Research articles for the 2020-04-07

Bootstraps Regularize Singular Correlation Matrices
Christian Bongiorno

I show analytically that the average of $k$ bootstrapped correlation matrices rapidly becomes positive-definite as $k$ increases, which provides a simple approach to regularize singular Pearson correlation matrices. If $n$ is the number of objects and $t$ the number of features, the averaged correlation matrix is almost surely positive-definite if $k> \frac{e}{e-1}\frac{n}{t}\simeq 1.58\frac{n}{t}$ in the limit of large $t$ and $n$. The probability of obtaining a positive-definite correlation matrix with $k$ bootstraps is also derived for finite $n$ and $t$. Finally, I demonstrate that the number of required bootstraps is always smaller than $n$. This method is particularly relevant in fields where $n$ is orders of magnitude larger than the size of data points $t$, e.g., in finance, genetics, social science, or image processing.

Collectivised Post-Retirement Investment
John Armstrong,Cristin Buescu

We quantify the benefit of collectivised investment funds, in which the assets of members who die are shared among the survivors. For our model, with realistic parameter choices, an annuity or individual fund requires approximately 20\% more initial capital to provide as good an outcome as a collectivised investment fund. We demonstrate the importance of the new concept of pension adequacy in defining investor preferences and determining optimal fund management. We show how to manage heterogeneous funds of investors with diverse needs. Our framework can be applied to existing pension products, such as Collective Defined Contribution schemes.

Controlling Human Utilization of Failure-Prone Systems via Taxes
Ashish R. Hota,Shreyas Sundaram

We consider a game-theoretic model where individuals compete over a shared failure-prone system or resource. We investigate the effectiveness of a taxation mechanism in controlling the utilization of the resource at the Nash equilibrium when the decision-makers have behavioral risk preferences, captured by prospect theory. We first observe that heterogeneous prospect-theoretic risk preferences can lead to counter-intuitive outcomes. In particular, for resources that exhibit network effects, utilization can increase under taxation and there may not exist a tax rate that achieves the socially optimal level of utilization. We identify conditions under which utilization is monotone and continuous, and then characterize the range of utilizations that can be achieved by a suitable choice of tax rate. We further show that resource utilization is higher when players are charged differentiated tax rates compared to the case when all players are charged an identical tax rate, under suitable assumptions.

Double continuation regions for American options under Poisson exercise opportunities
Zbigniew Palmowski,José Luis Pérez,Kazutoshi Yamazaki

We consider the L\'evy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case as in De Donno et al. [24], the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson arrival time to an interval via the scale function and then apply those identities to the computation of the optimal strategies. We also discuss the convergence of the optimal solutions to those in the continuous observation case as the rate of observation increases to infinity. Numerical experiments are also provided.

Instabilities in Multi-Asset and Multi-Agent Market Impact Games
Francesco Cordoni,Fabrizio Lillo

We consider the general problem of a set of agents trading a portfolio of assets in the presence of transient price impact and additional quadratic transaction costs and we study, with analytical and numerical methods, the resulting Nash equilibria. Extending significantly the framework of Schied and Zhang (2018), who considered two agents and one asset, we focus our attention on the conditions on the value of transaction cost making the trading profile of the agents, and as a consequence the price trajectory, wildly oscillating and the market unstable. We find that the presence of more assets, the heterogeneity of trading skills (e.g. speed or cost), and a large number of agents make the market more prone to large oscillations and instability. When the number of assets is fixed, a more complex structure of the cross-impact matrix, i.e. the existence of multiple factors for liquidity, makes the market less stable compared to the case when a single liquidity factor exists.

New fat-tail normality test based on conditional second moments with applications to finance
Damian Jelito,Marcin Pitera

In this paper we introduce an efficient fat-tail measurement framework that is based on the conditional second moments. We construct a goodness-of-fit statistic that has a direct interpretation and can be used to assess the impact of fat-tails on central data conditional dispersion. Next, we show how to use this framework to construct a powerful normality test. In particular, we compare our methodology to various popular normality tests, including the Jarque--Bera test that is based on third and fourth moments, and show that in many cases our framework outperforms all others, both on simulated and market stock data. Finally, we derive asymptotic distributions for conditional mean and variance estimators, and use this to show asymptotic normality of the proposed test statistic.

On the Evolution of Cryptocurrency Market Efficiency
Akihiko Noda

This study examines whether the efficiency of cryptocurrency markets (Bitcoin and Ethereum) evolve over time based on Lo's (2004) adaptive market hypothesis (AMH). In particular, we measure the degree of market efficiency using a generalized least squares-based time-varying model that does not depend on sample size, unlike previous studies that used conventional methods. The empirical results show that (1) the degree of market efficiency varies with time in the markets, (2) the degree of market efficiency varies with time, (2) Bitcoin's market efficiency level is higher than that of Ethereum over most periods, and (3) a market with high market liquidity has been evolving. We conclude that the results support the AMH for the most established cryptocurrency market.

Predicting tail events in a RIA-EVT-Copula framework
Wei-Zhen Lia,Jin-Rui Zhai,Zhi-Qiang Jiang,Gang-Jin Wang,Wei-Xing Zhou

Predicting the occurrence of tail events is of great importance in financial risk management. By employing the method of peak-over-threshold (POT) to identify the financial extremes, we perform a recurrence interval analysis (RIA) on these extremes. We find that the waiting time between consecutive extremes (recurrence interval) follow a $q$-exponential distribution and the sizes of extremes above the thresholds (exceeding size) conform to a generalized Pareto distribution. We also find that there is a significant correlation between recurrence intervals and exceeding sizes. We thus model the joint distribution of recurrence intervals and exceeding sizes through connecting the two corresponding marginal distributions with the Frank and AMH copula functions, and apply this joint distribution to estimate the hazard probability to observe another extreme in $\Delta t$ time since the last extreme happened $t$ time ago. Furthermore, an extreme predicting model based on RIA-EVT-Copula is proposed by applying a decision-making algorithm on the hazard probability. Both in-sample and out-of-sample tests reveal that this new extreme forecasting framework has better performance in prediction comparing with the forecasting model based on the hazard probability only estimated from the distribution of recurrence intervals. Our results not only shed a new light on understanding the occurring pattern of extremes in financial markets, but also improve the accuracy to predict financial extremes for risk management.

QuantNet: Transferring Learning Across Systematic Trading Strategies
Adriano Koshiyama,Sebastian Flennerhag,Stefano B. Blumberg,Nick Firoozye,Philip Treleaven

In this work we introduce QuantNet: an architecture that is capable of transferring knowledge over systematic trading strategies in several financial markets. By having a system that is able to leverage and share knowledge across them, our aim is two-fold: to circumvent the so-called Backtest Overfitting problem; and to generate higher risk-adjusted returns and fewer drawdowns. To do that, QuantNet exploits a form of modelling called Transfer Learning, where two layers are market-specific and another one is market-agnostic. This ensures that the transfer occurs across trading strategies, with the market-agnostic layer acting as a vehicle to share knowledge, cross-influence each strategy parameters, and ultimately the trading signal produced. In order to evaluate QuantNet, we compared its performance in relation to the option of not performing transfer learning, that is, using market-specific old-fashioned machine learning. In summary, our findings suggest that QuantNet performs better than non transfer-based trading strategies, improving Sharpe ratio in 15% and Calmar ratio in 41% across 3103 assets in 58 equity markets across the world. Code coming soon.

Robust estimation of superhedging prices
Jan Obloj,Johannes Wiesel

We consider statistical estimation of superhedging prices using historical stock returns in a frictionless market with d traded assets. We introduce a plugin estimator based on empirical measures and show it is consistent but lacks suitable robustness. To address this we propose novel estimators which use a larger set of martingale measures defined through a tradeoff between the radius of Wasserstein balls around the empirical measure and the allowed norm of martingale densities. We establish consistency and robustness of these estimators and argue that they offer a superior performance relative to the plugin estimator. We generalise the results by replacing the superhedging criterion with acceptance relative to a risk measure. We further extend our study, in part, to the case of markets with traded options, to a multiperiod setting and to settings with model uncertainty. We also study convergence rates of estimators and convergence of superhedging strategies.

The Economics of Social Data
Dirk Bergemann,Alessandro Bonatti,Tan Gan

A data intermediary pays consumers for information about their preferences and sells the information so acquired to firms that use it to tailor their products and prices. The social dimension of the individual data---whereby an individual's data are predictive of the behavior of others---generates a data externality that reduces the intermediary's cost of acquiring information. We derive the intermediary's optimal data policy and show that it preserves the privacy of the consumers' identities while providing precise information about market demand to the firms. This enables the intermediary to capture the entire value of information as the number of consumers grows large.

The market nanostructure origin of asset price time reversal asymmetry
Marcus Cordi,Damien Challet,Serge Kassibrakis

We introduce a framework to infer lead-lag networks between the states of elements of complex systems, determined at different timescales. As such networks encode the causal structure of a system, infering lead-lag networks for many pairs of timescales provides a global picture of the mutual influence between timescales. We apply our method to two trader-resolved FX data sets and document strong and complex asymmetric influence of timescales on the structure of lead-lag networks. Expectedly, this asymmetry extends to trader activity: for institutional clients in our dataset, past activity on timescales longer than 3 hours is more correlated with future activity at shorter timescales than the opposite (Zumbach effect), while a reverse Zumbach effect is found for past timescales shorter than 3 hours; retail clients have a totally different, and much more intricate, structure of asymmetric timescale influence. The causality structures are clearly caused by markedly different behaviors of the two types of traders. Hence, market nanostructure, i.e., market dynamics at the individual trader level, provides an unprecedented insight into the causality structure of financial markets, which is much more complex than previously thought.

Wavelet-based discrimination of isolated singularities masquerading as multifractals in detrended fluctuation analyses
Paweł Oświęcimka,Stanisław Drożdż,Mattia Frasca,Robert Gębarowski,Natsue Yoshimura,Luciano Zunino,Ludovico Minati

The robustness of two widespread multifractal analysis methods, one based on detrended fluctuation analysis and one on wavelet leaders, is discussed in the context of time-series containing non-uniform structures with only isolated singularities. Signals generated by simulated and experimentally-realized chaos generators, together with synthetic data addressing particular aspects, are taken into consideration. The results reveal essential limitations affecting the ability of both methods to correctly infer the non-multifractal nature of signals devoid of a cascade-like hierarchy of singularities. Namely, signals harboring only isolated singularities are found to artefactually give rise to broad multifractal spectra, resembling those expected in the presence of a well-developed underlying multifractal structure. Hence, there is a real risk of incorrectly inferring multifractality due to isolated singularities. The careful consideration of local scaling properties and the distribution of H\"older exponent obtained, for example, through wavelet analysis, is indispensable for rigorously assessing the presence or absence of multifractality.