Introduction
Robust optimization, an emerging branch in the field of optimization, offers vehicles to incorporate estimation risk into the decision making process in portfolio allocation. Generally, robust optimization refers to purse solutions to given optimization problems with uncertain input parameters that will achieve good objective values for all, or most, realizations of the uncertain input parameters.
KL divergence
If π , η \pi, \eta π,η are two probability vectors in R T \mathbb{R}^T RT, with η > 0 \eta>0 η>0 descrbing the nominal probability, the KL distance between π \pi π and η \eta η is defined as
K L ( π , η ) = ∑ k = 1 T π k log π k η k KL(\pi, \eta)=\sum_{k=1}^T\pi_k\log\frac{\pi_k}{\eta_k} KL(π,η)=k=1∑Tπklogηkπk
Assume that the true probability π \pi π is only known to lie within KL distance d ≥ 0 d\geq 0 d≥0 from η \eta η, i.e., π ∈ K ( η , d ) \pi\in K(\eta, d) π∈K(η,d), where
K ( η , d ) = { π ∈ Π : K L ( π , η ) ≤ d } K(\eta, d)=\{\pi\in\Pi: KL(\pi, \eta)\leq d\} K(η,d)={
π∈Π:KL(π,η)≤d}
Given the ambiguity model K ( η , d ) K(\eta, d) K(η,d) for the return distribution, we define the following worst-case (or robust) measure of risk for a portfolio with composition x x x
R w c : = max π ∈ K ( η , d ) ρ ( x , π ) − γ μ ( x , π ) \mathcal{R}_{wc}:=\max_{\pi\in K(\eta, d)}\rho(x, \pi)-\gamma \mu(x, \pi) Rwc:=π∈K(η,d)maxρ(x,π)−γμ(x,π)
worst-case VaR
The VaR looks at the probability of losses and is defined as the minial level γ \gamma γ such that the probability that the portfolio loss − l ( x , r ) -l(x, r) −l(x,r) exceeds γ \gamma γ is below ϵ \epsilon ϵ:
min x γ s . t . P r o b { γ ≤ − l ( x , r ) } ≤ ϵ \min_x \gamma\\ s.t.\quad Prob\{\gamma\leq -l(x, r)\}\leq \epsilon xminγs.t.Prob{
γ≤−l(x,r)}≤ϵ
With given distribution as Gaussian, mean as r r r and covariance matrix as Γ \Gamma Γ，the VaR can be expressed as
V ( x ) = − Φ ( ϵ ) − 1 x T Γ x − r T x V(x)=-\Phi(\epsilon)^{-1}\sqrt{x^T\Gamma x}-r^Tx V(x)=−Φ(ϵ)−1xTΓx−rTx
We define the worst-case VaR with respect to the set of probability distributions as
V π ( x ) = inf { γ ∣ λ ( γ , x , r ) ≤ ϵ } λ ( γ , x , r ) = sup { P r o b { γ ≤ − l ( x , r ) } } V_\pi(x)=\inf\{\gamma\mid \lambda(\gamma, x, r)\leq \epsilon\}\\ \lambda(\gamma, x, r)=\sup\{Prob\{\gamma\leq-l(x, r)\}\} Vπ(x)=inf{
γ∣λ(γ,x,r)≤ϵ}λ(γ,x,r)=sup{
Prob{
γ≤−l(x,r)}}
The robust portfolio optimization problem is given by
min − r T x + κ ( ϵ , d ) ∥ Γ 1 / 2 x ∥ 2 s . t . { 1 T x = 1 x ≥ 0 κ ( ϵ , d ) = − Φ − 1 ( f ( ϵ , d ) ) f ( ϵ , d ) = sup e − d ( 1 + v ) ϵ − 1 v \min -r^Tx+\kappa(\epsilon, d)\lVert\Gamma^{1/2}x\rVert_2\\ s.t. \begin{cases} \mathbf{1}^Tx=1\\ x\geq 0\\ \kappa(\epsilon, d)=-\Phi^{-1}(f(\epsilon, d))\\ f(\epsilon, d)=\sup\frac{e^{-d}(1+v)^\epsilon-1}{v} \end{cases} min−rTx+κ(ϵ,d)∥Γ1/2x∥2s.t.⎩⎪⎪⎪⎨⎪⎪⎪⎧1Tx=1x≥0κ(ϵ,d)=−Φ−1(f(ϵ,d))f(ϵ,d)=supve−d(1+v)ϵ−1
Algorithms
- Douglas Rachford Algorithm
- PG
- Nestrov
- FISTA