In equity markets, stocks are conventionally labeled by equity indices (company names). By relabeling stocks according to their ranks in capitalization, rather than their equity indices (company names), a different, more stable market structure can emerge. Specifically, we will gain a different perspective on market dynamics by focusing on the stock that occupies a certain rank in capitalization while the corresponding company name may change. We refer to a market labeled by the equity indices (company names) as a market in name space and one labeled by ranks in capitalization as a market in rank space . Market in rank space was explored by Fernholtz et al. who observed a stable distribution of capitalization across different ranks in the U.S. equity market over different time periods [11,16]. They further introduced an explanatory hybrid-Atlas model under stochastic portfolio theory, a framework that enables analyzing portfolios in rank space [5,15]. Empirically, B. Healy et al. analyzed the U.S. equity data and showed that the market in rank space is driven by a dominant single factor [14], in contrast to the multi-factor-driven market in name space [9,10,19]. While the primary market factor in rank space has been extensively studied, the residual returns - those not explained by this primary factor in stock returns - remain a fertile land of adventure.

We propose a new method for finding statistical arbitrages that can contain more assets than just the traditional pair. We formulate the problem as seeking a portfolio with the highest volatility, subject to its price remaining in a band and a leverage limit. This optimization problem is not convex, but can be approximately solved using the convex-concave procedure, a specific sequential convex programming method. We show how the method generalizes to finding moving-band statistical arbitrages, where the price band midpoint varies over time.

We present a machine learning approach for finding minimal equivalent martingale measures for markets simulators of tradable instruments, e.g. for a spot price and options written on the same underlying. We extend our results to markets with frictions, in which case we find "near-martingale measures" under which the prices of hedging instruments are martingales within their bid/ask spread. By removing the drift, we are then able to learn using Deep Hedging a "clean" hedge for an exotic payoff which is not polluted by the trading strategy trying to make money from statistical arbitrage opportunities. We correspondingly highlight the robustness of this hedge vs estimation error of the original market simulator. We discuss applications to two market simulators.

We present a numerically efficient approach for learning a risk-neutral measure for paths of simulated spot and option prices up to a finite horizon under convex transaction costs and convex trading constraints. This approach can then be used to implement a stochastic implied volatility model in the following two steps: 1. Train a market simulator for option prices, as discussed for example in our recent work Bai et al. (2019); 2. Find a risk-neutral density, specifically the minimal entropy martingale measure. The resulting model can be used for risk-neutral pricing, or for Deep Hedging (Buehler et al., 2019) in the case of transaction costs or trading constraints. To motivate the proposed approach, we also show that market dynamics are free from "statistical arbitrage" in the absence of transaction costs if and only if they follow a risk-neutral measure. We additionally provide a more general characterization in the presence of convex transaction costs and trading constraints. These results can be seen as an analogue of the fundamental theorem of asset pricing for statistical arbitrage under trading frictions and are of independent interest.

We use an adversarial expert based online learning algorithm to learn the optimal parameters required to maximise wealth trading zero-cost portfolio strategies. The learning algorithm is used to determine the relative population dynamics of technical trading strategies that can survive historical back-testing as well as form an overall aggregated portfolio trading strategy from the set of underlying trading strategies implemented on daily and intraday Johannesburg Stock Exchange data. The resulting population time-series are investigated using unsupervised learning for dimensionality reduction and visualisation. A key contribution is that the overall aggregated trading strategies are tested for statistical arbitrage using a novel hypothesis test proposed by Jarrow et al. on both daily sampled and intraday time-scales. The (low frequency) daily sampled strategies fail the arbitrage tests after costs, while the (high frequency) intraday sampled strategies are not falsified as statistical arbitrages after costs. The estimates of trading strategy success, cost of trading and slippage are considered along with an offline benchmark portfolio algorithm for performance comparison. In addition, the algorithms generalisation error is analysed by recovering a probability of back-test overfitting estimate using a nonparametric procedure introduced by Bailey et al.. The work aims to explore and better understand the interplay between different technical trading strategies from a data-informed perspective.

Arbitrage is the risk-free method of making profit from exploiting price differences in different markets. For example, if one stock is trading at a higher price in one market than another, one could buy the stock for the lower price on one market and sell it for the higher price on the other, thereby making profit without taking risks. These pricing disparities have become increasingly hard to capitalize on as they only appear for very short periods of time with the advancements in technology and highfrequency trading. Only those who can recognize and take advantage of arbitrage opportunities first can benefit, turning it into a winner-takes-all situation. This has made it difficult to make consistent profit from price discrepancies, as one needs to recognize them quickly and be the first to leverage them. Yet, arbitrage is a necessary tool in the marketplace as it quickly eliminates market inefficiencies and keeps prices uniform across markets [2, 5, 11, 6, 3, 17].