else:  # Left tail for losses
            z_score = stats.norm.ppf(1 - confidence_level)
        
        var_amount = z_score * std_dev
        portfolio_value = np.sum(weights)
        
        return {
            'var_amount': var_amount * portfolio_value,
            'var_percentage': var_amount,
            'total_variance': total_variance,
            'factor_contribution': portfolio_factor_variance / total_variance,
            'specific_contribution': specific_variance / total_variance,
            'confidence_level': confidence_level
        }
    
    def stress_test(self, weights, returns, shock_scenarios):
        """
        Perform stress testing on MEV portfolio
        """
        stress_results = {}
        
        for scenario_name, shocks in shock_scenarios.items():
            stressed_returns = returns.copy()
            
            # Apply factor shocks
            for factor_name, shock_magnitude in shocks.items():
                if factor_name in self.loadings.columns:
                    factor_index = self.loadings.columns.get_loc(factor_name)
                    
                    # Apply shock to factor returns
                    stressed_returns += self.loadings[factor_name].values.reshape(-1, 1) * shock_magnitude
            
            # Recalculate portfolio performance
            portfolio_returns = stressed_returns @ weights
            stress_results[scenario_name] = {
                'portfolio_returns': portfolio_returns,
                'mean_return': np.mean(portfolio_returns),
                'volatility': np.std(portfolio_returns),
                'worst_loss': np.min(portfolio_returns),
                'best_gain': np.max(portfolio_returns),
                'sharpe_ratio': np.mean(portfolio_returns) / np.std(portfolio_returns) if np.std(portfolio_returns) > 0 else 0
            }
        
        return stress_results
    
    def risk_attribution(self, weights):
        """
        Attribute portfolio risk to individual factors and strategies
        """
        if self.loadings is None:
            raise ValueError("Must identify factors first")
        
        # Portfolio factor exposure
        portfolio_factor_exposure = weights @ self.loadings.values
        
        # Factor risk contributions
        factor_covariance = np.cov(self.loadings.values.T)
        marginal_factor_risks = factor_covariance @ portfolio_factor_exposure
        
        factor_risks = portfolio_factor_exposure * marginal_factor_risks
        
        # Total portfolio variance
        total_variance = np.sum(factor_risks) + np.sum(weights**2 * self.specific_variance)
        
        # Risk attribution
        attribution = {
            'factor_attribution': {},
            'strategy_attribution': {}
        }
        
        # Factor risk attribution
        for i, factor in enumerate(self.loadings.columns):
            attribution['factor_attribution'][factor] = {
                'risk_contribution': factor_risks[i],
                'risk_percentage': factor_risks[i] / total_variance,
                'exposure': portfolio_factor_exposure[i]
            }
        
        # Strategy-specific risk attribution
        for i, strategy in enumerate(self.loadings.index):
            strategy_factor_risk = 0
            for j, factor in enumerate(self.loadings.columns):
                strategy_factor_risk += (weights[i] * self.loadings.iloc[i, j] * 
                                      portfolio_factor_exposure[j] * 
                                      factor_covariance[j, j])
            
            specific_risk = weights[i]**2 * self.specific_variance[i]
            total_strategy_risk = strategy_factor_risk + specific_risk
            
            attribution['strategy_attribution'][strategy] = {
                'total_risk': total_strategy_risk,
                'risk_percentage': total_strategy_risk / total_variance,
                'factor_risk': strategy_factor_risk,
                'specific_risk': specific_risk,
                'weight': weights[i]
            }
        
        return attribution

# Example usage with synthetic MEV data
np.random.seed(42)

# Generate synthetic MEV strategy returns
n_days = 500
n_strategies = 8

# Create correlated strategy returns with realistic MEV characteristics
# Arbitrage strategies (low correlation)
# Liquidation strategies (higher volatility)
# Cross-chain strategies (medium correlation)

strategies = [
    'ETH_USDC_Arbitrage',
    'UNI_SUSHI_Arbitrage', 
    'Aave_Liquidation',
    'Compound_Liquidation',
    'Cross_Chain_Bridge',
    'NFT_MEV_Arbitrage',
    'DeFi_Protocol_MEV',
    'MEV_Sandwich_Attack'
]

# Factor structure for returns
n_factors = 4
factor_loadings = np.random.randn(n_strategies, n_factors) * 0.6
specific_returns = np.random.randn(n_days, n_strategies) * 0.3

# Generate correlated factor returns
factor_returns = np.random.randn(n_days, n_factors)
strategy_returns = factor_returns @ factor_loadings.T + specific_returns

# Make returns more realistic for MEV
strategy_returns *= 0.02  # Scale to ~2% daily volatility
strategy_returns += 0.001  # Add small positive drift

returns_data = pd.DataFrame(strategy_returns, columns=strategies)

# Initialize risk model
risk_model = MEVRiskModel(confidence_level=0.95)

# Identify risk factors
factor_analysis = risk_model.identify_factors(returns_data)

print("MEV Risk Factor Analysis")
print("=" * 30)
print(f"Explained Variance by Factor:")
for i, variance in enumerate(factor_analysis['explained_variance']):
    print(f"  Factor {i+1}: {variance:.1%}")

print(f"\nFactor Interpretations:")
for factor, interpretation in factor_analysis['factor_interpretation'].items():
    print(f"  {factor}: {interpretation['interpretation']}")
    print(f"    Top strategies: {interpretation['top_strategies']}")

# Example portfolio
portfolio_weights = np.array([0.15, 0.12, 0.18, 0.15, 0.10, 0.08, 0.12, 0.10])
portfolio_weights /= np.sum(portfolio_weights)  # Normalize

# Calculate VaR
var_result = risk_model.calculate_portfolio_var(portfolio_weights, returns_data)

print(f"\nPortfolio VaR Analysis:")
print(f"  VaR (95%): {var_result['var_percentage']:.2%}")
print(f"  Factor Risk: {var_result['factor_contribution']:.1%}")
print(f"  Specific Risk: {var_result['specific_contribution']:.1%}")

# Stress testing scenarios
shock_scenarios = {
    'Market_Crash': {
        'Factor_1': -2.5,  # General market stress
        'Factor_2': -1.8   # Liquidity stress
    },
    'Gas_Price_Spike': {
        'Factor_3': 3.0,   # Gas cost increase
        'Factor_4': -1.2   # MEV profitability decrease
    },
    'DeFi_Protocol_Attack': {
        'Factor_2': -3.0,  # Protocol risk
        'Factor_4': 2.5    # MEV opportunities from chaos
    }
}

stress_results = risk_model.stress_test(portfolio_weights, returns_data.values, shock_scenarios)

print(f"\nStress Test Results:")
for scenario, results in stress_results.items():
    print(f"  {scenario}:")
    print(f"    Mean return: {results['mean_return']:.3f}")
    print(f"    Worst loss: {results['worst_loss']:.3f}")
    print(f"    Best gain: {results['best_gain']:.3f}")

# Risk attribution
attribution = risk_model.risk_attribution(portfolio_weights)

print(f"\nTop Risk Contributors:")
factor_attribution = attribution['factor_attribution']
sorted_factors = sorted(factor_attribution.items(), 
                       key=lambda x: x[1]['risk_percentage'], reverse=True)

for factor, attrs in sorted_factors[:3]:
    print(f"  {factor}: {attrs['risk_percentage']:.1%} of total risk")

# Visualize results
fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(15, 10))

# Factor loadings heatmap
import seaborn as sns
sns.heatmap(risk_model.loadings, annot=True, cmap='RdBu_r', center=0, ax=ax1)
ax1.set_title('Factor Loadings Matrix')

# Portfolio weights
ax2.bar(range(len(strategies)), portfolio_weights)
ax2.set_xticks(range(len(strategies)))
ax2.set_xticklabels([s.replace('_', '\n') for s in strategies], rotation=45, ha='right')
ax2.set_title('Portfolio Weights')
ax2.set_ylabel('Weight')

# VaR contributions
factor_names = list(factor_attribution.keys())
factor_risks = [factor_attribution[f]['risk_percentage'] for f in factor_names]
ax3.pie(factor_risks, labels=factor_names, autopct='%1.1f%%')
ax3.set_title('Risk Attribution by Factor')

# Stress test comparison
scenario_names = list(stress_results.keys())
scenario_returns = [stress_results[s]['mean_return'] for s in scenario_names]
ax4.bar(scenario_names, scenario_returns)
ax4.set_title('Stress Test: Expected Returns')
ax4.set_ylabel('Expected Return')
ax4.tick_params(axis='x', rotation=45)

plt.tight_layout()
plt.show()