wehub-resource-sync
96 lines
5.3 KiB
Python
96 lines
5.3 KiB
Python
# QUANTCONNECT.COM - Democratizing Finance, Empowering Individuals.
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# Lean Algorithmic Trading Engine v2.0. Copyright 2014 QuantConnect Corporation.
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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from AlgorithmImports import *
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from scipy.optimize import minimize
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### <summary>
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### Provides an implementation of a portfolio optimizer that maximizes the portfolio Sharpe Ratio.
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### The interval of weights in optimization method can be changed based on the long-short algorithm.
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### The default model uses flat risk free rate and weight for an individual security range from -1 to 1.'''
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### </summary>
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class MaximumSharpeRatioPortfolioOptimizer:
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'''Provides an implementation of a portfolio optimizer that maximizes the portfolio Sharpe Ratio.
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The interval of weights in optimization method can be changed based on the long-short algorithm.
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The default model uses flat risk free rate and weight for an individual security range from -1 to 1.'''
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def __init__(self,
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minimum_weight = -1,
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maximum_weight = 1,
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risk_free_rate = 0):
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'''Initialize the MaximumSharpeRatioPortfolioOptimizer
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Args:
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minimum_weight(float): The lower bounds on portfolio weights
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maximum_weight(float): The upper bounds on portfolio weights
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risk_free_rate(float): The risk free rate'''
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self.minimum_weight = minimum_weight
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self.maximum_weight = maximum_weight
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self.risk_free_rate = risk_free_rate
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self.expected_returns = []
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def optimize(self, historical_returns, expected_returns = None, covariance = None):
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'''
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Perform portfolio optimization for a provided matrix of historical returns and an array of expected returns
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args:
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historical_returns: Matrix of annualized historical returns where each column represents a security and each row returns for the given date/time (size: K x N).
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expected_returns: Array of double with the portfolio annualized expected returns (size: K x 1).
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covariance: Multi-dimensional array of double with the portfolio covariance of annualized returns (size: K x K).
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Returns:
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Array of double with the portfolio weights (size: K x 1)
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'''
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if covariance is None:
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covariance = historical_returns.cov()
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if expected_returns is None:
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expected_returns = historical_returns.mean()
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expected_returns = expected_returns - self.risk_free_rate
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size = covariance.columns.size # K x 1
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x0 = np.array(size * [1. / size])
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# SLSQP maximizes the Sharpe ratio (µ − r_f)^T w / √(w^T Σ w) directly, so the fractional
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# objective is optimized in place without any substitution. The budget constraint Σw = 1 and
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# the per-weight bounds lw ≤ w ≤ up are applied as-is. The previous implementation instead
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# fixed (µ − r_f)^T w to the equal-weight return, which collapsed the optimizer to minimum
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# variance. The C# implementation uses the Charnes-Cooper QP substitution because its solver
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# only handles quadratic objectives.
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# https://quant.stackexchange.com/questions/18521/sharpe-maximization-under-quadratic-constraints
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constraints = [
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# Σw = 1
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{'type': 'eq', 'fun': lambda weights: self.get_budget_constraint(weights)}]
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opt = minimize(lambda weights: -expected_returns.dot(weights) / np.sqrt(self.portfolio_variance(weights, covariance)), # Objective function: −Sharpe ratio
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x0, # Initial guess
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bounds = self.get_boundary_conditions(size), # Bounds for variables: lw ≤ w ≤ up
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constraints = constraints, # Constraints definition
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method='SLSQP') # Optimization method: Sequential Least SQuares Programming
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return opt['x'] if opt['success'] else x0
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def portfolio_variance(self, weights, covariance):
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'''Computes the portfolio variance
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Args:
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weighs: Portfolio weights
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covariance: Covariance matrix of historical returns'''
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variance = np.dot(weights.T, np.dot(covariance, weights))
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if variance == 0 and np.any(weights):
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# variance can't be zero, with non zero weights
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raise ValueError(f'MaximumSharpeRatioPortfolioOptimizer.portfolio_variance: Volatility cannot be zero. Weights: {weights}')
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return variance
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def get_boundary_conditions(self, size):
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'''Creates the boundary condition for the portfolio weights'''
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return tuple((self.minimum_weight, self.maximum_weight) for x in range(size))
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def get_budget_constraint(self, weights):
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'''Defines a budget constraint: the sum of the weights equals unity'''
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return np.sum(weights) - 1
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