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Mathematics
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Recent Advances on Nonlinear Models in Mathematical Finance
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Recent Advances on Nonlinear Models in Mathematical Finance

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

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 10943

Special Issue Editors

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
Prof. Dr. Maria Do Rosário Grossinho

E-Mail Website
Guest Editor
Department of Mathematics, Lisbon School of Economics and Management, and CEMAPRE-REM, Universidade de Lisboa, Lisbon, Portugal
Interests: nonlinear analysis; mathematical finance; pricing derivative securities
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

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 of 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. Matthias Ehrhardt
Prof. Dr. Maria Do Rosário Grossinho
Prof. Dr. Daniel Sevcovic
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

Published Papers (6 papers)

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Research

31 pages, 445 KiB  
Article
Nonlinear Valuation with XVAs: Two Converging Approaches
by Damiano Brigo, Cristin Buescu, Marco Francischello, Andrea Pallavicini and Marek Rutkowski
Mathematics 2022, 10(5), 791; https://doi.org/10.3390/math10050791 - 02 Mar 2022
Cited by 2 | Viewed by 2072
Abstract
When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while [...] Read more.
When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance)
10 pages, 254 KiB  
Article
New Approximations to Bond Prices in the Cox–Ingersoll–Ross Convergence Model with Dynamic Correlation
by Beáta Stehlíková
Mathematics 2021, 9(13), 1469; https://doi.org/10.3390/math9131469 - 23 Jun 2021
Viewed by 1183
Abstract
We study a particular case of a convergence model of interest rates. The bond prices are given as solutions of a parabolic partial differential equation and we consider different possibilities of approximating them, using approximate analytical solutions. We consider an approximation already suggested [...] Read more.
We study a particular case of a convergence model of interest rates. The bond prices are given as solutions of a parabolic partial differential equation and we consider different possibilities of approximating them, using approximate analytical solutions. We consider an approximation already suggested in the literature and compare it with a newly suggested one for which we derive the order of accuracy. Since the two formulae use different approaches and the resulting leading terms of the error depend on different parameter sets of the model, we propose their combination, which has a higher order of accuracy. Finally, we propose one more approach, which leads to higher accuracy of the resulting approximation formula. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance)
12 pages, 314 KiB  
Article
Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing
by Daniel Ševčovič and Cyril Izuchukwu Udeani
Mathematics 2021, 9(13), 1463; https://doi.org/10.3390/math9131463 - 22 Jun 2021
Cited by 1 | Viewed by 1156
Abstract
The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence [...] Read more.
The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance)
29 pages, 679 KiB  
Article
Residue Sum Formula for Pricing Options under the Variance Gamma Model
by Pedro Febrer and João Guerra
Mathematics 2021, 9(10), 1143; https://doi.org/10.3390/math9101143 - 18 May 2021
Cited by 3 | Viewed by 1799
Abstract
We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties [...] Read more.
We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties of multidimensional complex analysis, with particular emphasis on the multidimensional Jordan Lemma and the application of residue calculus to a Mellin–Barnes integral representation in C3, for the call option price. Moreover, we derive triple sum series formulas for some of the Greeks associated to the call option and we discuss the numerical accuracy and convergence of the main pricing formula. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance)
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8 pages, 352 KiB  
Article
The Heston Model with Time-Dependent Correlation Driven by Isospectral Flows
by Long Teng
Mathematics 2021, 9(9), 934; https://doi.org/10.3390/math9090934 - 22 Apr 2021
Cited by 1 | Viewed by 1775
Abstract
In this work, we extend the Heston stochastic volatility model by including a time-dependent correlation that is driven by isospectral flows instead of a constant correlation, being motivated by the fact that the correlation between, e.g., financial products and financial institutions is hardly [...] Read more.
In this work, we extend the Heston stochastic volatility model by including a time-dependent correlation that is driven by isospectral flows instead of a constant correlation, being motivated by the fact that the correlation between, e.g., financial products and financial institutions is hardly a fixed constant. We apply different numerical methods, including the method for backward stochastic differential equations (BSDEs) for a fast computation of the extended Heston model. An example of calibration to market data illustrates that our extended Heston model can provide a better volatility smile than the Heston model with other considered extensions. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance)
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19 pages, 677 KiB  
Article
Global Optimization for Automatic Model Points Selection in Life Insurance Portfolios
by Ana M. Ferreiro, Enrico Ferri, José A. García and Carlos Vázquez
Mathematics 2021, 9(5), 472; https://doi.org/10.3390/math9050472 - 25 Feb 2021
Viewed by 1442
Abstract
Starting from an original portfolio of life insurance policies, in this article we propose a methodology to select model points portfolios that reproduce the original one, preserving its market risk under a certain measure. In order to achieve this goal, we first define [...] Read more.
Starting from an original portfolio of life insurance policies, in this article we propose a methodology to select model points portfolios that reproduce the original one, preserving its market risk under a certain measure. In order to achieve this goal, we first define an appropriate risk functional that measures the market risk associated to the interest rates evolution. Although other alternative interest rate models could be considered, we have chosen the LIBOR (London Interbank Offered Rate) market model. Once we have selected the proper risk functional, the problem of finding the model points of the replicating portfolio is formulated as a problem of minimizing the distance between the original and the target model points portfolios, under the measure given by the proposed risk functional. In this way, a high-dimensional global optimization problem arises and a suitable hybrid global optimization algorithm is proposed for the efficient solution of this problem. Some examples illustrate the performance of a parallel multi-CPU implementation for the evaluation of the risk functional, as well as the efficiency of the hybrid Basin Hopping optimization algorithm to obtain the model points portfolio. Full article
(This article belongs to the Special Issue Recent Advances on Nonlinear Models in Mathematical Finance)
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