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Mathematics
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Stochastic Analysis and Applications in Financial Mathematics
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Stochastic Analysis and Applications in Financial Mathematics

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

Deadline for manuscript submissions: closed (28 February 2025) | Viewed by 6517

Special Issue Editors

Dr. Gerald H. L. Cheang

E-Mail Website
Guest Editor
Centre for Industrial and Applied Mathematics, UniSA STEM, University of South Australia, Adelaide 5001, Australia
Interests: stochastic processes; mathematical finance; non-parametric estimation; approximation theory
Dr. Len Patrick Garces

E-Mail Website
Guest Editor
School of Mathematical and Physical Sciences, University of Technology Sydney, Sydney 2007, Australia
Interests: mortality and longevity risk modelling; actuarial mathematics; financial mathematics; option pricing; stochastic calculus

Special Issue Information

Dear Colleagues,

The origin of the application of rigorous mathematical and stochastic methods for asset pricing can be traced back to Louis Bachelier’s 1900 doctoral thesis Théorie de la spéculation. From the early 1950s, economists, including Paul Samuelson, started to model asset prices using geometric Brownian motion. The Black and Scholes paper in 1973 is the first to show that the European option price is the solution of a partial differential equation that is derived from a self-financing hedging argument, which is also followed by some extensions by Merton in 1973. Merton, in 1976, further extended the Samuelson–Black–Scholes geometric Brownian motion asset pricing model to a geometric jump-diffusion model and derived a pricing formula for a European call option under a jump-diffusion model. Margrabe, in 1978, extended the Black–Scholes call/put option pricing formula under geometric Brownian motion dynamics to price a European-style exchange option. This was an early example of pricing an option written on multiple assets, which was extended by Carmona and Durrleman, in 2003, to more general spread options. The work by Harrison and Pliska, in 1981, also formalised the relationship between risk-neutral valuation and equivalent martingale measures, which also led to the Fundamental Theorems of Asset Pricing by Delbaen and Schachermayer in 1994. The original Black–Scholes 1973 model was also extended to include stochastic volatility (e.g., Hull and White in 1987 and Heston in 1993), both stochastic volatility and jumps (e.g., Bates in 1996). Other option pricing models that have been introduced include the Variance–Gamma model by Madan and Seneta in 1990 and Hidden Markov models by Elliott et al. in 1995. Apart from option pricing models, there is also a wealth of literature on interest rate term structure models and option pricing with stochastic interest rates.

In addition to stochastic analysis in asset pricing, option and derivatives pricing, and interest rate modelling, techniques around stochastic optimal control, forward–backward stochastic differential equations (FBSDEs), and stochastic filtering, among others, have gained greater traction in addressing financial problems, such as portfolio management and optimization, risk management and measurement, algorithmic trading and trading strategies, among many other areas of application.

In this Special Issue, we call for original papers that further extend the frontiers of the existing rich literature, as well as shorter insightful review and survey papers on the applications of stochastic analysis in financial mathematics.

Dr. Gerald H. L. Cheang
Dr. Len Patrick Garces
Guest Editors

Manuscript Submission Information

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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

  • financial mathematics
  • stochastic analysis
  • stochastic optimal control
  • asset pricing

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Published Papers (4 papers)

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Research

20 pages, 1481 KiB  
Article
Analytical Pricing of Commodity Futures with Correlated Jumps and Seasonal Effects: An Empirical Study of Thailand’s Natural Rubber Market
by Athinan Sutchada, Sanae Rujivan and Boualem Djehiche
Mathematics 2025, 13(5), 770; https://doi.org/10.3390/math13050770 - 26 Feb 2025
Viewed by 496
Abstract
This paper presents a novel multivariate mean-reverting jump-diffusion model that incorporates correlated jumps and seasonal effects to capture the complex dynamics of commodity prices. The model also accounts for the interplay between price volatility and convenience yield, offering a comprehensive framework for commodity [...] Read more.
This paper presents a novel multivariate mean-reverting jump-diffusion model that incorporates correlated jumps and seasonal effects to capture the complex dynamics of commodity prices. The model also accounts for the interplay between price volatility and convenience yield, offering a comprehensive framework for commodity futures pricing. By leveraging the Feynman–Kac theorem, we derive a partial integro-differential equation for the conditional moment generating function of the log price, enabling an analytical solution for pricing commodity futures. This solution is validated against Monte Carlo simulations, demonstrating high accuracy and computational efficiency. The model is empirically applied to historical futures prices of natural rubber from the Thailand Futures Exchange. Key parameters—including commodity price dynamics, convenience yields, and seasonal factors—are estimated, revealing the critical role of jumps and seasonality in influencing market behavior. Notably, our findings show that convenience yields are negative, reflecting higher inventory costs, and tend to increase with rising spot prices. These results provide actionable insights for traders, risk managers, and policymakers in commodity markets, emphasizing the importance of correlated jumps and seasonal patterns in pricing and risk assessment. Full article
(This article belongs to the Special Issue Stochastic Analysis and Applications in Financial Mathematics)
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19 pages, 311 KiB  
Article
Pricing a Defaultable Zero-Coupon Bond under Imperfect Information and Regime Switching
by Ashwaq Ali Zarban, David Colwell and Donna Mary Salopek
Mathematics 2024, 12(17), 2740; https://doi.org/10.3390/math12172740 - 2 Sep 2024
Cited by 2 | Viewed by 1398
Abstract
We propose a pricing formula for a defaultable zero-coupon bond with imperfect information under a regime switching model using a structural form of credit risk modelling. This paper provides explicit representations of risky debt under regime switching with a constant interest rate and [...] Read more.
We propose a pricing formula for a defaultable zero-coupon bond with imperfect information under a regime switching model using a structural form of credit risk modelling. This paper provides explicit representations of risky debt under regime switching with a constant interest rate and risky debt under regime switching with a regime switching interest rate. While the value of the firm’s equity is observed continuously, we assume that the total value of the firm is only observed at discrete times, such as the dates of the release of the firm’s annual reports, or quarterly reports. This uncertainty about the true value of the firm results in credit spreads that do not approach zero as the debt approaches maturity, which is a problem with many structural models. The firm’s value is typically decomposed into its equity and debt; however, we consider the asset–to–equity ratio, an accounting ratio used to examine a firm’s financial well-being. The parameters in our model are regime switching, where the regime can be thought of as the state of the economy. A Markov chain with a constant transition rate matrix produces the regime switching. Full article
(This article belongs to the Special Issue Stochastic Analysis and Applications in Financial Mathematics)
11 pages, 285 KiB  
Article
Valuation of Commodity-Linked Bond with Stochastic Convenience Yield, Stochastic Volatility, and Credit Risk in an Intensity-Based Model
by Junkee Jeon and Geonwoo Kim
Mathematics 2023, 11(24), 4969; https://doi.org/10.3390/math11244969 - 15 Dec 2023
Cited by 2 | Viewed by 1042
Abstract
In this study, we consider an intensity-based model for pricing a commodity-linked bond with credit risk. Recently, the pricing of a commodity-linked bond with credit risk under the structural model has been studied. We extend the result using an intensity-based model, stochastic volatility [...] Read more.
In this study, we consider an intensity-based model for pricing a commodity-linked bond with credit risk. Recently, the pricing of a commodity-linked bond with credit risk under the structural model has been studied. We extend the result using an intensity-based model, stochastic volatility model, and stochastic convenience yield model. In the intensity-based model, the credit event by the counterparty occurs at the time of first jump in a stochastic Poisson process, in which intensity is modeled as the sum of two CIR prosesses. We assume that the underlying asset follows the stochastic volatility and convenience yield models. Using the measure change technique, we explicitly derive the commodity-linked bond pricing formula in the proposed model. As a result, we provide the explicit solution for the price of the commodity-linked bond with stochastic convenience yield, stochastic volatility, and credit risk as single integrations. In addition, we present several examples to demonstrate the effects of significant parameters on the value of commodity-linked bond using numerical integration. In particular, examples are provided, focusing on the behavior of prices based on effects of recovery rate. Full article
(This article belongs to the Special Issue Stochastic Analysis and Applications in Financial Mathematics)
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20 pages, 331 KiB  
Article
A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes
by Emel Savku
Mathematics 2023, 11(14), 3043; https://doi.org/10.3390/math11143043 - 9 Jul 2023
Cited by 12 | Viewed by 2137
Abstract
We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems [...] Read more.
We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different types of constraints, which lead us to find real-valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with the stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type called Bancassurance. Full article
(This article belongs to the Special Issue Stochastic Analysis and Applications in Financial Mathematics)
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