Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the 'instantaneous arbitrage capability' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the (NFLVR) condition if and only if $0$ is in the discrete spectrum of the connection Laplacian on the cash flow bundle or of the Dirac Laplacian of the twisted cash flow bundle with the exterior algebra bundle. We apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles for complete arbitrage free markets to markets not satisfying the (NFLVR). Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler characteristic of the asset nominal space is a topological obstruction to the the (NFLVR) condition, and, by means of the Bochner-Weitzenb\"ock formula, the non vanishing of the homology group of the cash flow bundle is revealed to be a topological obstruction to (NFLVR), too. Asset bubbles are defined, classified and decomposed for markets allowing arbitrage.

Quality data is a fundamental contributor to success in statistics and machine learning. If a statistical assessment or machine learning leads to decisions that create value, data contributors may want a share of that value. This paper presents methods to assess the value of individual data samples, and of sets of samples, to apportion value among different data contributors. We use Shapley values for individual samples and Owen values for combined samples, and show that these values can be computed in polynomial time in spite of their definitions having numbers of terms that are exponential in the number of samples.

This paper establishes the existence of a unique nonnegative continuous viscosity solution to the HJB equation associated with a Markovian linear-quadratic control problems with singular terminal state constraint and possibly unbounded cost coefficients. The existence result is based on a novel comparison principle for semi-continuous viscosity sub- and supersolutions for PDEs with singular terminal value. Continuity of the viscosity solution is enough to carry out the verification argument.

A process is polynomial if its extended generator maps any polynomial to a polynomial of equal or lower degree. Then its conditional moments can be calculated in closed form, up to the computation of the exponential of the so-called generator matrix. In this article, we provide an explicit formula to the problem of computing correlators, that is, computing the expected value of moments of the process at different time points along its path. The strength of our formula is that it only involves linear combinations of the exponential of the generator matrix, as in the one-dimensional case. The framework developed allows then for easy-to-implement solutions when it comes to financial pricing, such as for path-dependent options or in a stochastic volatility models context.

We study Nash equilibria for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our set-up, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result which allows us to find Nash equilibria by solving suitable quasi-variational inequalities with some non-standard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.

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.

This paper employs machine learning algorithms to forecast German electricity spot market prices. The forecasts utilize in particular bid and ask order book data from the spot market but also fundamental market data like renewable infeed and expected demand. Appropriate feature extraction for the order book data is developed. Using cross-validation to optimise hyperparameters, neural networks and random forests are proposed and compared to statistical reference models. The machine learning models outperform traditional approaches.

We develop from the ground up a new market-making model tailor-made for high-frequency trading under a limit order book (LOB), based on the well-known classification of order types in market microstructure. Our flexible framework allows arbitrary volume, jump, and spread distributions as well as the use of market orders. It also honors the consistency of price movements upon arrivals of different order types (e.g. price never goes down on buy market order) in addition to respecting the price-time priority of LOB. In contrast to the approach of regular control on diffusion as in the classical Avellaneda and Stoikov market-making framework, we exploit the techniques of optimal switching and impulse control on marked point processes, which have proven to be very effective in modeling the order-book features. The Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) associated with the control problem can be solved numerically via the finite-difference method. We illustrate our optimal trading strategy with a full numerical analysis, calibrated to the order-book statistics of a popular ETF. Our simulation shows that the profit of market-making can be seriously overstated under LOBs with inconsistent price movements.

We introduce a novel application of Support Vector Machines (SVM), an important Machine Learning algorithm, to determine the beginning and end of recessions in real time. Nowcasting, "forecasting" a condition about the present time because the full information about it is not available until later, is key for recessions, which are only determined months after the fact. We show that SVM has excellent predictive performance for this task, and we provide implementation details to facilitate its use in similar problems in economics and finance.

We propose a simpler derivation of the probability density function of Feller Diffusion using the Fourier Transform and solving the resulting equation via the Method of Characteristics. We also discuss simulation algorithms and confirm key properties related to hitting time probabilities via the simulation.

Since the risk in financial markets has become much more uncertain and volatile than before, the usual risk measures may be limited when dealing with the risk management. In this paper, we will study several classes of 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. The dual representations for them are also provided.

We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for a generic market dynamics given by a multidimensional It\^o's process we specify and prove the equivalence between (NFLVR) and expected utility maximization. As a by-product we provide a geometric characterization of the (NUPBR) condition given by the zero curvature (ZC) condition. Finally, we extend the Black-Scholes PDE to markets allowing arbitrage.