Mathematical Finance Research Articles

Machine Learning-Driven Stock Selection System: A Proposed Multi-Factor Model-Based Navigation Tool for the Chinese Financial Market

In view of the constantly changing and volatile nature of the Chinese financial landscape, characterized by the introduction of high-tech and huge amounts of market data, there is a need for an improved method for stock selection. Conventional stock methods are, however, inadequate due to their tendency to ignore the high volatility of the market and the nonlinear relationship between many financial variables which are at play. They have not been able to deliver predictions that are strong and accurate. This research, therefore, aims to explore how machine learning algorithms can be utilized to improve the accuracy and adaptability of stock selection models, thereby addressing these limitations. Through empirical results, it can be claimed that ML models, especially tree-based algorithms like LightGBM and XGBoost, are superior to traditional models in predicting stock returns. In this case, rolling windows are very well adapted to machine learning methods, which stand out as superior in terms of the adaptability to the changing stock market conditions. In addition, the research reveals the necessity of integrating the market-driven data, for instance, the trading volume and the momentum, into the stock selection models can help to better capture short-term pricing dynamics. The dissertation concludes with a machine learning-based stock selection system that is optimized for the Chinese financial market. The results of this work will increase the knowledge of quantitative finance by showing the machine learning algorithms can be more accurate in stock selection, and at the same time, providing practical solutions to the investors and financial institutions. The research emphasizes the necessity of combining modern computational methods and traditional financial theories to devise more effective and adaptive investment strategies in a complicated financial environment.

Adaptive Hybrid Optimized LSTM Models: A Novel Computational Framework for Algorithmic Financial Trading System

In our modern society, with the development of Internet and information system, pre-programing algorithmic trading strategies for online automatic trading has also flourished, especially in the rapidly fluctuating trading market. Using quantitative trading programs that can automatically trade in response to market conditions has attracted the attention of major financial institutions and governments in the financial market. How to use self-adaptive programs to automatically trade in the ever-changing financial market has become a popular research topic for the development and pursuit of all financial markets in recent years. This research proposes an online self-adaptive trading algorithm that can be applied to financial markets such as stock market, currency markets, cryptocurrency markets, futures markets, etc. In the first part of the algorithm, the simplified swarm optimization will be used to optimize the parameters of the newly proposed flexible grid in this research. Then the data will be imported into the artificial neural network model for training in the latter part, helping the trading model automatically select the appropriate parameters for construction flexible grid corresponding to the market conditions. The greatest contribution of the research is to provide a whole new trading algorithm that can adapt to the dynamic trading market, automatically make suitable adjustments to current trading strategy and place real-time orders. The algorithm controlling both profit and risk, which can seem like a balanced trading algorithm that are robust and profitable.

Modelling and Forecasting of Exchange Rate Pairs Using the Kalman Filter

ABSTRACTDeveloping and employing practically useful and easy to calibrate models for prediction of exchange rates remains a challenging task, especially for highly volatile emerging market currencies. In this paper, we propose a novel approach for joint prediction of correlated exchange rates for two different currencies with respect to the same base currency. For this purpose, we reformulate a generalized version of a bivariate ARMA model into a state space model and use the Kalman filter for estimation and forecasting of the underlying exchange rates as latent variables. With extensive numerical experiments spanning 18 different exchange rates (across both emerging markets, developing and developed economies), we demonstrate that our approach consistently outperforms univariate ARMA models as well as the random walk model in short term out‐of‐sample prediction for various exchange rate pairs. Our study fills a gap in the empirical finance literature in terms of robust, explainable, accurate, and easy to calibrate models for forecasting correlated exchange rates. The proposed methodology has applications in exchange rate risk management as well as pricing of financial derivatives based on two exchange rates.

Applications of machine learning in quantitative trading

Abstract. This paper also addresses quantitative trading where machine learning techniques are applied. Quantitative trading uses historical data and makes future predictions that inform and optimise trading strategies. There are two machine learning techniques used in quantitative trading: supervised and reinforcement learning. In a supervised learning setting, machine learning techniques such as regression analysis, classification models and time series prediction are used to forecast future possible outcomes in the market. Data such as historical prices, volume and financial indicators are used to determine trends that inform and optimise trading decisions. Reinforcement learning is another machine learning technique in quantitative trading where adaptive strategies using a reward system are designed. These strategies then analyse market dynamics and optimise trades through interaction and feedback. The third technique, deep learning, takes neural networks and uses them to process large-scale, complex data. It's particularly useful in quantitative trading as it generates more accurate predictions; for instance, neural networks are capable of learning trading signals from price and volume data and even from news headlines. By enabling computers to analyse large quantities of training data, deep learning improves the predictive accuracy of future price trends. It also helps to generate trading signals that are more robust to market conditions.

AI-Driven Optimization of Financial Quantitative Trading Algorithms and Enhancement of Market Forecasting Capabilities

Abstract: The application of artificial intelligence (AI) into financial markets has revolutionised quantitative trading and market forecasting by increasing the efficiency of algorithmic trading, improving the accuracy of market predictions and facilitating real-time market decisions. This paper will provide an overview of the application of Al in the financial markets focusing on the use of machine learning (ML), deep learning (DL) and reinforcement learning (RL) in optimizing the trading algorithms, specifically the capability of Al to process very high data points and complex relationships that other quantitative models are unable to capture. We will discuss trading algorithms such as XGBoost, deep neural networks such as long short-term memory (LSTM) networks and convolutional neural networks (CNNs), how they can outperform traditional quantitative trading models and real-time decision making in stock price prediction, pattern recognition and trading strategy optimisation. We will also look at Al-enhanced predictive models that utilise deep learning and layered models, such as Natural language processing (NLP) sentiment analysis to capture the public sentiment in the market to forecast employing diverse datasets such as historical prices, market volatility, macroeconomic factors and social media sentiment to improve the forecasting accuracy. By going through several experiments and case studies, this paper will shed light on the impact of entrusting quantitative trading and market forecasting decisions to AI for improved performance and reduced errors. There are many challenges ahead but AI plays a constructive role in improving the trading strategies and forecasting market outcomes accurately.

Special Issue “Interplay Between Financial and Actuarial Mathematics II”

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Financial Mathematics in Risk Management: Construction and Optimization of Quantitative Models

Financial Mathematics in Risk Management: Construction and Optimization of Quantitative Models

Numerical Methods and Computation in Financial Mathematics: Solving PDEs, Monte Carlo Simulations, and Machine Learning Applications

Abstract. This paper delves into three fundamental numerical methods and computational techniques in financial mathematics: Finite Difference Methods (FDM), Monte Carlo Simulations (MCS), and Machine Learning (ML) applications. Finite Difference Methods are widely utilized for solving partial differential equations (PDEs) in option pricing, with various schemes offering different stability and convergence properties. Monte Carlo Simulations provide a powerful approach for pricing complex derivatives and risk management, addressing the challenges of high-dimensionality and computational complexity. Machine Learning has revolutionized predictive modeling in finance, enabling sophisticated analysis of large datasets to uncover hidden patterns and enhance trading strategies. Through a detailed examination of these methods, including specific examples and data, this paper highlights their theoretical foundations, practical implementations, and the advancements they bring to computational finance. By bridging theoretical approaches with practical applications, we aim to offer insights into the future directions and challenges in financial mathematics.

Research on the Application of Mathematics in Finance

Abstract. Mathematics is a fundamental science and a problem-solving method that plays an important role in revealing the essential laws of the development of certain things. It has rigorous logic, high abstractness, and wide applicability. This brings great difficulties to enterprises, especially high school mathematics because it involves a wide range of knowledge points and complex calculations. Among them, the formation of financial mathematics has greatly promoted the development of China's financial industry. Moreover, in solving financial problems, relevant knowledge and theories of mathematics can be used to analyze and explore financial laws, and the application effect is significant, for example, using deterministic mathematical method and nondeterministic mathematical method to solving question about financial investment and returns; building securities investment portfolio model or asset valuation model to calculating and expressing some economic models. Based on this, this article discusses the application of mathematical knowledge to several financial problems, analyzes the current problems, and looks forward to the future of financial mathematics.

Stochastic generalized standard materials and risk-averse effective behavior

In this work, we develop a theoretical formulation for describing dissipative material behaviors in a stochastic setting, using the framework of Generalized Standard Materials (GSM). Our goal is to capture the variability inherent in the material model while ensuring thermodynamic consistency, by employing the mathematical framework of stochastic programming. We first show how average behaviors can be computed using the expected value of the free energy and dissipation pseudo-potentials. We then introduce the concept of a risk-averse effective measure, which provides both an optimistic and a pessimistic estimate of the uncertain material behavior. To this end, we utilize the Conditional Value-at-Risk, a widely used risk measure in mathematical finance. We also demonstrate how these concepts can be extended to variational problems at the structure scale, allowing us to compute the effective response of a structure composed of a stochastic material.

Simpson�s Paradox: Conceptual Foundational Implications in Real-Life Decision-Making in Quantitative Investment and Trading

Pankaj Mani writes that a phenomenon in probability and statistics in which a trend appears in several groups of data but disappears or reverses when the groups are combined raises foundational issues in concluding the significance of statistical relationships among different variables reliably in real-world decision-making for trading, investing, risk management.

Interest Rate Risk Modelling Using Semi-Heavy Tail Distributions of Normal Variance-Mean Mixtures: A Study on Central Bank of Kenya Interest Rates

Derivative prices such as options and bond prices as well as swaps depend on the distributional assumptions of the underlying economic variables, normally, interest rates. The risk associated with changes in interest rates may worsen the value of the contract that depend on it since the values of these assets (derivative contracts) are affected directly by the fluctuations in interest rates. The distribution of interest rates, therefore, needs to be well understood to reduce the risks of losses associated with it. The Binomial Option pricing model assumes that interest rates are constant throughout the life of the option. Another common assumption of the underlying economic variables is that their returns are normally distributed with constant volatility. These assumptions have been used in pricing derivatives and currencies and has led to over-pricing and in some cases under-pricing. The models of the Normal Variance-Mean Mixtures used in this research shows better performance than the normal distribution. Weekly 91-day and commercial bank interest rates are used from 1991 to 2021. This study will provide a better model for interest rates to avoid mispricing. We find that the 91-day Treasury Bills interest rates follow a Generalized Hyperbolic distribution while the Commercial Bank interest rates follows a Normal Inverse Gaussian distribution. G.A.R.C.H model is then used in forecasting under each of the above findings. A 99pc Value at Risk is then computed for the two models and calculates the minimum expected returns in the subsequent months. This research forms a foundation for the development of advanced pricing models that incorporates the fluctuations of interest rate in the pricing industry. We conclude that the Normal Inverse Gaussian and Generalized hyperbolic distributions provide better fits than the assumed normal distribution.

Exploring AI's Role in Enhancing Risk Assessment Models in Financial Quantitative Trading

The application of Artificial Intelligence (AI) algorithms to financial risk management has fundamentally transformed the nature of risk and the tools being used to control and manage it. This paper examines how AI is deployed to manage market risk, including volatility forecasting, portfolio optimisation, stress testing, and sentiment analysis. While many of these models are well-accepted across the industry and underpin many front-office and risk-management processes, they are often limited in dynamic market environments where traditional linearity doesnt exist and real-time data is king. AI has brought advanced tools that can be applied to large and complex data sets, and that adapt quickly as market conditions change. Reinforcement learning algorithms are now deployed to optimise portfolio weights dynamically. Natural-language processing is also used to augment sentiment analysis, which can be leveraged to assess potential risk. Other forms of AI can be deployed to address challenges that come with more advanced AI models. Explainable AI models provide the opportunity to produce financial outcomes that are more transparent and less opaque while also helping to validate regulatory compliance. Analyzing the applications of AI case studies sheds light on why financial professionals must continue to embrace AI tools, many of which are designed to help them manage risk in an increasingly volatile market environment.

Portfolio Optimization Using Machine Learning Method and Monte Carlo Simulation

Investment portfolio optimization is a crucial aspect of quantitative finance, aiming to maximize returns and minimize risks. This study focuses on optimizing investment allocation among five selected stocks (BAIC Blue Valley, BYD, Chang’ an Automobile, Kweichow Moutai, Sunwoda) over 100 days using Monte Carlo simulation and machine learning methods. This study uses the sliding window approach with the linear regression model to predict future returns and calculate the variance-covariance matrix to determine the optimal portfolio weights, leading to two key strategies: maximizing the Sharpe ratio and minimizing the risk. The results reveal that the Max Sharpe Ratio portfolio achieved a cumulative return of 0.0909, significantly outperforming the CSI 300 index’s return of -0.0348. Additionally, the Min Risk portfolio exceeded the market index with a cumulative return of 0.0062. These findings demonstrate that both the Max Sharpe Ratio and Min Risk strategies are effective in achieving a balance between risk and return and surpassing the market, offering valuable insights for investors seeking optimized portfolio allocations.

Research on Quantitative Investment Strategies of Securities Investment Funds

This paper investigates the quantitative investment strategies of securities investment funds, aiming to explore their applications in financial markets and their impact on investment decision-making. Through a review of relevant literature and empirical data, the paper first outlines the basic concepts of quantitative investing and its importance in securities investment funds. Then, it discusses various quantitative strategies, including multi-factor models and machine learning algorithms, and validates their effectiveness through case studies. The research finds that quantitative investing can enhance the scientific and accurate nature of investment decisions while effectively reducing investment risks. Finally, the paper provides suggestions for optimizing quantitative strategies and discusses future trends, offering practical references for the investment industry.

Quantitative Finance and Information Technologies: A Comparative Analysis of Quantitative Trading and Cryptocurrency and Their Regulatory Challenges

As technology has improved in the last decade, financial institutions have developed new technologies, including quantitative trading and cryptocurrency, to enhance their financial products and services. This paper first provides a brief background of quantitative trading and argues for the transactional efficiency of quantitative trading over traditional trading practices; it characterizes quantitative trading as fast and precise. Meanwhile, the study also accounts for the regulatory concerns–including data leakage and platform security–that quantitative trading firms may encounter. This study then establishes a distinction between cryptocurrency and quantitative trading–the former is money-driven, and the latter is data-driven. This paper then discusses the speculative nature of cryptocurrency and addresses its financial concerns citing the FTX collapse. Overall, this paper establishes the argument that quantitative trading supported by technological experts and facilitators offers more advantages than disadvantages compared to cryptocurrency trading. This research concludes that since quantitative trading and cryptocurrency trading are conducted without consideration for international boundaries, they offer bold financial potential as alternatives to traditional banking practices, as long as specific international financial laws are complied with.

New control variates for pricing basket options

Abstract Accepted by: Giorgio Consigli Accurate pricing of basket options, which are financial derivatives on multiple underlying assets, is a challenging and practically important task for financial institutions. We propose several new control variates for accurate, fast and efficient pricing of basket options. The first approach to deriving new control variates is the use of Hermite polynomial approximation of appropriate function of the underlying asset prices, which leads to a Black–Scholes-like analytic solution. This approach is new in the option pricing context and opens up new possibilities in derivative pricing. Further control variates are analytically derived using Jensen’s inequality in one case, and distributional properties of multivariate Wiener processes in other cases. All the newly proposed control variates are shown to lead to excellent variance reduction in numerical experiments based on realistic data. The proposed methods are novel, computationally simple and have a strong potential to replace more conventional methods, such as the geometric lower bound in simulation-based pricing of basket options and similar products used in financial risk management.

Methods for estimating integrated variance: Jump robustness issues in high frequency time series

Integrated variance is a volatility measure of a process in continuous time, which is applicable in financial mathematics as a means to optimize a portfolio, project dynamics of a price of a financial asset. The consistency of an estimator for integrated variance of a random process is at the core of this article. A fundamental diffusion process is extended by adding a jump component as a means of improving the descriptive function of the process. It is activity of jumps that is a factor subject to which is the consistency of the estimator for integrated variance. Due to this fact, consistency is defined as an extent to which the estimator at hand is jump robust. Two main methods for estimating integrated variance are considered and the capacity of corresponding estimators to withstand the effect of jumps while converging is briefly analyzed. The arguments indicating a necessity for further research of the effect of jumps with reference to works of the authors who have established a ground for analysis of integrated variance and those works containing main asymptotic results for consistency of integrated variance estimators are elaborated. Based on this, avenue for further research and development of asymptotic theory for consistency of an estimator for integrated variance is identified.

Analytical computation of conditional moments in the extended Cox–Ingersoll–Ross process with regime switching: Hybrid PDE system solutions with financial applications

In this paper, we introduce a novel analytical approach for the computation of the nth conditional moments of an m-state regime-switching extended Cox–Ingersoll–Ross process driven by a continuous-time finite-state irreducible Markov chain. This approach is applicable for all integers n≥1 and m≥1, thereby ensuring wide-ranging utility. The key of our investigation is a complex hybrid system of inter-connected PDEs, derived through a utilization of the Feynman–Kac formula for regime-switching diffusion processes. Our exploration into the solutions of this hybrid PDE system culminates in the derivation of exact closed-form formulas for the conditional moments for diverse values of n and m. Additionally, we study the asymptotic characteristics of the first conditional moments for the 2-state regime-switching Cox–Ingersoll–Ross process, particularly focusing on the effects of the symmetry inherent in the Markov chain’s intensity matrix and the implications of various parameter configurations. Highlighting the practicality of our methodology, we conduct Monte Carlo simulations to not only corroborate the accuracy and computational efficacy of our proposed approach but also to demonstrate its applicability to real-world applications in financial markets. A principal application highlighted in our study is the valuation of VIX futures and VIX options within a dynamic, mean-reverting, hybrid regime-switching framework. This exemplifies the potential of our analytical method to significantly impact contemporary financial modeling and derivative pricing.

Stochastic Control Problems with State Reflections Arising from Relaxed Benchmark Tracking

This paper studies stochastic control problems motivated by optimal consumption with wealth benchmark tracking. The benchmark process is modeled by a combination of a geometric Brownian motion and a running maximum process, indicating its increasing trend in the long run. We consider a relaxed tracking formulation such that the wealth compensated by the injected capital always dominates the benchmark process. The stochastic control problem is to maximize the expected utility of consumption deducted by the cost of the capital injection under the dynamic floor constraint. By introducing two auxiliary state processes with reflections, an equivalent auxiliary control problem is formulated and studied, which leads to the Hamilton-Jacobi-Bellman equation with two Neumann boundary conditions. We establish the existence of a unique classical solution to the dual partial differential equation using some novel probabilistic representations involving the local time of some dual processes together with a tailor-made decomposition-homogenization technique. The proof of the verification theorem on the optimal feedback control can be carried out by some stochastic flow analysis and technical estimations of the optimal control. Funding: L. Bo and Y. Huang are supported by the National Natural Science Foundation of China [Grant 12471451], the Natural Science Basic Research Program of Shaanxi [Grant 2023-JC-JQ-05], the Shaanxi Fundamental Science Research Project for Mathematics and Physics [Grant 23JSZ010], and Fundamental Research Funds for the Central Universities [Grant 20199235177]. X. Yu is supported by the Hong Kong RGC General Research Fund (GRF) [Grant 15304122] and the Research Centre for Quantitative Finance at the Hong Kong Polytechnic University [Grant P0042708].