Statistical Tools for Finance and Insurance

Financial mathematics plays a pivotal role in various aspects of modern economics and finance. This paper provides an introduction to the fundamental concepts, theories, and applications of financial mathematics. It begins by outlining the basic principles of financial mathematics, including the time value of money, interest rates, and compounding. Computational programs enhance these mathematical models, offering robust solutions and efficient computation for complex financial problems. This study explores the integration of computational programs with financial mathematics, their methodologies, and applications in the finance sector. The results underscore the significance of computational methods in improving the accuracy, speed, and scalability of financial models, ultimately contributing to better decision-making and risk management. We explore fundamental concepts, models, and techniques employed in financial mathematics, aiming to provide a comprehensive understanding of their applications and significance in real-world financial scenarios. This paper provides a comprehensive overview of the application of differential equations in financial mathematics, highlighting key models such as the Black-Scholes model, interest rate models, and optimal investment strategies.

This chapter describes and illustrates the elements and mechanics of valuing mortgage-backed securities (MBS). It begins with the basics of valuation and then briefly reviews the sources of MBS prepayments. Next, it discusses the models and assumptions that go into generating a set of projected cash flows. Forecasting MBS prepayment speeds and, in turn, total cash flows, is a much more complex undertaking than predicting the timing of redemption of a callable corporate bond. Practitioners generally rely on econometric prepayment models and associated auxiliary models to generate speed and cash flow forecasts, which are then used to value the bond. The Monte Carlo simulation is the only viable methodology for valuing mortgage-backed securities, as closed-form solutions are unavailable, and the path-dependent nature of the embedded prepayment option generally precludes the use of lattice-based approaches.

We introduce a reduced-form modeling framework for mortgage- backed securities in which we solve for the implied prepayment function from the cross section of market prices. From the implied prepayment function, we find that prepayment rates are driven not only by interest rates, but also by two macroeconomic factors: turnover and rate response. Intuitively, turnover represents prepayments for exogenous reasons like employment-related moves, household income shocks, and foreclosures, while rate response reflects frictions faced by borrowers in refinancing into a lower rate. We find that the implied turnover and rate response measures are in fact significantly related to macroeconomic measures such as consumption growth, the unemployment rate, housing values, credit availability, and market uncertainty. Implied prepayments are substantially higher than actual prepayments, providing direct evidence of significant prepayment risk premia in mortgage-backed security prices. We analyze the properties of the prepayment risk premium and find that it is almost entirely due to compensation for turnover risk. We also find evidence that mortgage- backed security prices were significantly affected by Fannie Mae credit risk and the Federal Reserve’s Quantitative Easing Programs.

In this paper we develop a closed-form and thus computationally highly efficient formula to approximate the value of fixed-rate mortgage-backed securities (MBS). Our modelling framework is based on reduced-form and prepayment-risk-neutral valuation techniques and offers two major extensions compared with existing closed-form approximation approaches: we include a stochastic baseline prepayment factor in our model and we are able to capture the usual S-shaped curve of the refinancing incentive by a piecewise linear approximation. We apply our model to GNMA pass-through securities and test it on a 10-year sample of monthly GNMA MBS market prices for a wide range of coupons.

The quasi-Monte Carlo method for financial valuation and other integration problems has error bounds of size O((log N)k N-1), or even O((log N)k N-3/2), which suggests significantly better performance than the error size O(N-1/2) for standard Monte Carlo. But in high-dimensional problems, this benefit might not appear at feasible sample sizes. Substantial improvements from quasi-Monte Carlo integration have, however, been reported for problems such as the valuation of mortgage-backed securities, in dimensions as high as 360. The authors believe that this is due to a lower effective dimension of the integrand in those cases. This paper defines the effective dimension and shows in examples how the effective dimension may be reduced by using a Brownian bridge representation.

The 2024 3rd International Conference on Economics, Mathematical Finance and Risk Management (EMFRM 2024) was successfully held during December 07-08, 2024 in Dubai, United Arab Emirates. This conference aims to bring together innovative scholars and experts in economic system, business statistics, mathematical finance, corporate governance, and risk management in a common forum. The main objective of EMFRM 2024 is to promote economic development research, another objective is to facilitate the exchange of scientific information among researchers, developers, engineers, students, and practitioners around the world. We are confident that the participants benefited from the theoretical insights, practical knowledge, and personal contacts that will help them to build a long-term and sustainable communication among researchers and practitioners in related scientific areas. EMFRM 2024 invited scholars and experts from various universities to participate in the review and guidance of this conference. The conference focuses on the latest research fields, aims to provide an international cooperation and exchange platform for experts, scholars, and business managers to share their research achievements. This collection of proceedings compiles oral and paper presentations submitted by the authors and scrutinized by the special committee. EMFRM Organizing Committees Dubai, United Arab Emirates

Contributors

Along with the continuous development of capital markets and intelligent finance technologies, quantitative investment is entering into the most critical and challenging area – fundamental quantitative investment. So far, quantitative investment has been focused on automation of technical analysis and trading, while fundamental investment has been large discretionary. This paper provides an overview of quantitative investment and fundamental investment towards a fundamental quantitative investment theory and technical system based on multi-factor models. We start with reviewing relevant literature on modern financial quantitative investment and fundamental investment. Then we cover the theoretical basis and development of multi-factor models and their applications for stock selection, involving linear and non-linear relationships, machine learning, deep learning with neural networks, random forests, and Support Vector Machines (SVMs). We explore the frontiers of fundamental quantitative investment and shed light on the future research prospects.

Mathematical modeling and numerical methods play a key role in the field of quantitative finance, for example, for financial derivative pricing and for risk management purposes. Asset models of increasing complexity, like stochastic volatility models (local stochastic volatility, rough volatility based on fractional Brownian motion) require advanced, efficient numerical techniques to bring them successfully into practice. When computations take too long, an involved asset model is not a feasible option as practical considerations demand a balance between the model’s accuracy and the time it takes to compute prices and risk management measures. In the big data era, typical basic computational tasks in the financial industry are often involved and computationally intensive due to the large volumes of financial data that are generated nowadays. Besides the traditional numerical methods in financial derivatives pricing in quantitative finance (like partial differential equation (PDE) discretization and solution methods, Fourier methods, Monte Carlo simulation), recently deep machine learning techniques have emerged as powerful numerical approximation techniques within scientific computing. Following the so-called Universal Approximation Theory, we will employ deep neural networks for financial computations, either to speed up the solution processes or to solve highly complicated, highdimensional, problems in finance. Particularly, we will employ supervised machine learning techniques, based on intensive learning of so called labeled information (input-output relations, where sets of parameters form the input to a neural network, and the output to be learned is a solution to a financial problem).

This thesis has been written during my time as a Ph.D. candidate in Finance at Lancaster University and as a visiting Ph.D. scholar at the Econometrics Department of the University of Amsterdam. This Ph.D. thesis starts with an introduction to finance for a general audience. Followed by an extensive literature overview, which I have continued to enrich over the years, of interesting topics in the econometrics-intense field of finance so I felt the need to incorporate it into my Ph.D. – which is a documentation of my work over the last years. I then start with a very brief teaser regarding finite sample properties of the classical skewness estimator and a robust alternative. This paves the way to the first paper in this thesis on Realized Skewness, Asymmetric Volatility and Risk Management which is co-authored with my supervisor and was influenced by my time spent in Amsterdam. The next paper, asymmetric dynamics in index volatility and constituent correlation was my first work after completing my M.Sc. in Quantitative Finance at Lancaster University on jump detection methodologies. It largely developed during the first period of my Ph.D. with the great support of my supervisor and my thesis advisor. These two papers are linked by an overarching theme: the asymmetric effects of returns on future volatility and vice versa. The last chapter, co-authored with Cisil Sarisoy, developed more recently and concentrates on transformations of the covariance matrix, also known as the beta, in a noisy setting.

The main purpose of this chapter is to discuss important quantitative methods used to do the research in quantitative finance and risk management. We first discuss statistics theory and methods. Second, we discuss econometric methods. Third, we discuss mathematics. Finally, we discuss other methods such as operation research, stochastic process, computer science and technology, entropy, and fuzzy set theory.

Quantitative analysis of energy markets

Heavy-Tailed Distributions and Risk Management of Equity Market Tail Events

Contributors

Measuring and managing the financial sustainability of the borrowers is crucial to financial institutions for their risk management. As a result, building an effective corporate financial distress prediction model has been an important research topic for a long time. Recently, researchers are exerting themselves to improve the accuracy of financial distress prediction models by applying various business analytics approaches including statistical and artificial intelligence methods. Among them, support vector machines (SVMs) are becoming popular. SVMs require only small training samples and have little possibility of overfitting if model parameters are properly tuned. Nonetheless, SVMs generally show high prediction accuracy since it can deal with complex nonlinear patterns. Despite of these advantages, SVMs are often criticized because their architectural factors are determined by heuristics, such as the parameters of a kernel function and the subsets of appropriate features and instances. In this study, we propose globally optimized SVMs, denoted by GOSVM, a novel hybrid SVM model designed to optimize feature selection, instance selection, and kernel parameters altogether. This study introduces genetic algorithm (GA) in order to simultaneously optimize multiple heterogeneous design factors of SVMs. Our study applies the proposed model to the real-world case for predicting financial distress. Experiments show that the proposed model significantly improves the prediction accuracy of conventional SVMs.