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AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Optimization"
python "optimization_tutorial.py"

View source on GitHub

Quantitative Methods – Optimization

Overview

Optimization is the mathematical engine behind modern finance. From finding the best portfolio weights to calibrating complex models, optimization techniques are essential for quantitative analysts.

Key Concepts

Minimization

  • Objective Function: The quantity to minimize (e.g., risk, error)
  • Constraints: Limitations (e.g., weights sum to 1, long-only)
  • Bounds: Limits on individual variables (e.g., 0% to 100% allocation)
  • Algorithms: BFGS, SLSQP, Newton-CG

Portfolio Optimization

  • Mean-Variance Optimization: Markowitz Modern Portfolio Theory
  • Global Minimum Variance: Portfolio with lowest possible risk
  • Tangency Portfolio: Portfolio with highest Sharpe Ratio
  • Efficient Frontier: Set of optimal portfolios

Curve Fitting

  • Calibration: Fitting model parameters to market data
  • Least Squares: Minimizing the sum of squared errors
  • Yield Curve Modeling: Nelson-Siegel, Svensson models

Root Finding

  • Solving Equations: Finding x where f(x) = 0
  • Implied Volatility: Finding volatility that matches option price
  • IRR: Finding discount rate where NPV = 0

Key Examples

Basic Minimization

from scipy.optimize import minimize

def objective(x):
 return (x[0] - 3)**2 + 5

result = minimize(objective, [0])
print(result.x) # [3.0]

Portfolio Optimization

# Minimize volatility subject to constraints
def volatility(weights):
 return np.sqrt(weights.T @ cov_matrix @ weights)

constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})
bounds = ((0, 1), (0, 1), (0, 1))

result = minimize(volatility, init_guess,
 method='SLSQP', bounds=bounds, constraints=constraints)

Implied Volatility (Root Finding)

from scipy.optimize import newton

def objective(sigma):
 return bs_call(S, K, T, r, sigma) - market_price

iv = newton(objective, x0=0.2)

Files

  • optimization_tutorial.py: Comprehensive optimization tutorial

How to Run

pip install scipy numpy
python optimization_tutorial.py

Financial Applications

1. Portfolio Construction

  • Risk Parity: Equal risk contribution from each asset
  • Black-Litterman: Combining market equilibrium with views
  • Index Tracking: Minimizing tracking error vs benchmark

2. Model Calibration

  • Yield Curves: Fitting Nelson-Siegel to bond prices
  • Volatility Surfaces: Fitting SVI model to option chains
  • Local Volatility: Calibrating Dupire's formula

3. Algorithmic Trading

  • Parameter Tuning: Optimizing strategy parameters (stop loss, window size)
  • Execution Algorithms: Minimizing market impact (VWAP/TWAP)
  • Arbitrage: Finding optimal basket for stat arb

Best Practices

  • Convexity: Convex problems (like variance minimization) have unique global solutions. Non-convex problems may get stuck in local minima.
  • Scaling: Ensure variables are on similar scales for better convergence.
  • Constraints: Use 'SLSQP' or 'trust-constr' for constrained problems.
  • Initial Guess: A good starting point improves speed and reliability.

References

  • SciPy Optimize: https://docs.scipy.org/doc/scipy/reference/optimize.html
  • Convex Optimization: Boyd & Vandenberghe
  • Portfolio Optimization: Cornuejols & Tutuncu

Master optimization to solve the most challenging problems in quantitative finance!

Continue in Quantitative Methods

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