Modeling stock returns is not a new task for mathematicians, investors, and portfolio managers, but it remains a difficult objective due to the ebb and flow of stock markets. One common solution is to approximate the distribution of stock returns with a normal distribution. However, normal distributions place infinitesimal probabilities on extreme outliers, but these outliers are of particular importance in the practice of investing. In this paper, we investigate the normality of the distribution of daily returns of major stock market indices. We find that the normal distribution is not a good model for stock returns, even over several years' worth of data. Moreover, we propose using the Laplace distribution as a model for daily stock returns.

This note outlines a method for clustering time series based on a statistical model in which volatility shifts at unobserved change-points. The model accommodates some classical stylized features of returns and its relation to GARCH is discussed. Clustering is performed using a probability metric evaluated between posterior distributions of the most recent change-point associated with each series. This implies series are grouped together at a given time if there is evidence the most recent shifts in their respective volatilities were coincident or closely timed. The clustering method is dynamic, in that groupings may be updated in an online manner as data arrive. Numerical results are given analyzing daily returns of constituents of the S&P 500.

The prediction of stock prices is an important task in economics, investment and financial decision-making. It has for several decades, spurred the interest of many researchers to design stock price predictive models. In this paper, the symbiotic organisms search algorithm, a new metaheuristic algorithm is employed as an efficient method for training feedforward neural networks (FFNN). The training process is used to build a better stock price predictive model. The Straits Times Index, Nikkei 225, NASDAQ Composite, S&P 500, and Dow Jones Industrial Average indices were utilized as time series data sets for training and testing proposed predic-tive model. Three evaluation methods namely, Root Mean Squared Error, Mean Absolute Percentage Error and Mean Absolution Deviation are used to compare the results of the implemented model. The computational results obtained revealed that the hybrid Symbiotic Organisms Search Algorithm exhibited outstanding predictive performance when compared to the hybrid Particle Swarm Optimization, Genetic Algorithm, and ARIMA based models. The new model is a promising predictive technique for solving high dimensional nonlinear time series data that are difficult to capture by traditional models.

Lead-lag relationships among assets represent a useful tool for analyzing high frequency financial data. However, research on these relationships predominantly focuses on correlation analyses for the dynamics of stock prices, spots and futures on market indexes, whereas foreign exchange data have been less explored. To provide a valuable insight on the nature of the lead-lag relationships in foreign exchange markets here we perform a detailed study for the one-minute log returns on exchange rates through three different approaches: i) lagged correlations, ii) lagged partial correlations and iii) Granger causality. In all studies, we find that even though for most pairs of exchange rates lagged effects are absent, there are many pairs which pass statistical significance tests. Out of the statistically significant relationships, we construct directed networks and investigate the influence of individual exchange rates through the PageRank algorithm. The algorithm, in general, ranks stock market indexes quoted in their respective currencies, as most influential. Altogether, these findings suggest that all market information does not spread instantaneously, contrary to the claims of the efficient market hypothesis.

We introduce a computational framework for dynamic portfolio valuation and risk management building on machine learning with kernels. We learn the replicating martingale of a portfolio from a finite sample of its terminal cumulative cash flow. The learned replicating martingale is given in closed form thanks to a suitable choice of the kernel. We develop an asymptotic theory and prove convergence and a central limit theorem. We also derive finite sample error bounds and concentration inequalities. Numerical examples show good results for a relatively small training sample size.

We analyze the behavior of approximate Bayesian computation (ABC) when the model generating the simulated data differs from the actual data generating process; i.e., when the data simulator in ABC is misspecified. We demonstrate both theoretically and in simple, but practically relevant, examples that when the model is misspecified different versions of ABC can yield substantially different results. Our theoretical results demonstrate that even though the model is misspecified, under regularity conditions, the accept/reject ABC approach concentrates posterior mass on an appropriately defined pseudo-true parameter value. However, under model misspecification the ABC posterior does not yield credible sets with valid frequentist coverage and has non-standard asymptotic behavior. In addition, we examine the theoretical behavior of the popular local regression adjustment to ABC under model misspecification and demonstrate that this approach concentrates posterior mass on a completely different pseudo-true value than accept/reject ABC. Using our theoretical results, we suggest two approaches to diagnose model misspecification in ABC. All theoretical results and diagnostics are illustrated in a simple running example.

Attempts to allocate capital to a selection of different investment objects often face the problem that investors' decisions are made under limited information (no historical return data) and an extremely limited timeframe. Nevertheless, in some cases, rational investors with a certain level of experience are able to ordinally rank investment alternatives through relative assessments of the probability that an investment will be successful. However, to apply traditional portfolio optimization models, analysts must use historical (or simulated/expected) return data as the basis for their calculations. Our paper develops an alternative portfolio optimization framework that is able to handle this kind of information (given by the ordinal ranking of investment alternatives) and to calculate an optimal capital allocation based on a Cobb-Douglas function. Considering risk-neutral investors, we show that the results of this portfolio optimization model usually outperform the output generated by the (intuitive) Equally Weighted Portfolio (EWP) of the different investment alternatives, which is the result of optimization when one is unable to incorporate additional data (the ordinal ranking of the alternatives). In a further extension, we show that our model is also able to address risk-averse investors to capture diversification benefits.

Since the quasiconvex risk measures is a bigger class than the well known convex risk measures, the study of quasiconvex risk measures makes sense especially in the financial markets with volatility. In this paper, we will study the quasiconvex risk measures defined on a special space $L^{p(\cdot)}$ where the variable exponent $p(\cdot)$ is no longer a given real number like the space $L^{p}$, but a random variable, which reflects the possible volatility of the financial markets. The dual representation for this quasiconvex risk measures will also provided.

As systemic risk has become a hot topic in the financial markets, how to measure, allocate and regulate the systemic risk are becoming especially important. However, the financial markets are becoming more and more complicate, which makes the usual study of systemic risk to be restricted. In this paper, we will study the systemic risk measures on a special space $L^{p(\cdot)}$ where the variable exponent $p(\cdot)$ is no longer a given real number like the space $L^{p}$, but a random variable, which reflects the possible volatility of the financial markets. Finally, the dual representation for this new systemic risk measures will be studied. Our results show that every this new systemic risk measure can be decomposed into a convex certain function and a simple-systemic risk measure, which provides a new ideas for dealing with the systemic risk.