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PortfolioOpt

PortfolioOpt.BudgetSet — Type
BudgetSet{T<:Real, D<:ContinuousMultivariateSampleable} <: CenteredAmbiguitySet{T,D}

Bertsimas's uncertainty set:

\[\left\{ \mu \; \middle| \begin{array}{ll} s.t. \quad \mu_i \leq \hat{r}_i + z_i \Delta_i \quad \forall i = 1:\mathcal{N} \\ \quad \quad \mu_i \geq \hat{r}_i - z_i \Delta_i \quad \forall i = 1:\mathcal{N} \\ \quad \quad z_i \geq 0 \quad \forall i = 1:\mathcal{N} \\ \quad \quad z_i \leq 1 \quad \forall i = 1:\mathcal{N} \\ \quad \quad \sum_{i}^{\mathcal{N}} z_i \leq \Gamma \\ \end{array} \right\} \\\]

Attributes:

  • d::Sampleable{Multivariate, Continous}: The parent distribution with an uncertain mean
  • Δ::Array{Float64,1}: Uncertainty around mean. (default: std(d) / 5)
  • Γ::Float64: Number of assets in worst case. (default: 0.1 * length(d))

References:

  • Bertsimas, D. e Sim, M. (2004). The price of robustness. Operations research, 52(1):35–53.
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PortfolioOpt.CenteredAmbiguitySet — Type
CenteredAmbiguitySet

Defines the ambiguity related to the distribution of a random variable, often used in Robust Optimization (RO) and Distributionally Robust Optimization (DRO) problems. It represents a bounded infinite set of distributions.

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PortfolioOpt.DuWassersteinBall — Type
DuWassersteinBall <: CenteredAmbiguitySet

\[\left\{ r \; \middle| \begin{array}{ll} s.t. \quad d_w(P, \hat{P}_N) \leq \epsilon \\ \quad \quad || \xi || \leq \Lambda \\ \quad \quad [\xi; 0] + [0_{m x 1}; \Lambda] \in K \\ \quad \quad K = {[\omega, \pi] \in R^m x R: \pi \geq ||\omega||^*} \\ \end{array} \right\} \\\]

Atributes:

  • d::DeterministicSamples: Samples from the parent distribution
  • ϵ::Float64: Wasserstein distance from sampled distribution (has to be greater than 0). (default: 0.01)
  • Λ::Float64: Uncertainty around sampled values (has to be greater than 0). (default: maximum(d))

References:

  • NingNing paper on Wasserstein DRO (Corollary 1-3): https://ieeexplore.ieee.org/abstract/document/9311154
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PortfolioOpt.EllipticalSet — Type
EllipticalSet <: CenteredAmbiguitySet{T,D}

\[\left\{ \mu \; \middle| \begin{array}{ll} s.t. \quad \sqrt{(\hat{r} - \mu) ' \Sigma^{-1} (\hat{r} - \mu)} \leq \delta \\ \end{array} \right\} \\\]

Atributes:

  • d::Sampleable{Multivariate, Continous}: The parent distribution with an uncertain mean
  • Δ::Array{Float64,1}: Uniform uncertainty around mean. (default: 0.025)

References:

  • Ben-Tal, A. e Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical programming, 88(3):411–424.
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PortfolioOpt.MomentUncertainty — Type
MomentUncertainty <: CenteredAmbiguitySet

\[\left\{ r \; \middle| \begin{array}{ll} s.t. \quad (\mathbb{E} [r] - \hat{r}) ' \Sigma^{-1} (\mathbb{E} [r] - \hat{r}) \leq \gamma_1 \\ \quad \quad \mathbb{E} [ (r - \hat{r}) ' (r - \hat{r}) ] \leq \gamma_2 \Sigma \\ \end{array} \right\} \\\]

Atributes:

  • d::Sampleable{Multivariate, Continous}: The parent distribution with an uncertain mean
  • γ1::Float64: Uniform uncertainty around the mean (has to be greater than 0).
  • γ2::Float64: Uncertainty around the covariance (has to be greater than 1).

References:

  • Delage paper on moment uncertainty: https://www.researchgate.net/publication/220244490DistributionallyRobustOptimizationUnderMomentUncertaintywithApplicationtoData-Driven_Problems
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OptimalBids.change_bids! — Method
change_bids!(market::VolumeMarket, model::JuMP.Model, decision_variables)

Solves optimization model and set new bids to optimal values of the decision_variables.

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PortfolioOpt.calculate_measure! — Method
calculate_measure!(measure::ExpectedReturn{BudgetSet}, w)

Returns worst case return (WCR) in BudgetSet's uncertainty set (BudgetSet).

For further reading:

  • BudgetSet' paper: BudgetSet, D. e Sim, M. (2004). The price of robustness. Operations research, 52(1):35–53.
  • Original implementation from: https://github.com/andrewrosemberg/PortfolioOpt.jl.

WCR is defined by the following primal problem:

\[\begin{aligned} \min_{\mu, z} \quad & \mu ' w \\ s.t. \quad & \mu_i \leq \hat{r}_i + z_i \Delta_i \quad \forall i = 1:\mathcal{N} \quad &: \pi^-_i \\ & \mu_i \geq \hat{r}_i - z_i \Delta_i \quad \forall i = 1:\mathcal{N} \quad &: \pi^+_i \\ & z_i \geq 0 \quad \forall i = 1:\mathcal{N} \\ & z_i \leq 1 \quad \forall i = 1:\mathcal{N} \quad &: \theta_i \\ & \sum_{i}^{\mathcal{N}} z_i \leq \Gamma \quad : \lambda \\ \end{aligned}\]

Which is equivalent to the following dual problem:

\[\begin{aligned} \max_{\lambda, \pi^-, \pi^+, \theta} \quad \sum_{i}^{\mathcal{N}} (\hat{r}_i (\pi^+_i - \pi^-_i) - \theta_i ) - \Gamma \lambda\\ s.t. \quad & w_i = \pi^+_i - \pi^-_i \quad \forall i = 1:\mathcal{N} \\ & \Delta_i (\pi^+_i + \pi^-_i) - \theta_i \leq \lambda \quad \forall i = 1:\mathcal{N} \\ & \lambda \geq 0 , \; \pi^- \geq 0 , \; \pi^+ \geq 0 , \; \theta \geq 0 \\ \end{aligned}\]

To avoid solving an optimization problem we enforece the dual constraints in the upper level problem and return the objective expression (a lower bound of the optimum).

Arguments:

  • w: portfolio optimization investment variable ("weights").
  • s::BudgetSet: Struct containing atributes of BudgetSet's uncertainty set.
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PortfolioOpt.calculate_measure! — Method
calculate_measure!(measure::ExpectedReturn{EllipticalSet}, w)

Returns worst case return (WCR) in EllipticalSet's uncertainty set (EllipticalSet).

WCR is defined by the following primal problem:

\[\begin{aligned} \min_{\mu} \quad & \mu ' w \\ s.t. \quad & ||Σ^{-\frac{1}{2}} (\mu - \hat{r}) || \leq \delta \quad &: \theta \\ \end{aligned}\]

Which is equivalent to the following dual problem:

\[\begin{aligned} \max_{\theta} \quad & w ' \hat{r} - \theta \delta \\ s.t. \quad & ||Σ^{\frac{1}{2}} w || \leq \theta \\ \end{aligned}\]

To avoid solving an optimization problem we enforece the dual constraints in the upper level problem and return the objective expression (a lower bound of the optimum).

Arguments:

  • w: portfolio optimization investment variable ("weights").
  • s:: EllipticalSet: Struct containing atributes of EllipticalSet's uncertainty set.
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PortfolioOpt.calculate_measure! — Method
calculate_measure!(m::ExpectedUtility{U,S}, w) where {U<:PieceWiseUtility,S<:MomentUncertainty}

Returns worst case utility return (WCR) under distribution uncertainty defined by MomentUncertainty ambiguity set (MomentUncertainty).

Arguments:

  • w: portfolio optimization investment variable ("weights").
  • m::ExpectedUtility: Struct containing information about the utility and ambiguity set.
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PortfolioOpt.convex_hull — Method
calculate_measure!(measure::ConditionalExpectedReturn{0.0,S}, w)

Returns worst case return in the convex hull uncertainty set, defined by the following dual problem:

\[ \max_{\theta} \quad \theta \\ s.t. \quad \theta \leq r_s ' w \quad \forall s = 1:\mathcal{S} \\\]

Further information:

  • Fernandes, B., Street, A., ValladA˜ £o, D., e Fernandes, C. (2016). An adaptive robust portfolio optimization model with loss constraints based on data-driven polyhedral uncertainty sets. European Journal of Operational Research, 255(3):961 – 970. ISSN 0377-2217. URL.
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PortfolioOpt.market_model — Method
market_model(market::VolumeMarket{T,N}, optimizer_factory::Function) -> JuMP.Model, Vector{VariableRef}

Creates a JuMP model with appropriate PO variable and constraints: - Investment vector of variables w (portfolio weights if budget = 1). - Invested monney should be lower budget.

Returns the model and the reference to the vector of decision variables (length N).

Aditional arguments:

  • optimizer_factory: callable with zero arguments and return an empty MathOptInterface.AbstractOptimizer`.
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PortfolioOpt.sequential_backtest_market — Method
sequential_backtest_market(strategy_logic::Function, market_history::VolumeMarketHistory) -> Vector{Real}, Vector{Real}

Simple sequential backtest functionality for strategies that return an array of invested money per asset. Ideal for statefull strategies.

API:

wealth_strategy, returns_strategy = sequential_backtest_market(market_history; date_range=date_range) 
    do market, past_returns
    # ... Strategy definition ...
    return investment_decision
end

Arguments:

  • strategy_logic::Function: Function that represents the investment strategy.
  • market_history::MarketHistory: Market history.

Optional Keywork Arguments:

  • date_range: Dates to be simulated with corresponding entries in the market history.
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TestUtils

PortfolioOpt.TestUtils — Module
TestUtils

Some commom test utilities for portfolio optimization formulations.

Mainly: - get_test_data that returns a TimeArray of prices for 6 assets.

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