Applications of Quantum Annealing in Computational Finance. Dr. Phil Goddard Head of Research, 1QBit D-Wave User Conference, Santa Fe, Sept.

Transcription

1 Applications of Quantum Annealing in Computational Finance Dr. Phil Goddard Head of Research, 1QBit D-Wave User Conference, Santa Fe, Sept. 2016

2 Outline Where s my Babel Fish? Quantum-Ready Applications for Computational Finance Tools and Resources 2

3 Where s my Babel Fish?

4 Where s my Babel Fish? k-coloring Connected Dominating Set Job Shop Scheduling Portfolio Optimization Expected Return Mean-Variance-Efficient Frontier Job 1 Job 2 Job Risk (Standard Deviation) Asset Allocation Risk Management time Graph Similarity Graph Partitioning Option Pricing 4

5 Is that a QUBO? min - x. Qx x 0,1 & Q R & & x R & Q R & & Weight of assets in portfolio Covariance matrix 5

6 Quantum-Ready Applications for Computational Finance

7 Applications for Computational Finance A Multi-Period Optimal Trading Strategy Quantum-Ready Hierarchical Risk Parity (QHRP) Real-Time Optimization Framework Tax Loss Harvesting 7

8 8 Optimal Trading Trajectories

9 A Multi-Period Optimal Trading Strategy The optimal trading trajectory problem The mathematical formulation Considerations in using the quantum annealer Experimental Results 9

10 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 t h0.6 ig e a l W 0.5 tim p O Time of Rebalance Single-period Optimal Weight --- Multi-period Optimal Weight 10

11 Practical Considerations Friction: Transaction costs prevent portfolio managers from monetizing much of their forecasting power Basic Constraints Trading Constraints Efficient Frontier 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 Return Risk 11

12 Mathematical Formulation The multi-period integer optimization problem may be written as 12 w = argmax 5 6 μ 8. w 8 s.t.: mean-variance portfolio optimization. 8PQ 9:;<9=> γ 2 w 8. Σ 8 w 8 9C>D To solve, the problem must be converted to standard QUBO form: & t: 6 w U,8 K UPQ Δw 8. Λ 8 Δw 8 GC9:H; HJ>;>, ;:KL.CKLNH; ; t, n: w U,8 K O min - x. Qx x 0,1 &, Q R & & Δw 8. Λ 8 O Δw 8 L:9K.CKLNH; transaction cost and market impacts

13 Bit Encoding The integer variables of the optimization problem w Z annealer x Z. must be recast to the binary variables used by the 13

14 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 The annealer solution is typically within a small and acceptable margin of error of the exact globally minimal solution. 14

15 Quantum-Ready Hierarchical Risk Parity

16 16 Quantum-Ready Hierarchical Risk Parity

17 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 w = arg min w w. Σ^w s. t. 6 w Z = 1 Z Classical Build the final weight vector by recursively minimizing variance from the bottom of the tree to the top w 1,1 = arg min w w. Σ d Q,Qw QHRP 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 w 2,1 = arg min w w. Σ d c,qw w 2,2 = arg min w w. Σ d c,cw 17

18 (Out-of-Sample) Minimum Risk Portfolios 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 18

19 20 % Risk Improvement in Simulations Out-of-sample volatility is reduced by 20% in simulated examples using 10 assets with 10% volatility each, random shocks, random correlations. 19

20 Real-Time Optimization Framework

21 Real-Time Optimization Framework Stream Analytics Real-Time data signals Quantum powered analytics GOOG AAPL MSFT SBUX TSLA Correlation Graph Generator 1QBit SDK D-Wave 2X Processor Post Processing Stock Market Time Trading Signals 21

22 Real-Time Optimization Framework Stream Analytics Real-Time data signals Quantum powered analytics GOOG AAPL MSFT SBUX TSLA Correlation Graph Generator 1QBit SDK D-Wave 2X Processor Post Processing Stock Market Time Trading Signals 22

23 Real-Time Optimization Framework Stream Analytics Real-Time data signals Quantum powered analytics GOOG AAPL MSFT SBUX TSLA Correlation Graph Generator 1QBit SDK D-Wave 2X Processor Post Processing Stock Market Time Trading Signals 23

24 Tax Loss Harvesting

25 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. 25

26 Tools and Resources

27 Tools 1QBit s quantum-ready Software Development Kit: Explore: 1qbit.com Register for Beta Release: qdk.1qbit.com 27

28 28 Resources:

29 Phil Goddard QB Information Technologies. All rights reserved.