Mathematical Finance Research Articles

The role of risk attitudes in shaping digital privacy preferences: evidence from a large-scale survey

This paper investigates the relationship between individuals’ heterogeneous risk attitudes and privacy preferences using survey data from 24,899 respondents. It moves beyond demographic factors to examine how risk attitudes, the key psychological parameters in the utility function that determines consumer behaviors, related to consumers’ privacy decisions. Participants indicated their privacy preferences across eight dimensions, including age privacy, marital status privacy, and privacy preferences in the investment and financial obligation realms. We evaluated their risk attitudes through qualitative assessment and quantitative investment loss tolerances, as well as their risk attitudes towards rare and distant events. The study reveals that individuals with higher risk aversion exhibit more significant privacy concerns, especially regarding financial information. Individuals who perceive climate change—a non-immediate and rare risk—as a threat to their lives are also more likely to exhibit heightened concerns about their privacy. In addition, we observe that while these concerns translate into higher non-disclosure actions, the privacy paradox persists regardless of individuals’ risk attitudes. The findings suggest that privacy policies could be more effective through tailored approaches, including customized privacy settings and targeted educational campaigns that account for individual risk aversion.

Sustainability, Accuracy, Fairness, and Explainability (SAFE) Machine Learning in Quantitative Trading

The paper investigates the application of advanced machine learning (ML) methodologies, with a particular emphasis on state-of-the-art deep learning models, to predict financial market dynamics and maximize profitability through algorithmic trading strategies. The study compares the predictive capabilities and behavioral characteristics of traditional machine learning approaches, such as logistic regression and support vector machines, with those of highly sophisticated deep learning architectures, including Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM) networks, and Gated Recurrent Units (GRUs). The findings underscore the fundamental distinctions between these methodologies, with deeply trained models exhibiting markedly different predictive behaviors and performance, particularly in capturing complex temporal patterns within financial data. A cornerstone of the paper is the introduction and rigorous analysis of a framework to evaluate models, by means of the SAFE framework (Sustainability, Accuracy, Fairness, and Explainability). The framework is designed to address the opacity of black-box ML models by systematically evaluating their behavior across a set of critical dimensions. It also demonstrates how models’ predictive outputs align with the observed data, thereby reinforcing their reliability and robustness. The paper leverages historical stock price data from International Business Machines Corporation (IBM). The dataset is partitioned into a training phase during which the models are calibrated, and a validation phase, used to evaluate the predictive performance of the generated trading signals. The study addresses two primary machine learning tasks: regression and classification. Classical models are utilized for classification tasks, with their outputs directly interpreted as trading signals, while advanced deep learning models are employed for regression, with predictions of future stock prices further processed into actionable trading strategies. To evaluate the effectiveness of each strategy, rigorous backtesting is conducted, incorporating visual representations such as equity curves to assess profitability and key risk metrics like maximum drawdown for risk management. Supplementary performance indicators, including hit rates and the incidence of false positions, are analyzed alongside the equity curves to provide a holistic assessment of each model’s performance. This comprehensive evaluation not only highlights the superiority of cutting-edge deep learning models in predicting financial market trends but also demonstrates the pivotal role of the SAFE framework in ensuring that machine learning models remain trustworthy, interpretable, and aligned with ethical considerations.

Novel exact solutions for PDEs with mixed boundary conditions

We develop methods for the solution of inhomogeneous Robin-type boundary value problems (BVPs) that arise for certain linear parabolic partial differential equations (PDEs) on a half-line, as well as a second-order generalization. We are able to obtain nonstandard solutions to equations arising in a range of areas, including mathematical finance, stochastic analysis, hyperbolic geometry, and mathematical physics. Our approach uses the odd and even Hilbert transforms. The solutions we obtain and the method itself seem to be new.

The Price of Interest-rate Derivatives Based on Uncertain Mean-reverting Model

The Price of Interest-rate Derivatives Based on Uncertain Mean-reverting Model

Optimal Design of Multi-Asset Options

The combination of stochastic derivative pricing models and downside risk measures often leads to the paradox (risk, return) = (−infinity, +infinity) in a portfolio choice problem. The construction of a portfolio of derivatives with high expected returns and very negative downside risk (henceforth “golden strategy”) has only been studied if all the involved derivatives have the same underlying asset. This paper also considers multi-asset derivatives, gives practical methods to build multi-asset golden strategies for both the expected shortfall and the expectile risk measure, and shows that the use of multi-asset options makes the performance of the obtained golden strategy more efficient. Practical rules are given under the Black–Scholes–Merton multi-dimensional pricing model.

The Application of Stochastic Processes in Financial Mathematics

Stochastic processes are crucial in financial mathematics, providing a framework to model the inherent uncertainties of financial markets. This paper explores stochastic process application across various financial domains, such as option pricing, risk management, and financial engineering. Through case studies, literature review, and case analysis, the study demonstrates the practical effectiveness of stochastic processes in finance. The findings highlight the adaptability and robustness of these models in capturing market dynamics and optimizing financial strategies. This research particularly focuses on the implementation of Brownian motion and Its processes in derivative pricing, revealing significant improvements in accuracy compared to traditional methods. Additionally, we examine the integration of machine learning techniques with stochastic models to enhance predictive capabilities in risk assessment. This study also addresses the challenges and limitations of current stochastic approaches, proposing innovative solutions for practical implementation. These insights contribute to both theoretical understanding and practical applications in quantitative finance, offering valuable guidance for practitioners and researchers in the field

Solving Differential Equations with Physics-Informed Neural Networks

Solving differential equations is an extensive topic in various fields, such as fluid mechanics and mathematical finance. The recent resurgence in deep neural networks has opened up a brand new track for numerically solving these equations, with the potential to better deal with nonlinear problems and overcome the curse of dimensionality. The Physics-Informed Neural Network (PINN) is one of the fundamental attempts to solve differential equations using deep learning techniques. This paper aims to briefly review the application of PINNs and their variants in solving differential equations through a few simple examples, and to provide practitioners interested in this direction with a quick introduction to the relevant topic

Analysis of fractional Black-Scholes, Ornstein-Uhlenbeck, and Langevin models: a minimum distance estimation approach

The present research investigates parameter estimation in fractional stochastic models, specifically the Fractional Black-Scholes, Fractional Ornstein-Uhlenbeck, and Fractional Langevin models. These models, driven by fractional Brownian motion (fBm), capture long-memory effects and complex dependencies often observed in financial and physical systems. To efficiently estimate the parameters of these models, we apply the Minimum Distance Estimation (MDE) method. This approach tackles the challenges posed by long-range dependence, ensuring both asymptotic consistency and normality. We analyze the asymptotic properties of the estimators, including their consistency, asymptotic normality, and efficiency. By providing a unified theoretical framework, the current research highlights the robustness and applicability of the MDE method for parameter inference in fractional models. The findings contribute significantly to diverse fields such as quantitative finance, econometrics, and statistical physics.

Computational Reconstruction of the Volatility Term Structure in the General Hull–White Model

Volatility recovery is of paramount importance in contemporary finance. Volatility levels are heavily used in risk and portfolio management. We employ the Hull–White one- and two-factor models to describe the market condition. We computationally recover the volatility term structure as a piecewise-linear function of time. For every maturity, a cost functional, defined as the squared differences between theoretical and market prices, is minimized and the respective linear part is reconstructed. On the last time steps, before each maturity, the derivative price is decomposed in order to make the minimization problem analytically solvable. The procedure works fast since only scalar values are obtained on each minimization. However, the predictor–corrector nature of the algorithm allows for the precise recovery of very complex volatility functions. An implicit scheme is used to solve the PDEs on bounded domains. The computational simulations with artificial and real data show that the proposed algorithm is stable, accurate and efficient.

Measuring and reporting corporate actions in international financial statements: Combined instruments

Subject. This article discusses the mechanism for the formation of financial results of corporate actions with convertible bonds in the issuer's financial statements in accordance with the requirements of International Financial Reporting Standards (IFRS). Objectives. The article aims to improve and develop mathematical estimation tools for reflecting corporate actions with convertible bonds in the issuer's financial statements both on the part of the issuer and on the part of investors that meet the requirements of IFRS. Methods. For the study, I used models of financial mathematics, situational and dynamic approaches in systems analysis, as well as the methods of tabular inter­pretation and visualization. Results. The article proposes a formed set of calculation and estimation algorithms for combined financial instruments, based on the procedures of component-by-component assessment and the formation of a number of key accounting indicators in accordance with IFRS. The practical application of these algorithms based on mathematical estimation is illustrated by examples related to certain corporate actions of both the issuer and investors. Conclusions. The significance of this study lies in the development of scientific and applied tools, including estimation and accounting procedures that can be used by both Russian residents and international non-resident companies in the process of operating in the financial markets. The proposed economic and mathematical solutions are applicable for the purpose of preparing financial state­ments in accordance with IFRS at virtually any point in time, including cases where previous statements were not created.

Qualitative financial modelling in fractal dimensions

The Black–Scholes equation is one of the most important partial differential equations governing the value of financial derivatives in financial markets. The Black–Scholes model for pricing stock options has been applied to various payoff structures, and options trading is based on Black and Scholes’ principle of dynamic hedging to estimate and assess option prices over time. However, the Black–Scholes model requires severe constraints, assumptions, and conditions to be applied to real-life financial and economic problems. Several methods and approaches have been developed to approach these conditions, such as fractional Black–Scholes models based on fractional derivatives. These fractional models are expected since the Black–Scholes equation is derived using Ito’s lemma from stochastic calculus, where fractional derivatives play a leading role. Hence, a fractional stochastic model that includes the basic Black–Scholes model as a special case is expected. However, these fractional financial models require computational tools and advanced analytical methods to solve the associated fractional Black–Scholes equations. Nevertheless, it is believed that the fractal nature of economic processes permits to model economical and financial markets problems more accurately compared to the conventional model. The relationship between fractional calculus and fractals is well-known in the literature. This study introduces a generalized Black–Scholes equation in fractal dimensions and discusses its role in financial marketing. In our analysis, we consider power-laws properties for volatility, interest rated, and dividend payout, which emerge in several empirical regularities in quantitative finance and economics. We apply our model to study the problem of pricing barrier option and we estimate the values of fractal dimensions in both time and in space. Our model can be used to obtain the prices of many pay-off models. We observe that fractal dimensions considerably affect the solutions of the Black–Scholes equation and that, for fractal dimensions much smaller than unity, the call option increases significantly. We prove that fractal dimensions are a powerful tool to obtain new results. Further details are analyzed and discussed.

Analysis of Problems and Solutions of Quantitative Finance in UK

With the rapid development of the financial world, quantitative finance has become increasingly widespread in financial market. This paper illustrates the growing significance of quantitative finance and its impact on the financial market. This paper also explores in depth the challenges of quantitative analysis in the UK financial markets, with particular emphasis on three main problems: unpredictable market volatility, insufficient data, and an overreliance on historical data. By analyzing examples such as the volatility caused by Brexit and the rise of algorithmic trading, it is shown that these factors complicate the accuracy of financial models. Three solutions to the limitations are proposed, including integrating real-time information, adopting adaptive models that can respond to changing market conditions, and incorporating qualitative perspectives to improve forecasts. These methods are designed to improve the accuracy of financial forecasting, ultimately enabling financial professionals and policymakers to make more informed and accurate decisions under a dynamic and complex environment.

Some empirical studies for the applications of fractional $ G $-Brownian motion in finance

<p>Since the fractional $ G $-Brownian motion (fGBm) generalizes the concepts of the standard Brownian motion, fractional Brownian motion, and $ G $-Brownian motion, while it can exhibit long-range dependence or antipersistence and feature the volatility uncertainty simultaneously, it can be a better alternative stochastic process in the financial applications. Thus, in this paper, some empirical studies for the financial applications of the fGBm were carried out, where the recent high-frequency data for some selected assets in the financial market are from the Oxford-Man Institute of Quantitative Finance Realized Library. There are two main empirical findings. One was that the H-$ G $-normal distributions associated with the fGBm are more suitable in describing the dynamics of daily returns and increments of log-volatility for these assets than the usual distributions, since they not only characterize the properties of skewness, excess kurtosis, and long-range dependence $ \left(\frac{1}{2} < H < 1\right) $ or antipersistence $ \left(0 < H < \frac{1}{2}\right) $, but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different $ \underline{\sigma}^2 $ and $ \overline{\sigma}^2 $, but Hurst parameters $ H < \frac{1}{2} $, at any reasonable time scale. Then a generalized stochastic model for the dynamics of the assets called rough fractional stochastic volatility model driven by fGBm (RFSV-fGBm) was developed. Finally, some parameter estimates and numerical experiments for the RFSV-fGBm model were investigated and carried out.</p>

Building quantitative trading skills: integrating linear algebra and calculus into STEM programs

Building quantitative trading skills: integrating linear algebra and calculus into STEM programs

Competences development in financial mathematics at the secondary technical school: case study

Competences development in financial mathematics at the secondary technical school: case study

An Exploration of Different Strategies and Applications of Quantitative Investment under Different Models

With the rapid development of financial markets and the surge in data volume, quantitative investment has gradually become an important tool for investment decision-making. Traditional investment strategies often rely on experience and qualitative analysis, which is difficult to cope with complex market environments. Therefore, exploring quantitative investment models based on machine learning and deep learning has become a hot topic of research. This study aims to analyze the different applications of various models in quantitative investment, focusing on areas such as factor models, statistical arbitrage, and sentiment analysis. Through a comprehensive analysis of relevant literature in recent years, the results show that the future development of quantitative investment will increasingly rely on advanced algorithms and big data analysis techniques. Under the application of different models, there are diverse ideas and tools for the analysis and decision-making of financial markets. The combination of these models will promote the diversity and flexibility of investment strategies. At the same time, policymakers and financial institutions need to pay attention to the transparency and interpretability of the model to reduce potential risks. Looking forward, further research can focus on the combination optimization of models and the improvement of real-time data processing capabilities to cope with the changing market environment and provide stronger support for the practice of quantitative investment.

A Quantitative Strategy for value investing in Small Cap Stocks Based on Positive Feedback Asymmetry

Significant positive feedback trading exists in the Chinese market, and there is a significant difference between the extent of chasing up and killing down in the context of positive feedback trading. The purpose of this paper is to investigate whether a quantitative investment strategy with this asymmetric trait as the core factor can achieve significant excess returns and reduce the risk impact of irrational investors in the market. This paper is based on the CSI 500 index represented by small-cap stocks as the strategic stock pool, and further constructs a quantitative stock selection strategy, which firstly estimates the positive feedback trading asymmetry index of individual stocks in the market through the framework of SW, and at the same time, combines the Chinese three-factor model with Amihud's illiquidity index to construct the factor portfolios, and conducts a validity test to ultimately select the four portfolios with the highest cumulative returns as factor portfolios. This paper concludes that the strategy has a return of 316.73% during the back testing period, a weekly Sharpe ratio of 0.33, and a maximum backtest of 36.41%, realizing a significant excess return with controlled risk.

The Role of Artificial Intelligence and Machine Learning in Quantitative Finance and Stock Market Forecasting

The Role of Artificial Intelligence and Machine Learning in Quantitative Finance and Stock Market Forecasting

Preface: 3rd International Conference on Economics, Mathematical Finance and Risk Management (EMFRM 2024)

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

Estimation of the multivariate symmetric stable distribution using the method of moments

. The multivariate symmetric stable distribution is a heavy-tailed elliptically contoured law that has many important applications in signal processing and financial mathematics. The family includes the sub-Gaussian stable distribution as a special case. This work addresses the problem of feasible estimation of the parameters of the multivariate symmetric stable distribution. We present a method based on the application of the method of moments to the empirical characteristic function. Further, we show almost sure convergence of our estimators, discover their limiting distribution, and demonstrate their finite-sample performance.