06-portfolio

06 - Portfolio Optimization

Features: Multi-period, Transaction Costs, Short Selling

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Overview

This portfolio optimization problem extends classical Markowitz portfolio theory to include realistic financial market features: multiple time periods, transaction costs for buying/selling assets, borrowing costs for short positions, and capital constraints.

Problem Description

We aim to find the binary decision variables $x_i \in {0,1}$ for each asset (i) that

$$ \min_{\substack{ x \in {0,1}^{n \times t}\\ y \in {0,1}^{c \times t}\\ s \in {0,1}^{b \times t} }} \sum_{t=1}^{T} \biggl( \lambda \underbrace{\sum_{i,j} \tau_{i} p_{it} x_{it} \sigma_{ijt} \tau_{j} x_{jt} p_{jt}}_{\text{risk}} - \sum_i \bigl(\underbrace{\tau_{i}(p_{it+1} - p_{it}) x_{it}}_{\text{profit}} - \underbrace{\delta p_{it} (x_{it-1} + x_{it} - 2 x_{it-1} x_{it})}_{\text{transaction cost}}\bigr) - \underbrace{\rho_c u \sum_c 2^c y_{ct}}_{\text{cash interest}} + \underbrace{\rho_s \sum_{i \in S} p_{it} x_{it}}_{\text{short selling cost}}\biggr) + \underbrace{\delta \sum_i p_{iT} x_{iT}}_{\text{liquidation cost}} $$

subject to the constraints:

$$ \sum_i \tau_i x_{it} + \sum_c 2^c y_{ct} = C \quad \forall t \in {1,...,T} \quad \quad {\text{capital limit}} $$

$$ \sum_i x_{it} + \sum_b 2^b s_{bt} = B \quad \forall t \in {1,...,T} \quad \quad {\text{number of assets limit}} $$

Here, $p_{i,t}$ represents the price of one unit of asset $i$ at time $t$, and $\sigma_{ij,t}$ denotes the covariance between stocks $i$ and $j$ at time $t$. We denote by $\delta$ the transaction cost rate applied to both buying and selling. We have a short-selling indicator $\tau \in {-1, +1}$, where $-1$ indicates a short position and $+1$ a long position. Our model includes a borrowing cost rate for short sales, denoted by $\rho_s$. We introduce slack variables $s_{bt} \in {0,1}$ for $b \in {0, \ldots, \lfloor \log_2 B \rfloor}$ to cap the total number of assets, and use slack variables $y_{ct} \in {0,1}$ for $c \in {0, \ldots, \lfloor \log_2 C \rfloor}$ to help restrict the total available cash to not exceed $C$ units.

Directory Contents

• instances/ - Portfolio optimization instances with market data
• models/ - Mathematical model formulations
• solutions/ - Optimal or best-known solutions
• check/ - Solution verification tools
• info/ - Additional documentation and papers
• misc/ - Utility scripts and data generators
• submissions/ - Community solution submissions

Related Works

• Brandhofer et al. - Benchmarking the performance of portfolio optimization with QAOA
• Hodson et al. - Portfolio rebalancing experiments using the Quantum Alternating Operator Ansatz
• Baker & Radha - Wasserstein Solution Quality and the Quantum Approximate Optimization Algorithm: A Portfolio Optimization Case Study
• Mugel et al. - Dynamic Portfolio Optimization with Real Datasets Using Quantum Processors and Quantum-Inspired Tensor Networks
• Herman et al. - Constrained optimization via quantum Zeno dynamics
• Giron et al. - Approaching Collateral Optimization for NISQ and Quantum-Inspired Computing
• Woerner & Egger - Quantum risk analysis
• Egger et al. - Credit Risk Analysis using Quantum Computers
• Stamatopoulos et al. - Option Pricing using Quantum Computers
• Braine et al. - Quantum algorithms for mixed binary optimization applied to transaction settlement
• Calude et al. - QUBO formulations for the graph isomorphism problem and related problems
• Becker, Cheridito, Jentzen - Deep optimal stopping
• Felizardo, Matsumoto, Del-Moral-Hernandez - Solving the optimal stopping problem with reinforcement learning
• Soon & Ye - Currency Arbitrage Detection Using A Binary Integer Programming Model
• Díez-Valle et al. - Multi-objective variational quantum optimization for constrained problems: an application to Cash Management

References

• Brandhofer, S., et al. (2022). Benchmarking the performance of portfolio optimization with QAOA. Quantum Information Processing 22.1: 25.

• Mugel, S., et al. (2022). Dynamic portfolio optimization with real datasets using quantum processors and quantum-inspired tensor networks. Physical Review Research 4.1: 013006.