We fit a dynamic factor model: monthly inflation-adjusted S\&P 500 returns vs 10-year trailing earnings yield (the inverse of Shiller price-to-earnings ratio), 10-year trailing dividend yield, and 10-year real interest rate. We model these three factors as AR(1) in three dimensions. We use long-term data from 1881 compiled by Robert Shiller, available at multpl.com. However, in the short run, fluctuations have heavy tails and do not significantly depend on previous values. We use Bayesian regression with normal residuals. We show significant dependence of long-term returns on the initial factor values.

Long-term investors, different from short-term traders, focus on examining the underlying forces that affect the well-being of a company. They rely on fundamental analysis which attempts to measure the intrinsic value an equity. Quantitative investment researchers have identified some value factors to determine the cost of investment for a stock and compare different stocks. This paper proposes using sequence prediction models to forecast a value factor-the earning yield (EBIT/EV) of a company for stock selection. Two advanced sequence prediction models-Long Short-term Memory (LSTM) and Gated Recurrent Unit (GRU) networks are studied. These two models can overcome the inherent problems of a standard Recurrent Neural Network, i.e., vanishing and exploding gradients. This paper firstly introduces the theories of the networks. And then elaborates the workflow of stock pool creation, feature selection, data structuring, model setup and model evaluation. The LSTM and GRU models demonstrate superior performance of forecast accuracy over a traditional Feedforward Neural Network model. The GRU model slightly outperformed the LSTM model.

We propose a three-state microscopic opinion formation model for the purpose of simulating the dynamics of financial markets. In order to mimic the heterogeneous composition of the mass of investors in a market, the agent-based model considers two different types of traders: noise traders and contrarians. Agents are represented as nodes in a network of interactions and they can assume any of three distinct possible states (e.g. buy, sell or remain inactive). The time evolution of the state of an agent is dictated by probabilistic dynamics that include both local and global influences. A noise trader is subject to local interactions, tending to assume the majority state of its nearest neighbors, whilst a contrarian is subject to a global interaction with the behavior of the market as a whole, tending to assume the state of the global minority of the market. The model exhibits the typical qualitative and quantitative features of real financial time series, including distributions of returns with heavy tails, volatility clustering and long-time memory for the absolute values of the returns. The distributions of returns are fitted by means of coupled Gaussian distributions, quantitatively revealing transitions between leptokurtic, mesokurtic and platykurtic regimes in terms of a non-linear statistical coupling which describes the complexity of the system.

We study risk-sharing equilibria with general convex costs on the agents' trading rates. For an infinite-horizon model with linear state dynamics and exogenous volatilities, the equilibrium returns mean-revert around their frictionless counterparts -- the deviation has Ornstein-Uhlenbeck dynamics for quadratic costs whereas it follows a doubly-reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully-coupled forward-backward SDEs. These fall outside the scope of known wellposedness results, but can be solved numerically using the simulation-based deep-learning approach of \cite{han.al.17}. In a calibration to time series of returns, bid-ask spreads, and trading volume, transaction costs substantially affect equilibrium asset prices. In contrast, the effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.

Systematic trading strategies are rule-based procedures which choose portfolios and allocate assets. In order to attain certain desired return profiles, quantitative strategists must determine a large array of trading parameters. Backtesting, the attempt to identify the appropriate parameters using historical data available, has been highly criticized due to the abundance of misleading results. Hence, there is an increasing interest in devising procedures for the assessment and comparison of strategies, that is, devising schemes for preventing what is known as backtesting overfitting. So far, many financial researchers have proposed different ways to tackle this problem that can be broadly categorised in three types: Data Snooping, Overestimated Performance, and Cross-Validation Evaluation. In this paper, we propose a new approach to dealing with financial overfitting, a Covariance-Penalty Correction, in which a risk metric is lowered given the number of parameters and data used to underpins a trading strategy. We outlined the foundation and main results behind the Covariance-Penalty correction for trading strategies. After that, we pursue an empirical investigation, comparing its performance with some other approaches in the realm of Covariance-Penalties across more than 1300 assets, using Ordinary and Total Least Squares. Our results suggest that Covariance-Penalties are a suitable procedure to avoid Backtesting Overfitting, and Total Least Squares provides superior performance when compared to Ordinary Least Squares.

In the article "Stochastic evolution equations for large portfolios of Stochastic Volatility models" (Arxiv:1701.05640) there is a mistake in the proof of Theorem 3.1. In this erratum we establish a weaker version of this Theorem and then we redevelop the regularity theory for our problem accordingly. This means that most of our regularity results are replaced by slightly weaker ones. We also clarify a point in the proof of a correct result.

The notion of market impact is subtle and sometimes misinterpreted. Here we argue that impact should not be misconstrued as volatility. In particular, the so-called ``square-root impact law'', which states that impact grows as the square-root of traded volume, has nothing to do with price diffusion, i.e. that typical price changes grow as the square-root of time. We rationalise empirical findings on impact and volatility by introducing a simple scaling argument and confronting it to data.

This study investigates the impacts of the Automobile NOx Law of 1992 on four pollutants in Japan. By utilizing monitoring data between 1897 and 1997, we found that the regulation decreased the level of SO2 by approximately 12.4%. The improving effects on the levels of NOx, Ox, and suspended particulate matter are estimated at 5.8%, 3.1%, and 3.5%, respectively. The result from our flexible specification suggests that, although the regulated areas experienced improvements in concentration, the neighboring effects might have been negligible.

Rough volatility models are continuous time stochastic volatility models where the volatility process is driven by a fractional Brownian motion with the Hurst parameter less than half, and have attracted much attention since a seminal paper titled "Volatility is rough" was posted on SSRN in 2014 claiming that they explain a scaling property of realized variance time series. From our point of view, the analysis is not satisfactory because the estimation error of the latent volatility was not taken into account; we show by simulations that it in fact results in a fake scaling property. Motivated by this preliminary finding, we construct a quasi-likelihood estimator for a fractional stochastic volatility model and apply it to realized variance time series to examine whether the volatility is really rough. Our quasi-likelihood is based on a central limit theorem for the realized volatility estimation error and a Whittle-type approximation to the auto-covariance of the log-volatility process. We prove the consistency of our estimator under high frequency asymptotics, and examine by simulations the finite sample performance of our estimator. Our empirical study suggests that the volatility is indeed rough; actually it is even rougher than considered in the literature.

Optimal multi-asset trading with Markovian predictors is well understood in the case of quadratic transaction costs, but remains intractable when these costs are $L_1$. We present a mean-field approach that reduces the multi-asset problem to a single-asset problem, with an effective predictor that includes a risk averse component. We obtain a simple approximate solution in the case of Ornstein-Uhlenbeck predictors and maximum position constraints. The optimal strategy is of the "bang-bang" type similar to that obtained in [de Lataillade et al., 2012]. When the risk aversion parameter is small, we find that the trading threshold is an affine function of the instantaneous global position, with a slope coefficient that we compute exactly. We provide numerical simulations that support our analytical results.

We introduce two models of taxation, the latent and natural tax processes, which have both been used to represent loss-carry-forward taxation on the capital of an insurance company. In the natural tax process, the tax rate is a function of the current level of capital, whereas in the latent tax process, the tax rate is a function of the capital that would have resulted if no tax had been paid. Whereas up to now these two types of tax processes have been treated separately, we show that, in fact, they are essentially equivalent. This allows a unified treatment, translating results from one model to the other. Significantly, we solve the question of existence and uniqueness for the natural tax process, which is defined via an integral equation. Our results clarify the existing literature on processes with tax.

This study examines the role of pawnshops as a risk-coping strategy in Japan in the prewar period when poor people were highly vulnerable. Using data on pawnshop loans in more than 250 municipalities and the 1918--1920 influenza pandemic as a natural experiment, we find that the total loan amount increased because of the pandemic shock. Our results suggest that those who regularly relied on pawnshops borrowed from them more money than usual to cope with the adverse health shock, whereas others did not take out pawnshop loans. In addition, further analyses reveal that loans from pawnshops prevented an increase in the unemployment rate due to the pandemic. Pawnshops thus served as an informal social insurance mechanism in early twentieth-century Japan.