Quantitative Finance

We approach the continuous-time mean-variance (MV) portfolio selection with reinforcement learning (RL). The problem is to achieve the best tradeoff between exploration and exploitation, and is formulated as an entropy-regularized, relaxed stochastic control problem. We prove that the optimal feedback policy for this problem must be Gaussian, with time-decaying variance. We then establish connections between the entropy-regularized MV and the classical MV, including the solvability equivalence and the convergence as exploration weighting parameter decays to zero. Finally, we prove a policy improvement theorem, based on which we devise an implementable RL algorithm. We find that our algorithm outperforms both an adaptive control based method and a deep neural networks based algorithm by a large margin in our simulations.

We consider an incomplete market with a nontradable stochastic factor and a continuous time investment problem with an optimality criterion based on monotone mean-variance preferences. We formulate it as a stochastic differential game problem and use Hamilton-Jacobi-Bellman-Isaacs equations to find an optimal investment strategy and the value function. What is more, we show that our solution is also optimal for the classical Markowitz problem and every optimal solution for the classical Markowitz problem is optimal also for the monotone mean-variance preferences. These results are interesting because the original Markowitz functional is not monotone, and it was observed that in the case of a static one-period optimization problem the solutions for those two functionals are different. In addition, we determine explicit Markowitz strategies in the square root factor models.

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 utility function. Under additional assumptions, we show that this condition is indeed the borderline for attainability, in the sense that for slower convergence of the distortion function there does exist an optimal portfolio.

We study T. Cover's rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to a $1 deposit into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e. 200% stocks and −100% bonds) and then continuously executing rebalancing trades to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalancing rules (i.e. 2) has some advantages over the pioneering approach taken by Cover & Company in their brilliant theory of universal portfolios (1986, 1991, 1996, 1998), where one's on-line trading performance is benchmarked relative to the final wealth of the best unlevered rebalancing rule of any kind in hindsight. Our approach lets practitioners express an a priori view that one of the favored asset allocations ("bets") b∈{ b 1 ,..., b n } will turn out to have performed spectacularly well in hindsight. In limiting our robustness to some discrete set of asset allocations (rather than all possible asset allocations) we reduce the price of the rebalancing option and guarantee to achieve a correspondingly higher percentage of the hindsight-optimized wealth at the end of the planning period. A practitioner who lives to delta-hedge this variant of Cover's rebalancing option through several decades is guaranteed to see the day that his realized compound-annual capital growth rate is very close to that of the best b i in hindsight. Hence the point of the rock-bottom option price.

The paper is aware of the importance of certain figures that are essential to an understanding of Credit Scoring models in credit acceptance process optimization, namely if the power of discrimination measured by Gini value is increased by 5% then the profit of the process can be increased monthly by about 1 500 kPLN (300 kGBP, 500 kUSD, 350 kEUR). Simple business models of credit loans are also presented: acquisition - installment loan (low price) and cross-sell - cash loans (high price). Scoring models are used to optimize process, to become profitable. Various acceptance strategies with different cutoffs are presented, some are profitable and some are not. Moreover, in a time of prosperity some are preferable whilst the inverse is true during a period of high risk or crisis. To optimize the process four models are employed: three risk models, to predict the probability of default and one typical propensity model to predict the probability of response. It is a simple but very important example of the Customer Lifetime Value (CLTV or CLV) model business, where risk and response models are working together to become a profitable process.

We extend the analysis of investment strategies derived from penalized quantile regression models, introducing alternative approaches to improve state\textendash of\textendash art asset allocation rules. First, we use a post\textendash penalization procedure to deal with overshrinking and concentration issues. Second, we investigate whether and to what extent the performance changes when moving from convex to nonconvex penalty functions. Third, we compare different methods to select the optimal tuning parameter which controls the intensity of the penalization. Empirical analyses on real\textendash world data show that these alternative methods outperform the simple LASSO. This evidence becomes stronger when focusing on the extreme risk, which is strictly linked to the quantile regression method.

Stock price prediction has been an important research theme both academically and practically. Various methods to predict stock prices have been studied until now. The feature that explains the stock price by a cross-section analysis is called a "factor" in the field of finance. Many empirical studies in finance have identified which stocks having features in the cross-section relatively increase and which decrease in terms of price. Recently, stock price prediction methods using machine learning, especially deep learning, have been proposed since the relationship between these factors and stock prices is complex and non-linear. However, there are no practical examples for actual investment management. In this paper, therefore, we present a cross-sectional daily stock price prediction framework using deep learning for actual investment management. For example, we build a portfolio with information available at the time of market closing and invest at the time of market opening the next day. We perform empirical analysis in the Japanese stock market and confirm the profitability of our framework.

The global minimum-variance portfolio is a typical choice for investors because of its simplicity and broad applicability. Although it requires only one input, namely the covariance matrix of asset returns, estimating the optimal solution remains a challenge. In the presence of high-dimensionality in the data, the sample covariance estimator becomes ill-conditioned and leads to suboptimal portfolios out-of-sample. To address this issue, we review recently proposed efficient estimation methods for the covariance matrix and extend the literature by suggesting a multi-fold cross-validation technique for selecting the necessary tuning parameters within each method. Conducting an extensive empirical analysis with four datasets based on the S&P 500, we show that the data-driven choice of specific tuning parameters with the proposed cross-validation improves the out-of-sample performance of the global minimum-variance portfolio. In addition, we identify estimators that are strongly influenced by the choice of the tuning parameter and detect a clear relationship between the selection criterion within the cross-validation and the evaluated performance measure.

We propose factor models for the cross-section of daily cryptoasset returns and provide source code for data downloads, computing risk factors and backtesting them out-of-sample. In "cryptoassets" we include all cryptocurrencies and a host of various other digital assets (coins and tokens) for which exchange market data is available. Based on our empirical analysis, we identify the leading factor that appears to strongly contribute into daily cryptoasset returns. Our results suggest that cross-sectional statistical arbitrage trading may be possible for cryptoassets subject to efficient executions and shorting.

We study portfolio optimization of four major cryptocurrencies. Our time series model is a generalized autoregressive conditional heteroscedasticity (GARCH) model with multivariate normal tempered stable (MNTS) distributed residuals used to capture the non-Gaussian cryptocurrency return dynamics. Based on the time series model, we optimize the portfolio in terms of Foster-Hart risk. Those sophisticated techniques are not yet documented in the context of cryptocurrency. Statistical tests suggest that the MNTS distributed GARCH model fits better with cryptocurrency returns than the competing GARCH-type models. We find that Foster-Hart optimization yields a more profitable portfolio with better risk-return balance than the prevailing approach.