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Risks
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Stochastic Modelling in Financial Mathematics, 2nd Edition
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Stochastic Modelling in Financial Mathematics, 2nd Edition

A special issue of Risks (ISSN 2227-9091).

Deadline for manuscript submissions: closed (30 November 2025) | Viewed by 13759

Special Issue Editor

Prof. Dr. Anatoliy Swishchuk

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada
Interests: mathematical finance; energy finance; stochastic modelling; risk theory; random evolutions and their applications; modeling high-frequency and algorithmic trading; deep and machine learning in quantitative finance
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Financial mathematics (also known as mathematical finance and quantitative finance) is a field of applied mathematics, concerned with the mathematical and stochastic modelling of financial markets.

French mathematician Louis Bachelier is considered the author of the first scholarly work on mathematical finance, published in 1900. As a discipline, financial mathematics emerged in the 1970s, following the work of Fischer Black, Myron Scholes, and Robert Merton on the option pricing theory.

In financial mathematics, modelling entails the development of sophisticated mathematical and stochastic models, and one may take, for example, the share price as a given and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock. Thus, many problems, such as derivative pricing, portfolio optimization, risk modelling, etc., are generally stochastic in nature, and, hence, such models require complex stochastic analyses.

One contemporary example of such a problem is big data. Big data have now become a driver of model building and analysis in a number of areas, including finance, insurance, and energy markets, to name a few. For example, more than half of the markets in today’s highly competitive financial world now use a limit order book (LOB) mechanism to facilitate trade.

This current Special Issue is exactly devoted to modern trends in financial mathematics associated with stochastic modelling, including modelling of big data. Topics from many areas, such as high-frequency and algorithmic trading (limit order books), energy finance, regime switching, and stochastic volatility modelling, among others, have shown to have deep applicable values, which are useful for both academics and practitioners.

This Special Issue is a continuation of the previous successful Special Issue “Stochastic Modelling in Financial Mathematics".

Prof. Dr. Anatoliy Swishchuk
Guest Editor

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Keywords

  • stochastic modelling
  • mathematical finance
  • regime-switching models in finance
  • energy finance
  • limit order books
  • stochastic volatility modelling

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Published Papers (7 papers)

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Research

45 pages, 10369 KB  
Article
Evaluation and Prediction of Stock Market Crash Risk in Mexico Using Log-Periodic Power-Law Modeling
by Suryansh Sunil, Amit Kumar Goyal, Rajesh Mahadeva and Varun Sarda
Risks 2026, 14(1), 3; https://doi.org/10.3390/risks14010003 - 1 Jan 2026
Viewed by 284
Abstract
This study applies the Log-Periodic Power-Law (LPPL) framework to three major equity markets—Mexico (IPC), Brazil (IBOVESPA), and the United States (NYSE Composite)—using daily closes from 8 November 1991–30 January 2025 for IPC and NYSE, and 3 May 1993–30 January 2025 for IBOVESPA. Multi-window [...] Read more.
This study applies the Log-Periodic Power-Law (LPPL) framework to three major equity markets—Mexico (IPC), Brazil (IBOVESPA), and the United States (NYSE Composite)—using daily closes from 8 November 1991–30 January 2025 for IPC and NYSE, and 3 May 1993–30 January 2025 for IBOVESPA. Multi-window calibrations (Lϵ 180, 240, 300, 360, 420) are estimated in raw and log space to evaluate bubble signatures and the stability of the critical time tc. Across all indices, log-space fits consistently outperform raw fits in terms of RMSE and R2, and longer windows reduce parameter variability, yielding coherent clusters of tc. Under full-sample conditions, the LPPL structure points to March–April 2025 for NYSE, mid-October 2025 for IBOVESPA, and October–December 2025 for IPC, while shorter windows pull tc forward. A rolling early-warning ensemble translates these estimates into lead-based risk bands, with numerical reporting used when median leads fall just outside the 60-trading-day decision horizon. The early-2025 weakening in the U.S. market is consistent with the NYSE cluster, whereas Brazil and Mexico remain within their projected windows as of September 2025. The analysis highlights the strengths of LPPL—behavioral interpretability and hazard-based framing—while noting limitations such as window sensitivity and parameter sloppiness, reinforcing the need for conservative communication and the use of longer-window weighting in practical applications. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics, 2nd Edition)
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21 pages, 591 KB  
Article
From Stochastic Orders to Volatility Surfaces: Revisiting the One-X Property
by Zeyu Cao, Siqiao Zhao and Shaosai Huang
Risks 2025, 13(12), 252; https://doi.org/10.3390/risks13120252 - 15 Dec 2025
Viewed by 283
Abstract
The One-X property, introduced by Zetocha in a 2023 paper, provides a novel stochastic order with direct implications for constructing arbitrage-free implied volatility surfaces. The current work revisits its theoretical foundations and explores its connections with classical stochastic orders, thereby offering a deeper [...] Read more.
The One-X property, introduced by Zetocha in a 2023 paper, provides a novel stochastic order with direct implications for constructing arbitrage-free implied volatility surfaces. The current work revisits its theoretical foundations and explores its connections with classical stochastic orders, thereby offering a deeper understanding of its mathematical structure and practical significance in calendar-arbitrage-free modeling. We first present an explicit counterexample to a conjecture raised in Zetocha’s previous paper, and then provide a natural and valid enhancement of this conjecture. After discussing the inherent relations between the One-X property and properties such as TP2, RP2, and unimodality of density ratio (introcuded by Glasserman and Pirjol in their 2024 papers), we further explore some sufficient conditions to achieve the One-X property for random variables of certain mixture types that are frequently seen in applications. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics, 2nd Edition)
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20 pages, 1449 KB  
Article
Low Financial Risk of Default and Productive Use of Assets Through Hidden Markov Models
by Alexander Haro, Genaro Sandoval, María Rodríguez, Victor Armijo, Ivonne Arana, William Vasquez, Elizabeth Proaño and Amanda Martínez
Risks 2025, 13(12), 230; https://doi.org/10.3390/risks13120230 - 27 Nov 2025
Viewed by 797
Abstract
This paper analyzes solvency dynamics in Ecuador’s mutualist segment by modeling the joint behavior of the productive-assets-to-total-assets ratio (PATR) and portfolio-specific delinquency rates. Using monthly supervisory data from the Superintendencia de Economía Popular y Solidaria (SEPS) for the full universe of four mutualist [...] Read more.
This paper analyzes solvency dynamics in Ecuador’s mutualist segment by modeling the joint behavior of the productive-assets-to-total-assets ratio (PATR) and portfolio-specific delinquency rates. Using monthly supervisory data from the Superintendencia de Economía Popular y Solidaria (SEPS) for the full universe of four mutualist institutions (2022–2025), we estimate a multivariate Gaussian Hidden Markov Model on system-level aggregates. The model identifies latent regimes that summarize configurations of asset productivity and segmented credit risk, distinguishing relatively sound conditions from episodes of heightened vulnerability. Model selection is based on information criteria, complemented by convergence checks, distributional diagnostics, and alternative covariance specifications to assess robustness. The approach is explicitly framed as diagnostic rather than causal or prescriptive: it does not replace simple thresholds nor calibrate capital buffers, but organizes supervisory information into interpretable solvency states with associated frequencies and expected durations. The framework is transparent and reproducible and provides a baseline for future extensions with longer samples and richer covariates. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics, 2nd Edition)
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27 pages, 497 KB  
Article
Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy
by Moritz Sohns
Risks 2025, 13(4), 70; https://doi.org/10.3390/risks13040070 - 1 Apr 2025
Viewed by 2707
Abstract
We introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy σ-martingale measure—a concept inspired by the well-known minimal-entropy martingale measure used in option pricing. While the minimal-entropy martingale measure is commonly used for pricing and [...] Read more.
We introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy σ-martingale measure—a concept inspired by the well-known minimal-entropy martingale measure used in option pricing. While the minimal-entropy martingale measure is commonly used for pricing and hedging, the minimal-entropy σ-martingale measure has not previously been studied, nor has it been analyzed as a traditional risk measure. We address this gap by clearly defining this new risk measure and examining its fundamental properties. In addition, we revisit the entropic risk measure, typically expressed through an exponential formula. We provide an alternative definition using a supremum over Kullback–Leibler divergences, making its connection to entropy clearer. We verify important properties of both risk measures, such as convexity and coherence, and extend these concepts to dynamic situations. We also illustrate their behavior in scenarios involving optimal risk transfer. Our results link entropic concepts with incomplete-market pricing and demonstrate how both risk measures share a unified entropy-based foundation. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics, 2nd Edition)
22 pages, 1674 KB  
Article
Pricing of Averaged Variance, Volatility, Covariance and Correlation Swaps with Semi-Markov Volatilities
by Anatoliy Swishchuk and Sebastian Franco
Risks 2023, 11(9), 162; https://doi.org/10.3390/risks11090162 - 8 Sep 2023
Viewed by 2916
Abstract
In this paper, we consider the problem of pricing variance, volatility, covariance and correlation swaps for financial markets with semi-Markov volatilities. The paper’s motivation derives from the fact that in many financial markets, the inter-arrival times between book events are not independent or [...] Read more.
In this paper, we consider the problem of pricing variance, volatility, covariance and correlation swaps for financial markets with semi-Markov volatilities. The paper’s motivation derives from the fact that in many financial markets, the inter-arrival times between book events are not independent or exponentially distributed but instead have an arbitrary distribution, which means they can be accurately modelled using a semi-Markov process. Through the results of the paper, we hope to answer the following question: Is it possible to calculate averaged swap prices for financial markets with semi-Markov volatilities? This question has not been considered in the existing literature, which makes the paper’s results novel and significant, especially when one considers the increasing popularity of derivative securities such as swaps, futures and options written on the volatility index VIX. Within this paper, we model financial markets featuring semi-Markov volatilities and price-averaged variance, volatility, covariance and correlation swaps for these markets. Formulas used for the numerical evaluation of averaged variance, volatility, covariance and correlation swaps with semi-Markov volatilities are presented as well. The formulas that are detailed within the paper are innovative because they provide a new, simplified method to price averaged swaps, which has not been presented in the existing literature. A numerical example involving the pricing of averaged variance, volatility, covariance and correlation swaps in a market with a two-state semi-Markov process is presented, providing a detailed overview of how the model developed in the paper can be used with real-life data. The novelty of the paper lies in the closed-form formulas provided for the pricing of variance, volatility, covariance and correlation swaps with semi-Markov volatilities, as they can be directly applied by derivative practitioners and others in the financial industry to price variance, volatility, covariance and correlation swaps. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics, 2nd Edition)
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15 pages, 717 KB  
Article
Cox-Based and Elliptical Telegraph Processes and Their Applications
by Anatoliy Pogorui, Anatoly Swishchuk, Ramón M. Rodríguez-Dagnino and Alexander Sarana
Risks 2023, 11(7), 126; https://doi.org/10.3390/risks11070126 - 10 Jul 2023
Cited by 1 | Viewed by 1800
Abstract
This paper studies two new models for a telegraph process: Cox-based and elliptical telegraph processes. The paper deals with the stochastic motion of a particle on a straight line and on an ellipse with random positive velocity and two opposite directions of motion, [...] Read more.
This paper studies two new models for a telegraph process: Cox-based and elliptical telegraph processes. The paper deals with the stochastic motion of a particle on a straight line and on an ellipse with random positive velocity and two opposite directions of motion, which is governed by a telegraph–Cox switching process. A relevant result of our analysis on the straight line is obtaining a linear Volterra integral equation of the first kind for the characteristic function of the probability density function (PDF) of the particle position at a given time. We also generalize Kac’s condition for the telegraph process to the case of a telegraph–Cox switching process. We show some examples of random velocity where the distribution of the coordinate of a particle is expressed explicitly. In addition, we present some novel results related to the switched movement evolution of a particle according to a telegraph–Cox process on an ellipse. Numerical examples and applications are presented for a telegraph–Cox-based process (option pricing formulas) and elliptical telegraph process. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics, 2nd Edition)
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34 pages, 3929 KB  
Article
The SEV-SV Model—Applications in Portfolio Optimization
by Marcos Escobar-Anel and Weili Fan
Risks 2023, 11(2), 30; https://doi.org/10.3390/risks11020030 - 28 Jan 2023
Cited by 4 | Viewed by 3457
Abstract
This paper introduces and studies a new family of diffusion models for stock prices with applications in portfolio optimization. The diffusion model combines (stochastic) elasticity of volatility (EV) and stochastic volatility (SV) to create the SEV-SV model. In particular, we focus on the [...] Read more.
This paper introduces and studies a new family of diffusion models for stock prices with applications in portfolio optimization. The diffusion model combines (stochastic) elasticity of volatility (EV) and stochastic volatility (SV) to create the SEV-SV model. In particular, we focus on the SEV component, which is driven by an Ornstein–Uhlenbeck process via two separate functional choices, while the SV component features the state-of-the-art 4/2 model. We study an investment problem within expected utility theory (EUT) for incomplete markets, producing closed-form representations for the optimal strategy, value function, and optimal wealth process for two different cases of prices of risk on the stock. We find that when EV reverts to a GBM model, the volatility and speed of reversion of the EV have a strong impact on optimal allocations, and more aggressive (bull markets) or cautious (bear markets) strategies are hence recommended. For a model when EV reverts away from GBM, only the mean reverting level of the EV plays a role. Moreover, the presence of SV leads mainly to more conservative investment decisions for short horizons. Overall, the SEV plays a more significant role than SV in the optimal allocation. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics, 2nd Edition)
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