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HUMAN-FINANCE INTERACTION: A MULTIDISCIPLINARY APPROACH
TO FINANCE PROBLEM SOLVING IN THE ERA OF CHATGPT, AI, AND
BLOCKCHAIN
RIZWAN JAHANGIR AND DAISUKE ISHII
ABS TR ACT.
This paper investigates the interplay between mathematical theories and
human behavior in financial markets, focusing on group theory, game theory, and partial
differential equations (PDE) in the context of human finance interaction (HFI). It integrates
mathematical models with behavioral insights to address the limitations of traditional
financial models that often overlook emotional and irrational decision-making. Group
theory plays a crucial role in our approach, particularly in identifying and assessing
risks, managing portfolios, and conducting thorough scenario analyses. To model the
equations derived from these observations, we employ Partial Differential Equations
(PDEs), which provide a robust mathematical framework. Additionally, game theory
significantly enhances our understanding of the intricate relationships and dynamics
inherent in financial interactions. This integrated approach, combining group theory, PDEs,
and game theory, allows for a more comprehensive and nuanced understanding of the
financial landscape.
1. Introduction
Financial markets’ complex dynamics have been studied for years, focusing on quantitative
and mathematical financial models to predict market trends, optimize investment strategies,
and assess risks. However, these models often overlook human behavior, a crucial element
in the financial world. This paper examines group theory, game theory, and partial
differential equations (PDE) in financial contexts, with a special focus on Human Finance
Interaction (HFI).
Group theory has been instrumental in enhancing our understanding of various aspects
of human financial interactions. Specifically, it has aided in identifying and managing
risks, optimizing investment portfolios, and conducting detailed scenario analyses. By
leveraging the principles of group theory, we have been able to systematically explore and
model complex market dynamics, which are crucial for effective risk assessment. This has
significantly improved our ability to prepare for diverse market scenarios, ensuring more
robust and resilient financial strategies.
In our case, we observed that permutations, a key concept in group theory, provided
valuable insights, especially in the realm of portfolio management and market analysis.
Through permutations, we could analyze different combinations and sequences of assets,
allowing for the creation of optimized investment portfolios. This approach not only
maximized returns but also minimized risk by considering various potential market
conditions. Additionally, in algorithmic trading and financial modeling, permutations
enabled us to quickly navigate through vast data sets, uncovering effective trading patterns
Key words and phrases. Finance, AI, Emotional decisions, Mathematics for finance, Automatons.
1
2 RIZWAN JAHANGIR AND DAISUKE ISHII
and tailoring financial products to specific market needs. In the Figure 1 we illustrate
the complex interactions between Market Conditions (MC), Financial Assets (FA), and
Financial Outcomes (FO) within the realm of financial decision-making. Each cluster
of nodes, distinctly color-coded, represents a different category: red squares for Market
Conditions, blue circles for Financial Assets, and green triangles for Financial Outcomes.
The dotted lines connecting these nodes symbolize potential permutations, demonstrating
how variations in market conditions and asset combinations can lead to diverse financial
outcomes. This dynamic representation serves to underscore the multifaceted nature of
financial permutations and their pivotal role in shaping investment strategies and results.
FIGURE 1. Visualization of permutations in financial decision-making.
The purpose of this paper is to demonstrate how group theory, game theory, and PDE, when
combined with insights into human behavior, can enhance our understanding of financial
markets. We delve into the applications and limitations of these mathematical theories in
finance, highlighting their potential in improving market prediction, investment strategy,
and risk management. Additionally, the importance of Human Finance Interaction (HFI)
becomes even more pronounced in the context of emerging technologies like ChatGPT
AI and blockchain. This integrated approach can create more adaptive, secure, and user-
centered financial systems in the digital economy.
To facilitate a comprehensive understanding among our diverse readership, this paper
includes a foundational overview of mathematical concepts pertinent to the discussed
topics.
1.1. Group Theory in Financial Markets. This section delineates the fundamental
principles of group theory. We outline these details in the following bullet points for better
understanding.
HUMAN-FINANCE INTERACTION: A MULTIDISCIPLINARY APPROACH 3
•
Group theory examines the structure and behavior of sets under a specific operation,
akin to analyzing teams abiding by certain rules.
•
Consider a group as a collection of elements, where each element can interact with
another through a defined operation.
•
This operation, analogous to a specific way of interaction like a handshake, adheres
to particular mathematical rules.
•The fundamental rules of group theory include:
–
Existence of an identity element that, when combined with any element, leaves
it unchanged, similar to a neutral interaction in a social context.
–
Closure property ensuring that the interaction of any two elements produces
another element within the same group.
–
Each element having an inverse such that their combined interaction
neutralizes their effect, akin to reversing an action.
–
Associativity, where the order of interaction does not affect the final outcome.
•
Mastery of group theory enables the discernment of underlying patterns in complex
systems, enhancing problem-solving capabilities.
For an extensive, formal and detailed exploration of these concepts, readers are encouraged
to consult seminal works in the field, such as Dummit and Foote’s "Abstract Algebra" [
8
],
Artin’s "Algebra" [1], and Fraleigh’s "A First Course in Abstract Algebra" [12].
We now provide an overview of group theory in financial markets. Financial market
symmetries and structures can be analyzed mathematically using group theory. Studies
show it can identify market patterns, optimize investment portfolios, and manage financial
risks. Using group theory in finance provides a unique perspective on market dynamics and
decision-making [
13
]. New group theory applications to financial markets have produced
sophisticated time series prediction models. GMDH-type neural networks, based on group
theory, can improve stock market informational efficiency. These models use financial data
symmetries to predict market trends and movements more accurately [16].
Group theory’s role in financial market structural dynamics is also becoming clearer.
Researchers have found hidden patterns and relationships in financial market group
structures. Risk management benefits from a deeper understanding of market volatility and
instability factors. Additionally, group theory in portfolio optimization is gaining popularity.
Group-theoretic methods help investors and financial analysts find asset allocations that
maximize returns and minimize risk. This approach considers the symmetrical relationships
between financial instruments and market sectors, making investment strategies more
efficient and effective. The incorporation of group theory into financial market analysis
advances financial engineering. It helps understand and navigate complex financial market
dynamics and promises to build more resilient financial systems.
1.2. Game Theory in Financial Decision-Making. Same as in case of group theory we
provide the following bullet points for the understanding of young reader about game
theory.
4 RIZWAN JAHANGIR AND DAISUKE ISHII
•
Game theory explores strategic decision-making, focusing on predicting and
analyzing the choices of different participants.
•
The field extends beyond mere victory in competitive scenarios, emphasizing
optimal decision-making for collective benefit.
•
Consider an example where two individuals must choose how to share resources.
Cooperative choices can lead to mutually beneficial outcomes.
•
Conversely, competitive or selfish strategies may result in suboptimal outcomes for
all parties involved. Game theory provides a framework for understanding such
dynamics.
•
This theory finds practical applications in diverse areas, ranging from international
diplomacy and economic policy to business strategy and market pricing.
•
Central to game theory is the concept that collaborative strategies often yield more
favorable outcomes for all involved parties.
The preceding discussion offers a preliminary, non-technical overview of game theory.
For a more rigorous and comprehensive exploration of the subject, readers are directed to
seminal texts in the field. Osborne’s "An Introduction to Game Theory" [
20
] provides
an accessible yet thorough introduction to the fundamental concepts. Luce and Raiffa’s
"Games and Decisions: Introduction and Critical Survey" [
19
], and von Neumann and
Morgenstern’s "Theory of Games and Economic Behavior" [
25
] are foundational works,
offering in-depth analyses and historical context vital for a profound understanding of
game theory.
We now provide some literature review for game theory in financial decision making.
Game theory, which studies strategic decision-making, illuminates financial interactions.
Financial decisions made by individuals and institutions in competitive and cooperative
environments are examined in this study. This study examined how strategic interactions,
trust, cooperation, and financial incentives affect human behavior. Recent research has
expanded game theory’s use in financial decision-making. Song and Wu [
24
] use game-
theoretic models to examine how financial forecasting affects financial market decision-
making by econometricians and individual participants. An empirical study shows the
complexity of financial forecasting and its impact on market participants.
Cheng et al. [
5
] examined corporate-independent director dynamics in financial fraud using
evolutionary game theory. This study examines strategic decision-making in corporate
governance and financial integrity. In addition, Xiao et al. [
27
] proposed a new financial
institution decision-making method that upgrades industrial clusters. The method uses
interval-valued intuitionistic trapezoidal fuzzy number game matrices. Financial decision-
making is unclear and unpredictable, so this method addresses it. It provides a better
understanding of industrial cluster financial support strategies and conditions. El Fakir
and Tkiouat [
10
] used game theory and agent-based simulation to study entrepreneurial
financing profit ratio negotiability. Islamic finance profit-and-loss contracts are the focus
of this study. As El Fakir stated, it provides a comprehensive theoretical and practical
framework for negotiating ratios and mitigating moral hazards.
HUMAN-FINANCE INTERACTION: A MULTIDISCIPLINARY APPROACH 5
The above advances in game theory in financial decision-making highlight strategic
cognition and the complexity of financial market behavior.
1.3. Partial Differential Equations in Financial Modeling. Partial differential equations
(PDEs) can be understood basically by the followings.
•
Partial Differential Equations (PDEs) are a class of mathematical equations used to
describe phenomena involving changes across multiple dimensions and over time.
•
PDEs are distinguished by their involvement of partial derivatives, which represent
rates of change with respect to multiple variables.
•
These equations are fundamental in expressing physical laws and principles,
especially those concerning wave propagation, heat diffusion, fluid dynamics,
and quantum mechanics.
•
The complexity of PDEs arises from their ability to model systems with varying
conditions across different spatial dimensions and temporal intervals.
•
Solutions to PDEs often require advanced mathematical techniques and are crucial
for simulations and predictions in engineering, physics, and other applied sciences.
•
The study of PDEs not only provides insights into specific physical systems but
also contributes to the advancement of mathematical theory and computational
methods.
The above points provide a cursory overview of the fundamental concepts inherent in
Partial Differential Equations. For an exhaustive and nuanced study, esteemed sources such
as "Partial Differential Equations for Scientists and Engineers" by Farlow [
11
], "Applied
Partial Differential Equations" by Logan [
18
], "An Introduction to Partial Differential
Equations" by Pinchover and Rubinstein [
21
], and "Partial Differential Equations" by Evans
[
9
], are invaluable. These texts offer profound insights and a comprehensive treatment of
the subject, encompassing both theoretical and applied dimensions of PDEs.
Partial differential equations (PDEs) are utilized to represent and analyze various aspects
of market dynamics, risk assessment, and the pricing of financial derivatives. The
utilization of a quantitative methodology for comprehending intricate financial systems
has been applied in the domains of digital finance, human rights in finance, and human
resource accounting.The advancement of partial differential equation (PDE) modeling
has contributed to the enhancement of financial analysis. Duck et al. [
7
] assert that their
partial differential equation (PDE) system, designed for modeling stochastic storage in
physical and financial systems, exhibits significantly improved computational efficiency
compared to conventional simulation techniques. This enhancement enhances the analysis
of system design and operation in various financial applications.The damped diffusion
framework developed by Minqiang Li aims to tackle the issue of asset price bubbles within
continuous-time diffusion processes in financial modeling.
According to Li [
17
], the inclusion of damping in the diffusion or drift function enhances
the precision of financial models and maintains the integrity of the martingale pricing
approach.Efficient implementation on high-performance computing platforms such as
GPUs is of utmost importance in Duy Minh Dang’s thesis, which focuses on the
6 RIZWAN JAHANGIR AND DAISUKE ISHII
modeling of multi-factor financial derivatives using partial differential equations (PDEs).
According to Dang [
6
], this approach demonstrates effectiveness in handling intricate
financial instruments such as cross-currency interest rate derivatives and multi-asset
options.Binder et al. also proposed a framework for model order reduction in high-
dimensional financial risk analysis models. The approach proposed by Binder [
3
] utilizes
orthogonal decomposition to enhance the efficiency of analysis without compromising
accuracy. This methodology proves to be particularly advantageous in the context of
historical or Monte Carlo Value-at-Risk calculations.The aforementioned advancements
serve as evidence of the versatility and efficacy of partial differential equations (PDEs) in
comprehending the intricacies of financial markets and offering reliable tools for financial
analysis and decision-making.
2. Methodology
This study employs a comprehensive qualitative research methodology, synthesizing
information from a wide array of academic journals, articles, and studies. The primary
focus is on analyzing the applications of group theory, game theory, and Partial Differential
Equations (PDE) in financial contexts, particularly emphasizing their implications for
Human Finance Interaction (HFI).
2.1. Data Collection and Synthesis. The process of collecting data entailed a thorough
examination of scholarly literature from diverse academic databases and journals. Our
focus was on scholarly articles that have undergone peer review, influential papers that
have made significant contributions to the field, and recent studies that offer valuable
perspectives on the utilization of mathematical theories in the domain of finance. The
criteria for selection were determined based on the significance to group theory, game
theory, and partial differential equations (PDE), as well as their potential for practical
application in financial decision-making, market dynamics, and risk assessment.In order
to obtain financial data, we utilize the
=googlefinance("currency:ABCXYZ")
formula within Google Sheets [
14
], which enables us to generate historical data for multiple
currencies. The formula exhibited significant efficacy, as it demonstrated its applicability
in the domain of cryptography and certain stock prices. One advantageous aspect of this
formula is its ability to automatically exclude data values that are blank, such as those
occurring during weekends, particularly in the context of stock prices.
2.2. Focus on Human Finance Interaction. Human Finance Interaction (HFI) bridges
the divide between conventional financial theories and the complexities of real-world
financial scenarios. This approach recognizes the influence of human behaviors, emotions,
and irrational decisions in financial markets. To gain a deeper insight into market dynamics
and decision-making processes, HFI employs methodologies such as group theory, game
theory, and Partial Differential Equations (PDEs). In practice, this involves analyzing
financial data curated in Google Sheets. The primary tool for this analysis is the linear
graph functionality in Google Sheets. Furthermore, HFI utilizes statistical methods, notably
HUMAN-FINANCE INTERACTION: A MULTIDISCIPLINARY APPROACH 7
standard deviation, along with derivatives to refine and enhance the accuracy of financial
models developed from historical data.
2.3. Limitations and Future Research. This qualitative research approach offers
valuable insights but is limited by the subjective nature of data interpretation and the
potential for bias in source selection. To strengthen the study, incorporating quantitative
methods could validate and complement the qualitative findings. Additionally, integrating
advanced computational techniques with existing mathematical theories may provide
innovative approaches to financial modeling and analysis.
Currently, our analysis is constrained by the data availability in Google Sheets, which has
limitations, such as incomplete coverage of cryptocurrencies and certain stock prices. For
future research, we suggest developing a tool or incorporating programming within Google
Sheets. This enhancement would enable linking to external data sources, thereby expanding
the scope and improving the functionality of Google Sheets for more comprehensive
financial analysis.
3. Analysis and Discussion
The study utilized data from academic journals, articles, and platforms like Google Finance,
enriched with economic indicators from the World Bank and IMF, and behavioral data
from the Behavioral Risk Factor Surveillance System. Group theory was key in identifying
market patterns and informing investment strategies, revealing repetitive patterns in stock
indices. Game theory provided insights into strategic decision-making in competitive
finance sectors, like investment banking. Partial Differential Equations (PDE) were crucial
in risk assessment and derivative pricing, simulating market scenarios to evaluate risks and
returns.
The concept of Human Finance Interaction (HFI) highlighted the importance of integrating
human behavior into financial models, as behavioral finance data showed the significant
impact of irrational decision-making on markets. This research informs financial practice
and policy, suggesting that a comprehensive understanding of group theory, game theory,
PDE, and human behavior can enhance investment strategies and regulatory frameworks,
accounting for both rational and irrational market dynamics.
4. Results
This study highlights the integration of group theory, game theory, and Partial Differential
Equations (PDE) in financial analysis, emphasizing the importance of Human Finance
Interaction (HFI). Group theory helped identify investment strategies through patterns
in stock market indices. Game theory was crucial in understanding strategic dynamics
in financial markets. PDEs provided insights into risk assessment and derivative pricing.
Integrating human behavior into these models is vital, as irrational and emotional investor
decisions significantly impact market dynamics, advocating for a holistic approach in
financial analysis that combines mathematical precision with psychological aspects.
As a result we are able to get some prediction models, portfolio as well as buy and sell
indicators based on the analysis. We need to introduce/recall some terminology from
8 RIZWAN JAHANGIR AND DAISUKE ISHII
financial mathematics for the better understanding of the equation that we give later. In the
realm of financial mathematics, several key terms play pivotal roles in the formulation and
interpretation of models:
•
Stock Price (
S
): Represents the current market value of a single share of a
company’s stock, fluctuating based on market demand and supply [15].
•Drift Rate (µ): Denotes the expected rate of return of a stock [4].
•
Volatility (
σ
): Measures the degree of variation in a stock’s price over time,
indicating risk and potential for rapid price changes [15].
•
Stochastic Term (
ϵ(t)
): Introduces randomness into financial models to capture
market fluctuations [23].
•Investor Sentiment or Behavior Index (I): Reflects investors’ attitudes towards
the market, influencing market trends [2].
•
Baseline (
I0
) and Minimum Thresholds (
Imin
) of Investor Sentiment: Important
for understanding market reactions under varying sentiment levels [2].
•Constants αand β: Quantify changes in investor behavior [22].
•
Price of the Derivative (
V
): The current market price of a derivative instrument
[15].
•
Risk-Free Interest Rate (
r
): A hypothetical rate used in financial models as a
benchmark [4].
•
Game Theory Terms - Company’s Market Position or Strategy (
P
), Optimal
Position (
Pmax
), Competitor’s Position (
Pcomp
): Crucial in strategic financial
decision-making [26].
•
Constants
γ
and
δ
: Represent factors in strategic financial decisions in game
theory [26].
We are able to obtain following equations based on our observation and the experiments
using the available historical data.
(1)
Group Theoretic Asset Price Modeling Equation: We propose the Group
Theoretic Asset Price Modeling Equation, a modified version of the Black-Scholes
equation, incorporating a Market Influence Factor (MIF) based on experimental
data insights. The equation is given by:
∂V
∂t +1
2σ2S2∂2V
∂S2+rS ∂V
∂S
−rV =M I F ·S
The derivation is based on the following assumptions and modifications:
(a)
The stock price dynamics still follow a geometric Brownian motion, as in the
standard model.
(b)
The MIF is derived from data analytics, possibly gathered from financial
databases and sentiment analysis tools, to reflect additional market conditions
or external influences.
(c)
The risk-free hedge portfolio construction and no-arbitrage principle
application remain as in the classic Black-Scholes model derivation.
HUMAN-FINANCE INTERACTION: A MULTIDISCIPLINARY APPROACH 9
(d)
The MIF term is added to the standard Black-Scholes equation to adjust the
option pricing to reflect the additional market insights. This term represents
the additional market influence on the stock price, which is a product of the
MIF and the stock price itself.
This modified equation aims to provide a more comprehensive tool for option
pricing by considering additional market factors that influence stock price dynamics.
We give a visual representation of this modified equation in Figure 2.
FIGURE 2. Time-series plot of stock price dynamics.
(2)
Strategic Investment Dynamics Equation: In our derivation of the Strategic
Investment Dynamics Equation, we begin by acknowledging the dynamic nature of
investor sentiment. Recognizing that investor behavior is influenced by a blend of
rational decision-making and psychological factors, we define investor sentiment,
denoted as I, as a variable subject to change over time.
We then introduce the concept of strategic interaction among investors, a
fundamental principle of game theory. In a financial market comprising
n
investors,
each investor’s strategy, denoted as
Sj
, influences and is influenced by the others.
This interdependence is key to understanding the collective dynamics of market
behavior.
Next, we formulate the rate of change of an individual investor’s sentiment.
We express this as
dI
dt
and model it as a function of the strategies of other market
participants. Specifically, we use the term
γPn
j=1 Sj(Ij−I)
to represent the
aggregated impact of the strategies of other investors on an individual’s sentiment,
where Ijstands for the sentiment index of the j-th investor.
To capture the influence of group dynamics on individual sentiment, we introduce
the term
δΦ(I, G)
. Here,
Φ(I, G)
is a function that encapsulates the complex
10 RIZWAN JAHANGIR AND DAISUKE ISHII
interaction between an investor’s sentiment and the overall group behavior, denoted
by G.
Bringing these elements together, we arrive at the Strategic Investment Dynamics
Equation:
dI
dt =γ
n
X
j=1
Sj(Ij−I)−δΦ(I, G)
This equation aims to model the evolution of an investor’s sentiment in a dynamic
system, influenced by both individual strategic decisions and the broader market
behavior. In Figure 3 shows a visualization of this equation.
FIGURE 3. Visualization of strategic investment dynamics.
(3)
Game Theory in Financial Markets: The application of game theory in financial
markets examines the strategic interaction among market participants. The
differential equation:
dP
dt =γ(Pmax −P)−δ(P−Pcomp)
captures the dynamics of a company’s market position
P
over time.
Pmax
denotes
the company’s optimal market position, and
Pcomp
represents the competitor’s
market position. Constants
γ
and
δ
reflect the company’s sensitivity to achieving its
optimal position and to the actions of its competitors, respectively. This model is an
abstraction of strategic decision-making processes, where companies continuously
adjust their strategies in response to the market and competitors.
Each of these models represents a blend of mathematical rigor and practical applicability,
providing valuable insights into the complex dynamics of financial markets. They illustrate
how quantitative methods can be used to understand and predict market behaviors, investor
psychology, and strategic decision-making in finance.
HUMAN-FINANCE INTERACTION: A MULTIDISCIPLINARY APPROACH 11
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KIA RA IN C. TOKYO, 150-0041, JAPAN
Email address:rizwan@kiara.team
KIA RA IN C. TOKYO, 150-0041, JAPAN
Email address:dai@kiara.team
