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Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition
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Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E5: Financial Mathematics".

Deadline for manuscript submissions: closed (30 November 2024) | Viewed by 7759

Special Issue Editors

Prof. Dr. Daniel Sevcovic

E-Mail Website
Guest Editor
Department of Applied Mathematics and Statistics Faculty of Mathematics, Physics and Informatics Comenius University, Bratislava, Slovakia
Interests: partial differential equations and their applications; curvature driven flows of curves and interfaces; financial mathematics; pricing derivative securities
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Matthias Ehrhardt

E-Mail Website
Guest Editor
Applied Mathematics and Numeical Analysis, University of Wuppertal, Wuppertal, Germany
Interests: computational finance; sparse grids; artificial boundaries; splitting methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical finance provides a large set of theoretical and applied tools that range from pure branches of mathematics to more applied areas with the objective to deeply understand and solve current problems in finance.

In fact, a rigorous analysis of contemporary finance relies on advanced analytical and numerical methods and needs high-level mathematical skills with special emphasis on stochastic calculus with its rich mathematical structure, partial differential equations, partial integrodifferential equations, and fractional diffusion equations. Robust techniques of numerical analysis and computation are also required.

The aim of this Special Issue is to contribute to the enrichment of mathematical finance by broadening the knowledge in this area, with research papers on the following potential topics:

  • Stochastic analysis and control theory in finance;
  • fPDE and fractional models in finance;
  • PIDE and Lévy processes in finance;
  • Interest rate and credit risk modeling;
  • Models for portfolio and risk management;
  • Big data analytics;
  • Computational methods and high-performance computing in finance;
  • Machine learning in finance;
  • High-performance computing in finance;
  • Mathematical modeling in energy markets;
  • Methods for high-dimensional problems;
  • Applications of forward–backward stochastic differential equations;
  • Risk modeling and risk management;
  • Stochastic dynamic programming and Hamilton–Jacobi–Bellman equations;
  • Nonlinear option pricing models, generalizations of the Black–Scholes equation.

Prof. Dr. Daniel Sevcovic
Prof. Dr. Matthias Ehrhardt
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional models in finance
  • PIDE, and Lévy processes in finance
  • interest rate and credit risk modeling
  • portfolio management
  • risk management
  • big data analytics
  • stochastic dynamic programming
  • Hamilton–Jacobi–Bellman equations
  • nonlinear option pricing models

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (6 papers)

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Research

23 pages, 434 KiB  
Article
Portfolio Selection Based on Modified CoVaR in Gaussian Framework
by Piotr Jaworski and Anna Zalewska
Mathematics 2024, 12(23), 3766; https://doi.org/10.3390/math12233766 - 29 Nov 2024
Viewed by 492
Abstract
We study a Mean-Risk model, where risk is measured by a Modified CoVaR (Conditional Value at Risk): [...] Read more.
We study a Mean-Risk model, where risk is measured by a Modified CoVaR (Conditional Value at Risk): CoVaRα,β≤(X|Y)=VaRβ(X|Y+VaRα(Y)≤0). We prove that in a Gaussian setting, for a sufficiently small β, such a model has a solution. There exists a portfolio that fulfills the given constraints and for which the risk is minimal. This is shown in relation to the mean–standard deviation portfolio, and numerical examples are provided. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition)
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12 pages, 321 KiB  
Article
Calibration of the Ueno’s Shadow Rate Model of Interest Rates
by Lenka Košútová and Beáta Stehlíková
Mathematics 2024, 12(22), 3564; https://doi.org/10.3390/math12223564 - 14 Nov 2024
Cited by 1 | Viewed by 768
Abstract
Shadow rate models of interest rates are based on the assumption that the interest rates are determined by an unobservable shadow rate. This idea dates back to Fischer Black, who understood the interest rate as an option that cannot become negative. Its possible [...] Read more.
Shadow rate models of interest rates are based on the assumption that the interest rates are determined by an unobservable shadow rate. This idea dates back to Fischer Black, who understood the interest rate as an option that cannot become negative. Its possible zero values are consequences of negative values of the shadow rate. In recent years, however, the negative interest rates have become a reality. To capture this behavior, shadow rate models need to be adjusted. In this paper, we study Ueno’s model, which uses the Vasicek process for the shadow rate and adjusts its negative values when constructing the short rate. We derive the probability properties of the short rate in this model and apply the maximum likelihood estimation method to obtain the parameters from the real data. The other interest rates are—after a specification of the market price of risk—solutions to a parabolic partial differential equation. We solve the equation numerically and use the long-term rates to fit the market price of risk. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition)
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17 pages, 791 KiB  
Article
A Stock Optimization Problem in Finance: Understanding Financial and Economic Indicators through Analytical Predictive Modeling
by Aditya Chakraborty and Chris Tsokos
Mathematics 2024, 12(15), 2407; https://doi.org/10.3390/math12152407 - 2 Aug 2024
Cited by 1 | Viewed by 1240
Abstract
Given the significant impact of healthcare stock changes on the global economy, including its GDP and other financial factors, we endeavored to create an analytical prediction model for forecasting the annual percentage change of these stocks. Our model, which is nonlinear, incorporates five [...] Read more.
Given the significant impact of healthcare stock changes on the global economy, including its GDP and other financial factors, we endeavored to create an analytical prediction model for forecasting the annual percentage change of these stocks. Our model, which is nonlinear, incorporates five key discoveries. We focused on predicting the average weekly closing price (pWCP) of AbbVie Inc. (North Chicago, IL, USA)’s healthcare stock (ABBV) from 1 August 2017 to 31 December 2019. The stock was chosen based on the low beta risk, high dividend yield, and high yearly percentage return criteria. Alongside predicting the weekly stock price, our model identifies the individual indicators and their interactions that notably influence the response. These indicators were ranked based on their contribution percentages to the response. The validity of the model was justified based on the root mean square error (RMSE) and R2 value by performing 10-fold cross-validation. Furthermore, an optimization process using the desirability function was implemented to determine the optimal values of the indicators that maximize the response, along with the 95% confidence and 95% prediction interval. We also visually depicted the optimal ranges of any two indicators that affect the response AWCP. In our evaluation, we compared the original and predicted responses of AWCP using our analytical model. The results demonstrated a close alignment between the two sets of observations, highlighting the high accuracy of our model. Beyond these findings, our model provides additional valuable insights into the subject area. It has undergone thorough validation and testing, confirming its high quality and the precision of our weekly stock price predictions. The information derived from the modeling and analysis is important for constructive and accurate decision-making for individual investors, portfolio managers, and financial institutions concerning the financial and economic aspects of the healthcare industry. By identifying the optimum values of the controllable contributors through the optimization process, financial institutions can make the strategic changes needed for the company’s long-term viability. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition)
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13 pages, 4333 KiB  
Article
Projection and Contraction Method for Pricing American Bond Options
by Qi Zhang, Qi Wang, Ping Zuo, Hongbo Du and Fangfang Wu
Mathematics 2023, 11(22), 4689; https://doi.org/10.3390/math11224689 - 18 Nov 2023
Cited by 2 | Viewed by 1315
Abstract
In this paper, an effective numerical method is proposed for a linear complementarity problem (LCP) arising in the valuation of American bond options under the Cox–Ingersoll–Ross (CIR) model. Firstly, a variable substitution is used to simplify the linear complementary model. Secondly, the finite [...] Read more.
In this paper, an effective numerical method is proposed for a linear complementarity problem (LCP) arising in the valuation of American bond options under the Cox–Ingersoll–Ross (CIR) model. Firstly, a variable substitution is used to simplify the linear complementary model. Secondly, the finite difference method is adopted to discretize the simplified model, and an equivalent variational form is obtained. Based on the positive definiteness of the discretized matrix, a projection and contraction method (PCM) is adopted for the resulting discretized variational problem. Finally, numerical experiments highlight the effectiveness and performance of the proposed algorithm. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition)
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28 pages, 786 KiB  
Article
Optimal Consumption and Robust Portfolio Choice for the 3/2 and 4/2 Stochastic Volatility Models
by Yuyang Cheng and Marcos Escobar-Anel
Mathematics 2023, 11(18), 4020; https://doi.org/10.3390/math11184020 - 21 Sep 2023
Cited by 2 | Viewed by 1365
Abstract
This manuscript derives optimal consumption and investment strategies for risk-averse investors under the 4/2 stochastic volatility class of models. We work under an expected utility (EUT) framework and consider a Constant Relative Risk Aversion (CRRA) investor, who may also be ambiguity-averse. The corresponding [...] Read more.
This manuscript derives optimal consumption and investment strategies for risk-averse investors under the 4/2 stochastic volatility class of models. We work under an expected utility (EUT) framework and consider a Constant Relative Risk Aversion (CRRA) investor, who may also be ambiguity-averse. The corresponding Hamilton–Jacobi–Bellman (HJB) and HJB–Isaacs (HJBI) equations are solved in closed-form for a subset of the parametric space and under some restrictions on the portfolio setting, for complete markets. Conditions for proper changes of measure and well-defined solutions are provided. These are the first analytical solutions for the 4/2 stochastic volatility model and the embedded 3/2 model for the type of excess returns established in the literature. We numerically illustrate the differences between the 4/2 model and the embedded cases of the 1/2 model (Heston) as well as the 3/2 model under the same data, and for two main cases: risk-averse investor in a complete market with consumption, and ambiguity-averse investor in a complete market with no consumption. In general, the 4/2 and 1/2 models recommend similar levels of consumption and exposure, while the 3/2 leads to significantly different recommendations. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition)
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27 pages, 4683 KiB  
Article
Contagion Patterns Classification in Stock Indices: A Functional Clustering Analysis Using Decision Trees
by Jorge Omar Razo-De-Anda, Luis Lorenzo Romero-Castro and Francisco Venegas-Martínez
Mathematics 2023, 11(13), 2961; https://doi.org/10.3390/math11132961 - 3 Jul 2023
Cited by 2 | Viewed by 1466
Abstract
This paper aims to identify the main determinants of the countries that present contagion during the period 2000–2021, based on the determination of the behavior patterns of 18 stock market indices of 15 of the main economies. To do that, first, the B-spline [...] Read more.
This paper aims to identify the main determinants of the countries that present contagion during the period 2000–2021, based on the determination of the behavior patterns of 18 stock market indices of 15 of the main economies. To do that, first, the B-spline method and Bezier curves are used to smooth observations by minimizing the noise. Subsequently, the Functional Principal Component Analysis (FPCA) methodology is applied. Then, the K-means clustering algorithm is used to determine the main groups using the silhouette method and cross-validation, considering the sum of squares of the distances as the function to minimize. Finally, classification trees and macroeconomic and financial analyses are used to determine the rules of variables that give a direct explanation of the contagion (clustering) between the stock indices. The main empirical results obtained suggest that the most significant macroeconomic variables are the Gross Domestic Product, the Consumer Price Index, and Foreign Direct Investment, while in the financial aspect and the most representative are Domestic Credit and number of companies listed on the stock market. It is worth noticing that government spending does not have a significant effect at any time as a determinant of contagion. Finally, it is important to mention, and surprising, that Mexico’s IPC was not clustered in the same group of US stock market indices anytime, despite the strong commercial relationship and the geographical closeness. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance, 2nd Edition)
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