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Stochastic Modelling in Financial Mathematics
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Stochastic Modelling in Financial Mathematics

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

Deadline for manuscript submissions: closed (10 November 2021) | Viewed by 22724

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 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 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, are shown to have deep applicable values which are useful for both academics and practitioners.

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 (6 papers)

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Research

16 pages, 535 KiB  
Article
A Novel Implementation of Siamese Type Neural Networks in Predicting Rare Fluctuations in Financial Time Series
by Treena Basu, Olaf Menzer, Joshua Ward and Indranil SenGupta
Risks 2022, 10(2), 39; https://doi.org/10.3390/risks10020039 - 11 Feb 2022
Cited by 6 | Viewed by 3593
Abstract
Stock trading has tremendous importance not just as a profession but also as an income source for individuals. Many investment account holders use the appreciation of their portfolio (as a combination of stocks or indexes) as income for their retirement years, mostly betting [...] Read more.
Stock trading has tremendous importance not just as a profession but also as an income source for individuals. Many investment account holders use the appreciation of their portfolio (as a combination of stocks or indexes) as income for their retirement years, mostly betting on stocks or indexes with low risk/low volatility. However, every stock-based investment portfolio has an inherent risk to lose money through negative progression and crash. This study presents a novel technique to predict such rare negative events in financial time series (e.g., a drop in the S&P 500 by a certain percent in a designated period of time). We use a time series of approximately seven years (2517 values) of the S&P 500 index stocks with publicly available features: the high, low and close price (HLC). We utilize a Siamese type neural network for pattern recognition in images followed by a bootstrapped image similarity distribution to predict rare events as they pertain to financial market analysis. Extending on literature about rare event classification and stochastic modeling in financial analytics, the proposed method uses a sliding window to store the input features as tabular data (HLC price), creates an image of the time series window, and then uses the feature vector of a pre-trained convolutional neural network (CNN) to leverage pre-event images and predict rare events. This research does not just indicate that our proposed method is capable of distinguishing event images from non-event images, but more importantly, the method is effective even when only limited and strongly imbalanced data is available. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)
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21 pages, 1309 KiB  
Article
Transformations of Telegraph Processes and Their Financial Applications
by Anatoliy A. Pogorui, Anatoliy Swishchuk and Ramón M. Rodríguez-Dagnino
Risks 2021, 9(8), 147; https://doi.org/10.3390/risks9080147 - 17 Aug 2021
Cited by 4 | Viewed by 2515
Abstract
In this paper, we consider non-linear transformations of classical telegraph process. The main results consist of deriving a general partial differential Equation (PDE) for the probability density (pdf) of the transformed telegraph process, and then presenting the limiting PDE under Kac’s conditions, which [...] Read more.
In this paper, we consider non-linear transformations of classical telegraph process. The main results consist of deriving a general partial differential Equation (PDE) for the probability density (pdf) of the transformed telegraph process, and then presenting the limiting PDE under Kac’s conditions, which may be interpreted as the equation for a diffusion process on a circle. This general case includes, for example, classical cases, such as limiting diffusion and geometric Brownian motion under some specifications of non-linear transformations (i.e., linear, exponential, etc.). We also give three applications of non-linear transformed telegraph process in finance: (1) application of classical telegraph process in the case of balance, (2) application of classical telegraph process in the case of dis-balance, and (3) application of asymmetric telegraph process. For these three cases, we present European call and put option prices. The novelty of the paper consists of new results for non-linear transformed classical telegraph process, new models for stock prices based on transformed telegraph process, and new applications of these models to option pricing. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)
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13 pages, 397 KiB  
Article
Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models
by Anatoliy Swishchuk
Risks 2021, 9(6), 108; https://doi.org/10.3390/risks9060108 - 3 Jun 2021
Cited by 4 | Viewed by 3239
Abstract
We show how to solve Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance (Propositions 1 and 2), i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound [...] Read more.
We show how to solve Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance (Propositions 1 and 2), i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound Hawkes process (GCHP), and for a capital R(t) (risk process) of an insurance company with the amount of claims described by the risk model based on GCHP. The main approach in both cases is to use functional central limit theorem for the GCHP to approximate it with a diffusion process. Then we construct and solve Hamilton–Jacobi–Bellman (HJB) equation for the expected utility function. The novelty of the results consists of the new Hawkes-based models and in the new optimal investment results in finance and insurance for those models. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)
20 pages, 693 KiB  
Article
Multivariate General Compound Point Processes in Limit Order Books
by Qi Guo, Bruno Remillard and Anatoliy Swishchuk
Risks 2020, 8(3), 98; https://doi.org/10.3390/risks8030098 - 11 Sep 2020
Cited by 3 | Viewed by 3357
Abstract
In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the [...] Read more.
In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. The law of large numbers (LLN) and two functional central limit theorems (FCLTs) for the MGCPP were proved in this work. Applications of the MGCPP in the limit order market were also considered. We provided numerical simulations and comparisons for the MGCPP and MGCHP by applying Google, Apple, Microsoft, Amazon, and Intel trading data. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)
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25 pages, 1782 KiB  
Article
General Compound Hawkes Processes in Limit Order Books
by Anatoliy Swishchuk and Aiden Huffman
Risks 2020, 8(1), 28; https://doi.org/10.3390/risks8010028 - 14 Mar 2020
Cited by 10 | Viewed by 5505
Abstract
In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regard to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional [...] Read more.
In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regard to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for several specific variations. We apply several of these FCLTs to limit order books to study the link between price volatility and order flow, where the volatility in mid-price changes is expressed in terms of parameters describing the arrival rates and mid-price process. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)
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29 pages, 507 KiB  
Article
General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory
by Yuliya Mishura, Kostiantyn Ralchenko and Sergiy Shklyar
Risks 2020, 8(1), 11; https://doi.org/10.3390/risks8010011 - 30 Jan 2020
Cited by 4 | Viewed by 2518
Abstract
We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D [ 0 , T ] . Then we investigate the convergence of the related multiplicative scheme to a process that [...] Read more.
We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D [ 0 , T ] . Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)
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