The first quantum computers are expected to perform well at quadratic optimisation problems. In this paper a quadratic problem in finance is taken, the Portfolio Optimisation problem. Here, a set of assets is chosen for investment, such that the total risk is minimised, a minimum return is realised and a budget constraint is met. This problem is solved for several instances in two main indices, the Nikkei225 and the S\&P500 index, using the state-of-the-art implementation of D-Wave's quantum annealer and its hybrid solvers. The results are benchmarked against conventional, state-of-the-art, commercially available tooling. Results show that for problems of the size of the used instances, the D-Wave solution, in its current, still limited size, comes already close to the performance of commercial solvers.
Regression is widely used by practioners across many disciplines. We reformulate the underlying optimisation problem as a second-order conic program providing the flexibility often needed in applications. Using examples from portfolio management and quantitative trading we solve regression problems with and without constraints. Several Python code fragments are given. The code and data are available online at http://www.github.com/tschm/MosekRegression.
We extend Relative Robust Portfolio Optimisation models to allow portfolios to optimise their distance to a set of benchmarks. Portfolio managers are also given the option of computing regret in a way which is more in line with market practices than other approaches suggested in the literature. In addition, they are given the choice of simply adding an extra constraint to their optimisation problem instead of outright changing the objective function, as is commonly suggested in the literature. We illustrate the benefits of this approach by applying it to equity portfolios in a variety of regio…
We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may…
We investigate a hybrid quantum-classical solution method to the mean-variance portfolio optimization problems. Starting from real financial data statistics and following the principles of the Modern Portfolio Theory, we generate parametrized samples of portfolio optimization problems that can be related to quadratic binary optimization forms programmable in the analog D-Wave Quantum Annealer 2000Q. The instances are also solvable by an industry-established Genetic Algorithm approach, which we use as a classical benchmark. We investigate several options to run the quantum computation optimally…
Performance analysis, from the external point of view of a client who would only have access to returns and holdings of a fund, evolved towards exact attribution made in the context of portfolio optimisation, which is the internal point of view of a manager controlling all the parameters of this optimisation. Attribution is exact, that-is-to-say no residual "interaction" term remains, and various contributions to the optimal portfolio can be identified: predictive signals, constraints, benchmark. However constraints are identified as a separate portfolio and attribution for each signal that ar…
We employ perturbation analysis technique to study multi-asset portfolio optimisation with transaction cost. We allow for correlations in risky assets and obtain optimal trading methods for general utility functions. Our analytical results are supported by numerical simulations in the context of the Long Term Growth Model.
Portfolio optimization requires sophisticated covariance estimators that are able to filter out estimation noise. Non-linear shrinkage is a popular estimator based on how the Oracle eigenvalues can be computed using only data from the calibration window. Contrary to common belief, NLS is not optimal for portfolio optimization because it does not minimize the right cost function when the asset dependence structure is non-stationary. We instead derive the optimal target. Using historical data, we quantify by how much NLS can be improved. Our findings reopen the question of how to build the covar…
A novel optimisation framework through quadratic nonlinear projection is introduced for credit portfolio when the portfolio risk is measured by Conditional Value-at-Risk (CVaR). The whole optimisation procedure to search toward the optimal portfolio state is conducted by a series of single-step optimisations under the local constraints described in the multi-dimensional constraint parameter space as functions of the total amount of portfolio adjustment. Each single-step optimisation is approximated by the first-order variation of the weight increments with respect to the total amount of portfo…
The portfolio optimisation problem, first raised by Harry Markowitz in 1952, has been a fundamental and central topic to understanding the stock market and making decisions. There has been plenty of works contributing to development of the mean-variance optimisation (MVO) so far. In this paper, one kind of them, namely, dynamic mean-variance optimisation (DMVO) is mainly discussed. One can apply either precommitment or game-theoritical approach to address time-inconsistency in DMVO. We use the second approach to seek for a time-consistent strategy. After obtaining the optimal strategy, we exte…
For portfolio optimisation under proportional transaction costs, we provide a duality theory for general cadlag price processes. In this setting, we prove the existence of a dual optimiser as well as a shadow price process in a generalised sense. This shadow price is defined via a "sandwiched" process consisting of a predictable and an optional strong supermartingale and pertains to all strategies which remain solvent under transaction costs. We provide examples showing that in the present general setting the shadow price process has to be of this generalised form.
In this paper we consider the problem of minimising drawdown in a portfolio of financial assets. Here drawdown represents the relative opportunity cost of the single best missed trading opportunity over a specified time period. We formulate the problem (minimising average drawdown, maximum drawdown, or a weighted combination of the two) as a nonlinear program and show how it can be partially linearised by replacing one of the nonlinear constraints by equivalent linear constraints. Computational results are presented (generated using the nonlinear solver SCIP) for three test instances drawn fro…
Portfolio optimisation typically aims to provide an optimal allocation that minimises risk, at a given return target, by diversifying over different investments. However, the potential scope of such risk diversification can be limited if investments are concentrated in only one country, or more specifically one currency. Multi-currency portfolio is an alternative to achieve higher returns and more diversified portfolios but it requires a careful management of the entailed risks from changes in exchange rates. The deviation between asset and currency exposures in a portfolio is defined as the "…
The debate between active and passive investment strategies has been ongoing for many years and is far from being over. In this paper, we show that the choice of an optimal portfolio management strategy depends on an investment climate, which we measure via the parameters of the Heston model calibrated to the real stock market data. Depending on the values of those parameters, the passive strategy may namely outperform the active ones or vice versa. The method is tested on three stock market indices: S\&P500, DAX and WIG20.
A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function …
We consider an investor, whose portfolio consists of a single risky asset and a risk free asset, who wants to maximize his expected utility of the portfolio subject to the Value at Risk assuming a heavy tail distribution of the stock prices return. We use Markov Decision Process and dynamic programming principle to get the optimal strategies and the value function which maximize the expected utility for parametric as well as non parametric distributions. Due to lack of explicit solution in the non parametric case, we use numerical integration for optimization
This paper examines an optimal investment problem in a continuous-time (essentially) complete financial market with a finite horizon. We deal with an investor who behaves consistently with principles of Cumulative Prospect Theory, and whose utility function on gains is bounded above. The well-posedness of the optimisation problem is trivial, and a necessary condition for the existence of an optimal trading strategy is derived. This condition requires that the investor's probability distortion function on losses does not tend to 0 near 0 faster than a given rate, which is determined by the util…
In this paper we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of backward SDEs. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively $m$-stable dual sets. Using this notion …
Large-scale Time-Varying Portfolio Optimisation using Graph Attention Networks
University of Southampton（南安普顿大学）；Southampton Business School（南安普顿商学院）；Centre for Operational Research, Management Sciences and Information Systems (CORMSIS)（运筹学、管理科学与信息系统中心）；The University of Western Ontario（西安大略大学）
Signals coming from multivariate higher order conditional moments as well as the information contained in exogenous covariates, can be effectively exploited by rational investors to allocate their wealth among different risky investment opportunities. This paper proposes a new flexible dynamic copula model being able to explain and forecast the time-varying shape of large dimensional asset returns distributions. Moreover, we let the univariate marginal distributions to be driven by an updating mechanism based on the scaled score of the conditional distribution. This framework allows us to intr…
Traditional portfolio management methods can incorporate specific investor preferences but rely on accurate forecasts of asset returns and covariances. Reinforcement learning (RL) methods do not rely on these explicit forecasts and are better suited for multi-stage decision processes. To address limitations of the evaluated research, experiments were conducted on three markets in different economies with different overall trends. By incorporating specific investor preferences into our RL models' reward functions, a more comprehensive comparison could be made to traditional methods in risk-retu…
The proprietary nature of Hedge Fund investing means that it is common practise for managers to release minimal information about their returns. The construction of a Fund of Hedge Funds portfolio requires a correlation matrix which often has to be estimated using a relatively small sample of monthly returns data which induces noise. In this paper random matrix theory (RMT) is applied to a cross-correlation matrix C, constructed using hedge fund returns data. The analysis reveals a number of eigenvalues that deviate from the spectrum suggested by RMT. The components of the deviating eigenvecto…
We study the problem of active portfolio management where an investor aims to outperform a benchmark strategy's risk profile while not deviating too far from it. Specifically, an investor considers alternative strategies whose terminal wealth lie within a Wasserstein ball surrounding a benchmark's -- being distributionally close -- and that have a specified dependence/copula -- tying state-by-state outcomes -- to it. The investor then chooses the alternative strategy that minimises a distortion risk measure of terminal wealth. In a general (complete) market model, we prove that an optimal dyna…
In this paper we show how to implement in a simple way some complex real-life constraints on the portfolio optimization problem, so that it becomes amenable to quantum optimization algorithms. Specifically, first we explain how to obtain the best investment portfolio with a given target risk. This is important in order to produce portfolios with different risk profiles, as typically offered by financial institutions. Second, we show how to implement individual investment bands, i.e., minimum and maximum possible investments for each asset. This is also important in order to impose diversificat…
While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading …
Partial (replication) index tracking is a popular passive investment strategy. It aims to replicate the performance of a given index by constructing a tracking portfolio which contains some constituents of the index. The tracking error optimisation is quadratic and NP-hard when taking the L0 constraint into account so it is usually solved by heuristic methods such as evolutionary algorithms. This paper introduces a simple, efficient and scalable connectionist model as an alternative. We propose a novel reparametrisation method and then solve the optimisation problem with stochastic neural netw…
