This Julia package provides functionality for generating scenarios for stochastic programs which use a tail risk measure such as the conditional value-at-risk. For problems involving tail risk measure, the value of the tail risk measure depends only a subset of the support of the distribution called the risk region. By prioritising the generation of scenarios in the risk region, one can approximate much better the value of a tail risk measure in a stochastic program.
This package allows users to construct risk regions and use these scenario generaion and reduction. Currently risk regions are provided for two types of problems:
- Portfolio selection problem with Elliptical return distributions
- Stochastic programs with monotonic loss/recourse functions.
In the former case it has been shown that the methodology still performs well when the returns have near-elliptical distributions.
For more details on this methodology see the following two papers:
- Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure
- Scenario Generation for Single-Period Portfolio Selection Problems with Tail Risk Measures: Coping with High Dimensions and Integer Variables
The risk region for our class of problems have the following form:
Given this formulation It is difficult to test whether a scenario y is in the risk region. When K is a convex cone, the above expression can be rewritten as follows:
where K' = PK and p_k(y) is the projection of the point y onto the cone K.
The construction of a risk region requires four parameters: a vector μ, a positive definite matrix Σ=PᵀP, a convex cone K and positive scalar α. The package currently contains two types of cone objects: a FiniteCone and a PolyhedralCone. A FiniteCone object is constructed from a matrix of cone generators where each column corresponds to a cone generator.
K = FiniteCone(eye(2)) # Constructs cone of positive quadrant in R^2
We now construct a RiskRegion for Normally distributed returns at the level β = 0.95:
μ = [0.05, 0.02]
Σ = [[0.5, 0.2] [0.2, 0.5]]
β = 0.95
Ω = RiskRegion(μ, Σ, K, quantile(Normal(), β))
Convenience constructors are provided for multivariate normal and t distributions:
Y = MvNormal(μ, Σ)
Ω = RiskRegion(Y, K, β)
Scenario reduction requires a matrix of (equiprobable) scenarios, where each column corresponds to one scenario, and a risk region. The aggregate_scenarios function outputs a scenario matrix where all scenarios in the non-risk region have been aggregated into a single point, and the corresponding vector of scenario probabilities.
n = 1000 # Initial number of scenarios
scenarios = rand(Y, num_s) # Generate a matrix of scenarios
reduced_set, reduced_probs = aggregate_scenarios(scenarios, Ω) # Reduce scenario set
A scenario set of specified size can be generated via the aggregation sampling algorithm. This takes a risk region and a distribution, and samples points from this, stores any point in the risk region and aggregates any in the non-risk region. The algorithm terminates when the required number of points in the risk region have been sampled.
n = 500 # Scenario set size
scenarios, probs = aggregation_sampling(Y, Ω, n)