Mathematical Finance - Science topic

The Dupire formula plays a significant role in pricing financial derivatives. This paper is devoted to deriving a generalised version of the Dupire formula for Margrabe options and its mathematically rigorous proof. Financial derivatives of this type are not special cases of plain vanilla options. Moreover, application of Lévy-type stochastic integ...

In mathematical finance, many derivatives from markets with frictions can be formulated as optimal control problems in the HJB framework. Analytical optimal control can result in highly nonlinear PDEs, which might yield unstable numerical results. Accurate and convergent numerical schemes are essential to leverage the benefits of the hedging proces...

Although financial models violate ergodicity in general, observing the ergodic behavior in the markets is not rare. Policymakers and market participants control the market behavior in critical and emergency states, which leads to some degree of ergodicity as their actions are intentional. In this paper, we define a parametric operator that acts on...

This paper investigates the well-posedness of singular mean-field backward stochastic Volterra integral equations (MF-BSVIEs) in infinite-dimensional spaces. We consider the equation: \[ X(t) = \Psi(t) + \int_t^b F\big(t, s, X(s), Z(t, s), Z(s, t), \mathbb{E}[X(s)], \mathbb{E}[Z(t, s)], \mathbb{E}[Z(s, t)]\big) ds - \int_t^b Z(t, s) dB_s, \] where...

We consider the development of unbiased estimators, to approximate the stationary distribution of Mckean-Vlasov stochastic differential equations (MVSDEs). These are an important class of processes, which frequently appear in applications such as mathematical finance, biology and opinion dynamics. Typically the stationary distribution is unknown an...

Market turnover levels and liquidity changes across various territories significantly influence currency prices, leading to continuous fluctuations. Consequently, traders and investors constantly seek strategies to mitigate exchange rate risks. This study aimed to measure and assess foreign exchange risk utilizing Neural Networks and ARMA-GARCH mod...

Forward-backwards stochastic differential equations (FBSDEs) are central in optimal control, game theory, economics, and mathematical finance. Unfortunately, the available FBSDE solvers operate on individual FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs as these solvers must be rer...

We study the strong existence and uniqueness of solutions within a Weyl chamber for a class of time-dependent particle systems driven by multiplicative noise. This class includes well-known processes in physics and mathematical finance. We propose a method to prove the existence of negative moments for the solutions. This result allows us to analyz...

This novel work is the first study in India to incorporate the Human capital (HC) factor as a six-factor asset-pricing model and presents a robust methodology. The aim of this work is to examine the ability of the six-factor model to capture excess returns using a GMM framework with time periods that were missing in previous studies. Therefore, dat...

We investigate the feasibility of integrating quantum algorithms as subroutines of simulation-based optimisation problems with relevance to and potential applications in mathematical finance. To this end, we conduct a thorough analysis of all systematic errors arising in the formulation of quantum Monte Carlo integration in order to better understa...

This research examines the co-movement between exchange rates and equity prices in a selection of frontier African markets (Ghana, Mauritius, and Tunisia). The analysis encompasses data from 4 January 2010 to 31 March 2023. Employing advanced econometric techniques, the study investigates the interconnectedness of frontier markets and the direction...

We study the causal distributionally robust optimization (DRO) in both discrete- and continuous- time settings. The framework captures model uncertainty, with potential models penalized in function of their adapted Wasserstein distance to a given reference model. Strength of the penalty is controlled using a real-valued parameter, which is simply t...

We derive a Clark-Ocone-Haussmann (COH) type formula under a change of measure for $$ L^1 $$ L 1 -canonical additive processes, providing a tool for representing financial derivatives under a risk-neutral probability measure. COH formulas are fundamental in stochastic analysis, providing explicit martingale representations of random variables in te...

This article introduces a novel distributionally robust model predictive control (DRMPC) algorithm for a specific class of controlled dynamical systems where the disturbance multiplies the state and control variables. These classes of systems arise in mathematical finance, where the paradigm of distributionally robust optimization (DRO) fits perfec...

Incorporating (near) real-time detection data into active sonar risk assessments has been difficult due to challenges in collecting the data and fusing multiple sensing modalities. A mixture model is developed in this work which splits marine mammals in the area into known and unknown mammals according to the detections made by sensors. The spatial...

The application of data science in the financial field has become an important trend in the financial industry. This article reviews the current application status of data science in the financial field, including application scenarios in risk management, credit assessment, market forecasting, etc. Through the analysis of existing research, this ar...

We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology, and other fields. We prove a general result on the existence of a family...

This study presents an efficient method using the local radial basis function finite difference scheme (RBF-FD). The innovative coefficients are derived from the integrals of the multiquadric (MQ) function. Theoretical convergence rates for the coefficients used in function derivative approximation are provided. The proposed scheme utilizes RBF-FD...

The present article aims to design and analyze efficient first-order strong schemes for a generalized Aït-Sahalia type model arising in mathematical finance and evolving in a positive domain (0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsf...

Fuzzy random variables were studied by Puri and Ralescu [1] which was used by Yoshida [2] in the field of Mathematical Finance. Since then, several researchers (to cite a few, [3], [4], [5], [6], [7], [8], [9], [10]) use fuzzy set theory application to Mathematical Finance. Later, he ([2], [10], [11]) applied fuzzy binomial tree model based on Cox...

This paper provides a Feller's test for explosions of one-dimensional continuous stochastic Volterra processes of convolution type. The study focuses on dynamics governed by nonsingular kernels, which preserve the semimartingale property of the processes and introduce memory features through a path-dependent drift. In contrast to the classical path...

Sessions Submission of Session Closed The submission of sessions for the conference closed on April 30, 2024. We have received a significant number of submissions and are pleased to present the following 21 sessions along with one open session for contributed talks. 1. Advances in Nonlinear Analysis 2. Analysis and Mathematical Finance 3. Applicat...

We introduce and study geometric Bass martingales. Bass martingales were introduced in \cite{Ba83} and studied recently in a series of works, including \cite{BaBeHuKa20,BaBeScTs23}, where they appear as solutions to the martingale version of the Benamou-Brenier optimal transport formulation. These arithmetic, as well as our novel geometric, Bass ma...

This paper proposes to parameterize open loop controls in stochastic optimal control problems via suitable classes of functionals depending on the driver's path signature, a concept adopted from rough path integration theory. We rigorously prove that these controls are dense in the class of progressively measurable controls and use rough path metho...

The Standard Quadratic optimization Problem (StQP), arguably the simplest among all classes of NP-hard optimization problems, consists of extremizing a quadratic form (the simplest nonlinear polynomial) over the standard simplex (the simplest polytope/compact feasible set). As a problem class, StQPs may be nonconvex with an exponential number of in...

Predicting the conditional evolution of Volterra processes with stochastic volatility is a crucial challenge in mathematical finance. While deep neural network models offer promise in approximating the conditional law of such processes, their effectiveness is hindered by the curse of dimensionality caused by the infinite dimensionality and non-smoo...

The main aim of this article is to demonstrate the collocation method based on the barycentric rational interpolation function to solve nonlinear stochastic differential equations driven by fractional Brownian motion. First of all, the corresponding integral form of the nonlinear stochastic differential equations driven by fractional Brownian motio...

This explanatory article provides an introductory overview to Geometric Arbitrage Theory, a reformulation of mathematical finance in terms of stochastic differential geometry, allowing for the formalization of arbitrage measure and minimal arbitrage probability.

Keywords: Factor-Based Modeling Multi-level Multivariate Clustering Cluster Volatility Fractal Structure of the Market(FSM) ABSTRACT Cluster fluctuations and fractal structures are important features of space-time correlation in complex financial systems. However, the microscopic mechanism of creation and expansion of these two features in financia...

In today's world, financial markets, as the main arteries of any country's economic system, have created an attractive environment for investors, and therefore it is necessary to identify the behavior of investors in this space and variables affecting prices and stock returns in these markets. The purpose of this study is to use the method of rando...

La matemática financiera, también conocidas como finanzas cuantitativas, son un campo de las matemáticas aplicadas que se ocupa de la modelización matemática de los mercados financieros. En general, existen dos ramas separadas de las finanzas que requieren técnicas cuantitativas avanzadas: la fijación de precios derivados por un lado y la gestión d...

The authors of Kaur and Natesan 2023 [A novel numerical scheme for time-fractional Black-Scholes PDE governing European options in mathematical finance, (Numerical Algorithms, 94, (2023) 1519–1549)] proposed a numerical scheme, which is based on a combination of L1 scheme for time discretization and spine method for spatial discretization, for solv...

The research community’s treatise on computational economics and financial models has promising interest for the exploration and exploitation of artificial intelligence (AI)-based computing paradigm to offer enriched efficacies for business stratagems, consumer utility, and scarce resource management for enriched society evolution. In this study, A...

The present work is devoted to strong approximations of a generalized Aït-Sahalia model arising from mathematical finance. The numerical study of the considered model faces essential difficulties caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. In this paper, a n...

Investment serves as a means to cultivate assets and secure long-term profits. However, inadequate investment planning poses a substantial risk of significant losses. Consequently, precise analysis becomes imperative for making intelligent investment decisions. One effective analytical approach involves the application of mathematical concepts in i...

The present article aims to design and analyze efficient first-order strong schemes for a generalized A ̈ıt-Sahalia type model arising in mathematical finance and evolving in a positive domain (0, ∞), which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of th...

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find applicati...

Studies on the tripartite nexuses among information and communication technology (ICT), financial development (FD), and banking sector efficiency have largely produced mixed findings. More importantly, how countries’ levels of ICT advancement moderate FD-banks efficiency interlinkages need to be re-assessed. By utilizing data from 48 African countr...

This paper examines the stochastic maximum principle (SMP) for a forward-backward stochastic control system where the backward state equation is characterized by the backward stochastic differential equation (BSDE) with quadratic growth and the forward state at the terminal time is constrained in a convex set with probability one. With the help of...

We consider one-dimensional stochastic Volterra equations with jumps for which we establish conditions upon the convolution kernel and coefficients for the strong existence and path-wise uniqueness of a non-negative càdlàg solution. By using the approach recently developed by arXiv:2302.07758, we show the strong existence by using a nonnegative app...

The accounting software is considered to be of the most critical components of accounting information system, with particular significance as of accounting and financial systems. the most important problems with accounting education systems is that students do not adequately learn the financial software required by the accounting profession, which,...

In this article we consider Bayesian parameter inference for a type of partially observed stochastic Volterra equation (SVE). SVEs are found in many areas such as physics and mathematical finance. In the latter field they can be used to represent long memory in unobserved volatility processes. In many cases of practical interest, SVEs must be time-...

The relationship between expectation and price is commonly established with two principles: no-arbitrage, which asserts that both maps are positive; and equivalence, which asserts that the maps share the same null events. Constructed from the Arrow-Debreu securities, classical and quantum models of economics are then distinguished by their respecti...

We show in a simulation when economic agents are subject to evolution (random change and selection based on the success in the estimation of the result of the gamble) they acquire risk aversive behavior. This behavior appears in the form of adjustment of their estimation of probabilities when calculating the expected value (ensemble average). It me...

A linear map Φ between matrix spaces is called cross-positive if it is positive on orthogonal pairs (U, V) of positive semidefinite matrices in the sense that U, V := tr(U V) = 0 implies Φ(U), V ≥ 0, and is completely cross-positive if all its ampliations I n ⊗Φ are cross-positive. (Completely) cross-positive maps arise in the theory of operator se...

In this paper, we study the optimization problem of an economic agent who chooses the best time for retirement as well as consumption and investment in the presence of a mandatory retirement date. Moreover, the agent faces the borrowing constraint which is constrained in the ability to borrow against future income during working. By utilizing the d...

The classical bond-pricing models, as important financial tools, show strong vitality in bond pricing. However, these models also expose their theoretical defects, which leads to inconsistencies with the actual observation results and usually causes the theoretical prices of bonds to be lower than the actual market prices in the financial market. I...

This paper presents a novel integration of Machine Learning (ML) models with Monte Carlo simulations to enhance financial forecasting and risk assessments in dynamic market environments. Traditional financial forecasting methods, which primarily rely on linear statistical and econometric models, face limitations in addressing the complexities of mo...

This work delves into the intricacies of financial partial differential equations (PDEs), emphasizing their pivotal role in modeling financial derivatives' behaviors. Through detailed exploration of seminal models like Black-Scholes, CIR, and Merton, it unveils the multidimensional realm of financial mathematics, from stock option valuation to inte...

This research delves into the multifaceted applications of transformation semigroups, leveraging insights from algebraic cryptography, group theory, blockchain technology, and computational mathematics. Through a comprehensive exploration, we unveil novel cryptographic protocols, enhance blockchain consensus algorithms, develop efficient computatio...

Hamza and Klebaner (2007) [10] posed the problem of constructing martingales with one-dimensional Brownian marginals that differ from Brownian motion, so-called fake Brownian motions. Besides its theoretical appeal, this problem represents the quintessential version of the ubiquitous fitting problem in mathematical finance where the task is to cons...

Financial literacy refers to skills and knowledge needed to make responsible financial decisions. It, therefore, consists of a set of components, including basic knowledge, money management skills, understanding of financial products, risk awareness, planning and self-control. It can be broken down into three main components: financial knowledge, f...

This work delves into the intricacies of financial partial differential equations (PDEs), emphasizing their pivotal role in modeling financial derivatives' behaviors. Through detailed exploration of seminal models like Black-Scholes, CIR, and Merton, it unveils the multidimensional realm of financial mathematics, from stock option valuation to inte...

In this paper, by extending the classic stochastic integrals, we investigate three kinds of more general stochastic integrals: Lebesgue-Stieltjes integrals on predictable sets of interval type (in short: PSITs), stochastic integrals on PSITs of predictable processes with respect to local martingales, and stochastic integrals on PSITs of predictable...

Deep Kalman filters (DKFs) are a class of neural network models that generate Gaussian probability measures from sequential data. Though DKFs are inspired by the Kalman filter, they lack concrete theoretical ties to the stochastic filtering problem, thus limiting their applicability to areas where traditional model-based filters have been used, e.g...

For probability measures μ , ν \mu ,\nu , and ρ \rho , define the cost functionals C ( μ , ρ ) ≔ sup π ∈ Π ( μ , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) and C ( ν , ρ ) ≔ sup π ∈ Π ( ν , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) , C\left(\mu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\mu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y)\hspace{1...

Consider the following two-person mixed strategy game of a probabilist against Nature with respect to the parameters $(f, \mathcal{B},\pi)$, where $f$ is a convex function satisfying certain regularity conditions, $\mathcal{B}$ is either the set $\{L_i\}_{i=1}^n$ or its convex hull with each $L_i$ being a Markov infinitesimal generator on a finite...

In this article we consider Bayesian parameter inference for a type of partially observed stochastic Volterra equation (SVE). SVEs are found in many areas such as physics and mathematical finance. In the latter field they can be used to represent long memory in unobserved volatility processes. In many cases of practical interest SVEs must be time-d...

In this paper, we attempt to provide an elementary introduction to the excursion risk theory by constructing a discrete-time analogue. The excursion risk theory is a theory of calculating risks in investments that use mean-reverting trading signals. We consider the case where the trading signal is a simple symmetric random walk (RW) and use the exc...

This article celebrates the legacy of Tomas Björk. As we reflect on Tomas’ life and career, we explore his personal and professional journey, highlighting his most significant contributions to mathematical finance.

Expected shortfall (ES, also known as CVaR) is the most important coherent risk measure in finance, insurance, risk management, and engineering. Recently, Wang and Zitikis (2021) put forward four economic axioms for portfolio risk assessment and provide the first economic axiomatic foundation for the family of ES. In particular , the axiom of no re...

In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to...

Article: DEEP EMPIRICAL RISK MINIMIZATION IN FINANCE: LOOKING INTO THE FUTURE Authors: Anders Max Reppen and Halil Mete Soner Journal: Mathematical Finance (2023), Vol. 33 Issue 1 Field: Mathematical Finance and Deep Learning. Date: 3rd August 2023. Courtesy of American Mathematical Society - MathSciNet - Mathematical Reviews

This research article addresses the close relationship between mathematics and finance, highlighting its importance in understanding and applying financial concepts. It was recognized that financial knowledge is crucial in daily life and the business world. However, the need for a renewal in the teaching of financial mathematics has been raised due...

ChatGPT, an implementation and application of large language models, has gained significant popularity since its initial release. Researchers have been exploring ways to harness the practical benefits of ChatGPT in real-world scenarios. Educational researchers have investigated its potential in various subjects, e.g., programming, mathematics, fina...

This paper is concerned with the quadratic volatility family of driftless stochastic differential equations (SDEs), also known in the literature as Quadratic Normal Volatility models (QNV), which have found applications primarily in mathematical finance, but can also model dynamics of stochastic processes in other fields such as mathematical biolog...

Deep hedging is a deep-learning-based framework for derivative hedging in incomplete markets. The advantage of deep hedging lies in its ability to handle various realistic market conditions, such as market frictions, which are challenging to address within the traditional mathematical finance framework. Since deep hedging relies on market simulatio...

The Skorokhod embedding problem (SEP) is to represent a given probability measure as a Brownian motion $B$ at a particular stopping time. In recent years particular attention has gone to solutions which exhibit additional optimality properties due to applications to martingale inequalities and robust pricing in mathematical finance. Among these sol...

Article: ON THE CONVERGENCE SCHEME IN THE CRR MODEL Author: Tomasz Kostrzewa Journal: Applicationes Mathematicae 49 (2022), no. 2 Field: Mathematical Finance. Date: 30th June 2023. Courtesy of American Mathematical Society - MathSciNet - Mathematical Reviews

Banks play a crucial role in the growth and development of an economy. A profitable banking system contributes to economic stability and efficiency, helping to mitigate the impacts of sudden macroeconomic shocks. In order to improve efficiency and profitability, banks need to identify the factors that influence their performance. Accrued liabilitie...

Computational models, a.k.a. simulators, are used in all fields of engineering and applied sciences to help design and assess complex systems. Advanced analyses such as optimization or uncertainty quantification, which require repeated runs by varying input parameters, cannot be carried out with brute force methods such as Monte Carlo simulation du...

We investigate analytical solvability of models with affine stochastic volatility (SV) and Lévy jumps by deriving a unified formula for the conditional moment generating function of the log-asset price and providing the condition under which this new formula is explicit. The results lay a foundation for a range of valuation, calibration, and econom...

The aim of this short note is to present a solution to the discrete time exponential utility maximization problem in a case where the underlying asset has a multivariate normal distribution. In addition to the usual setting considered in Mathematical Finance, we also consider an investor who is informed about the risky asset's price changes with a...

In the financial market, the change in price of underlying stock following fractal transmission system is modeled by time-fractional Black-Scholes partial differential equations (PDEs). In this paper, we propose a numerical scheme on uniform mesh for solving time-fractional Black-Scholes PDEs governing European options. The fractional time derivati...

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump L\'evy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find applica...

We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of Xt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \beg...

Digital transformation has pass through aspects of social livelihood and manufacture services, and digital finance has become a new motor for progress with quality as the main goal of China’s financial condition. Here first elaborates on the connotation of progress with quality as the main goal, selects 20 indicators from five aspects of progress w...

We introduce a numerical methodology, referred to as the transport-based mesh-free method, which allows us to deal with continuous, discrete, or statistical models in the same unified framework, and leads us to a broad class of numerical algorithms recently implemented in a Python library (namely, CodPy). Specifically, we propose a mesh-free discre...

We introduce a numerical methodology, referred to as the transport-based mesh-free method, which allows us to deal with continuous, discrete, or statistical models in the same unified framework, and leads us to a broad class of numerical algorithms recently implemented in a Python library (namely, CodPy). Specifically, we propose a mesh-free discre...

In this paper, we solve an optimal reinsurance problem in the mathematical finance area. We assume that the surplus process of the insurance company follows a controlled diffusion process and the constant interest rate is involved in the financial model. During the whole optimization period, the company has a choice to buy reinsurance contract and...

This work is concerned with the computational solution of the time-dependent 3D parabolic Heston–Cox–Ingersoll–Ross (HCIR) PDE, which is of practical importance in mathematical finance. The HCIR dynamic states that the model follows randomness for the underlying asset, the volatility and the rate of interest. Since the PDE formulation has degenerac...

Distribution Regression on path-space refers to the task of learning functions mapping the law of a stochastic process to a scalar target. The learning procedure based on the notion of path-signature, i.e. a classical transform from rough path theory, was widely used to approximate weakly continuous functionals, such as the pricing functionals of p...