Leveraging Quantum Annealing for Large MIMO Processing in ...
Applications of Quantum Annealing in Computational Finance
Transcript of Applications of Quantum Annealing in Computational Finance
Applications of Quantum Annealing in Computational FinanceDr. Phil GoddardHead of Research, 1QBitD-Wave User Conference, Santa Fe, Sept. 2016
• Where’s my Babel Fish?
• Quantum-Ready Applications for Computational Finance
• Tools and Resources
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Outline
Where’s my Babel Fish?
Where’s my Babel Fish?
• k-Coloring • Connected Dominating Set• Job Shop Scheduling
• Graph Similarity• Graph Partitioning
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• Portfolio Optimization
• Asset Allocation• Risk Management
• Option Pricing
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Job 1Job 2Job 3
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1.2Mean-Variance-Efficient Frontier
Risk (Standard Deviation)
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Is that a QUBO?
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𝑥 ∈ 0,1 &
𝑄 ∈ ℝ&×&
min-
𝑥.𝑄𝑥
𝑥 ∈ ℝ&𝑄 ∈ ℝ&×&
Weightofassetsinportfolio
Covariancematrix
Quantum-Ready Applicationsfor Computational Finance
• A Multi-Period Optimal Trading Strategy
• Quantum-Ready Hierarchical Risk Parity (QHRP)
• Real-Time Optimization Framework
• Tax Loss Harvesting
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Applications for Computational Finance
Optimal Trading Trajectories
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• The optimal trading trajectory problem
• The mathematical formulation
• Considerations in using the quantum annealer
• Experimental Results
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A Multi-Period Optimal Trading Strategy
The Optimal Trading Trajectory Problem
• Managers of large portfolios typically need to optimize their portfolios over a multiple period horizon.
• A sequence of single-period optimal positons is rarely multi-period optimal.
• Rebalancing the portfolio to align with each single period optimal weight is typically prohibitively expensive.
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OptimalWeight
TimeofRebalance
Single-periodOptimalWeight --- Multi-periodOptimalWeight
Practical Considerations
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• Friction: Transaction costs prevent portfolio managers from monetizing much of their forecasting power
• Market Impact: The sale or purchase of large blocks of a given asset may result in temporary and/or permanent price movements
• Constraints: Some positions can only be traded in blocks (e.g. real-estate, private offerings, fixed lot sizes,…), thus requiring integer solutions
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0.14Efficient Frontier
Risk
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Basic ConstraintsTrading Constraints
Mathematical Formulation• The multi-period integer optimization problem may be written as
• To solve, the problem must be converted to standard QUBO form:
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𝑤 = argmax5 6 𝜇8.𝑤89:;<9=>
−𝛾2𝑤8
.Σ8𝑤89C>D
− Δ𝑤8.Λ8Δ𝑤8GC9:H;HJ>;>,;:KL.CKLNH;
− Δ𝑤8.Λ8O Δ𝑤8L:9K.CKLNH;
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8PQ
s.t.:
∀𝑡:6𝑤U,8 ≤ 𝐾&
UPQ
; ∀𝑡, ∀𝑛:𝑤U,8 ≤ 𝐾O
mean-varianceportfolio optimization
transactioncostandmarketimpacts
min-𝑥.𝑄𝑥
𝑥 ∈ 0,1 &, 𝑄 ∈ ℝ&×&
Bit Encoding• The integer variables of the optimization problem 𝑤Z must be recast to the binary variables used by the
annealer 𝑥Z .
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Experimental Procedure and Results• Generate a random problem: number of assets; time horizon; and total investable assets
• Solve using quantum annealer
• Find exact minimum solution using an exhaustive search
• Evaluate performance by how far the quantum solution is from the exact solution
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• The annealer solution is typically within a small and acceptable margin of error of the exact globally minimal solution.
Quantum-Ready Hierarchical Risk Parity
Quantum-Ready Hierarchical Risk Parity
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The Approach
• Use tree structure to reduce connections between assets as well as estimation error
• Instead of minimizing variance via all correlations using all assets at once, solve the minimum variance problem at each node of the tree
• Build the final weight vector by recursively minimizing variance from the bottom of the tree to the top
• Ignore correlations in determining weights, use only in calculating variance of sub-trees
• Effect: Improve out-of-sample realized volatility
• Can also be applied to linear regression
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𝒘∗ = argmin𝒘
𝒘.�̂�𝒘 𝑠. 𝑡.6𝑤Z = 1�
Z
𝒘𝟐,𝟏∗ = argmin
𝒘𝒘.Σc,Qd 𝒘 𝒘𝟐,𝟐
∗ = argmin𝒘
𝒘.Σc,cd 𝒘
𝒘𝟏,𝟏∗ = argmin
𝒘𝒘.ΣQ,Qd 𝒘
Classical
QHRP
(Out-of-Sample) Minimum Risk Portfolios
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• Classical algorithm minimizes in-sample risk, but sometimes missed the point
• In this example, it invests over 50% of the portfolio in just McDonalds, Coca-Cola and Johnson & Johnson
• QHRP provides more diversification
Risk Improvement in SimulationsOut-of-sample volatility is reduced by 20% in simulated examples using 10 assets with 10% volatility each, random shocks, random correlations.
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Real-Time Optimization Framework
Real-Time Optimization Framework
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• Stream Analytics
• Real-Time data signals• Quantum powered analytics
Stock Market
1QBit SDKD-Wave 2X Processor
Post Processing
Trading Signals
GOOG
AAPL
MSFT
SBUX
TSLA
CorrelationGraph
Generator
Time
Real-Time Optimization Framework
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• Stream Analytics
• Real-Time data signals• Quantum powered analytics
Stock Market
1QBit SDKD-Wave 2X Processor
Post Processing
Trading Signals
GOOG
AAPL
MSFT
SBUX
TSLA
CorrelationGraph
Generator
Time
Real-Time Optimization Framework
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• Stream Analytics
• Real-Time data signals• Quantum powered analytics
Stock Market
1QBit SDKD-Wave 2X Processor
Post Processing
Trading Signals
GOOG
AAPL
MSFT
SBUX
TSLA
CorrelationGraph
Generator
Time
Tax Loss Harvesting
Tax Loss Harvesting
• Track an index using a subset of the components of the index while selling losses and moving into unowned components
• Integer Quadratic Programming: sell quantities from owned lots; buy new lots (avoiding wash sales)
• Offset gains and income; carry losses forward; create a tax buffer
• Quantum annealer used to find the optimal share quantities to buy and sell to generate the greatest tax benefit while staying within a specified tracking error.
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Tools and Resources
Tools
1QBit’squantum-readySoftwareDevelopmentKit:Explore: 1qbit.comRegister for Beta Release: qdk.1qbit.com
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Resources: www.quantumforquants.org
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Phil [email protected]
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