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