chore: import upstream snapshot with attribution
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wehub-resource-sync
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/*
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* 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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*/
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using System.Collections.Generic;
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using System.Linq;
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using Accord.Math;
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using Accord.Math.Optimization;
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using Accord.Statistics;
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namespace QuantConnect.Algorithm.Framework.Portfolio
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{
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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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public class MaximumSharpeRatioPortfolioOptimizer : IPortfolioOptimizer
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{
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private double _lower;
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private double _upper;
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private double _riskFreeRate;
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/// <summary>
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/// Initialize a new instance of <see cref="MaximumSharpeRatioPortfolioOptimizer"/>
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/// </summary>
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/// <param name="lower">Lower constraint</param>
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/// <param name="upper">Upper constraint</param>
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/// <param name="riskFreeRate"></param>
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public MaximumSharpeRatioPortfolioOptimizer(double lower = -1, double upper = 1, double riskFreeRate = 0.0)
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{
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_lower = lower;
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_upper = upper;
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_riskFreeRate = riskFreeRate;
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}
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/// <summary>
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/// Boundary constraints on weights: lw ≤ w ≤ up
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/// </summary>
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/// <remarks>
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/// Expressed in the substituted variable y = κw (κ = 1ᵀy > 0), the per-weight bounds
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/// become linear: yᵢ − up·(1ᵀy) ≤ 0 and yᵢ − lw·(1ᵀy) ≥ 0.
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/// </remarks>
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/// <param name="size">number of variables</param>
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/// <returns>enumeration of linear constraint objects</returns>
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protected IEnumerable<LinearConstraint> GetBoundaryConditions(int size)
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{
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for (int i = 0; i < size; i++)
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{
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// yᵢ − up·(1ᵀy) ≤ 0
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var upper = Vector.Create(size, -_upper);
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upper[i] += 1.0;
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yield return new LinearConstraint(size)
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{
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CombinedAs = upper,
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ShouldBe = ConstraintType.LesserThanOrEqualTo,
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Value = 0.0
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};
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// yᵢ − lw·(1ᵀy) ≥ 0
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var lower = Vector.Create(size, -_lower);
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lower[i] += 1.0;
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yield return new LinearConstraint(size)
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{
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CombinedAs = lower,
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ShouldBe = ConstraintType.GreaterThanOrEqualTo,
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Value = 0.0
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};
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}
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}
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/// <summary>
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/// Perform portfolio optimization for a provided matrix of historical returns and an array of expected returns
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/// </summary>
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/// <param name="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).</param>
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/// <param name="expectedReturns">Array of double with the portfolio annualized expected returns (size: K x 1).</param>
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/// <param name="covariance">Multi-dimensional array of double with the portfolio covariance of annualized returns (size: K x K).</param>
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/// <returns>Array of double with the portfolio weights (size: K x 1)</returns>
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public double[] Optimize(double[,] historicalReturns, double[] expectedReturns = null, double[,] covariance = null)
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{
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covariance = covariance ?? historicalReturns.Covariance();
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var returns = (expectedReturns ?? historicalReturns.Mean(0)).Subtract(_riskFreeRate);
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var size = covariance.GetLength(0);
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var equalWeights = Vector.Create(size, 1.0 / size);
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// The Charnes-Cooper substitution needs a portfolio with positive expected excess
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// return to exist, otherwise the Sharpe ratio cannot be maximized.
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var feasible = _lower >= 0 ? returns.Any(x => x > 0) : returns.Any(x => x != 0);
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if (!feasible)
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{
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return equalWeights;
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}
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// Charnes-Cooper substitution y = κw (κ = 1ᵀy): maximizing the Sharpe ratio
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// (µ − r_f)ᵀw / √(wᵀΣw) becomes minimizing wᵀΣw subject to (µ − r_f)ᵀy = 1,
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// recovering the weights afterwards as w = y / (1ᵀy).
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// https://quant.stackexchange.com/questions/18521/sharpe-maximization-under-quadratic-constraints
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var constraints = new List<LinearConstraint>
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{
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// (µ − r_f)ᵀy = 1
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new LinearConstraint(size)
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{
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CombinedAs = returns,
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ShouldBe = ConstraintType.EqualTo,
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Value = 1.0
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}
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};
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// lw ≤ w ≤ up
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constraints.AddRange(GetBoundaryConditions(size));
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// Setup solver: minimize yᵀΣy
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var optfunc = new QuadraticObjectiveFunction(covariance, Vector.Create(size, 0.0));
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var solver = new GoldfarbIdnani(optfunc, constraints);
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// Solve problem
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var success = solver.Minimize(Vector.Copy(equalWeights));
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if (!success)
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{
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return equalWeights;
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}
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// Recover the portfolio weights: w = y / (1ᵀy)
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var y = solver.Solution;
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var sum = y.Sum();
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return sum > 0 ? y.Divide(sum) : equalWeights;
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}
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}
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}
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