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\textsc{\LARGE Raghav Goyal, Harsh, Mohit Goenka }\\[1.5cm] % Name of your university/college
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\textsc{\large Indian Institute of Technology Kanpur}\\[0.5cm] % Minor heading such as course title
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{ \huge \bfseries Term Project}\\[0.4cm] % Title of your document
{ \large \bfseries Price Dynamics in Financial Markets : A Kinetic Approach}\\[0.2cm]
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\begin{center} \large
\emph{Course Instructor: Dr. Rakesh Mathpal}
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\begin{center}
\today
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\begin{center}
\section*{Abstract}
\end{center}
Econophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dynamics. Some of its application to the study of financial markets has also been termed statistical finance referring to its roots in statistical physics.\\
The recent events of the 2008 world’s financial crisis and
its uncontrolled effect propagated among the global economic
system, has produced a deep rethink of some paradigm and
fundamentals in economic modeling of financial markets.\\
Any reasonable model need to rely on some fundamental
hypotheses and to rest on a theoretical framework, which
should be able to provide some basic and universal principles,
this is the way all the models arising from the physical world
are build up.\\
One of the most classical approach has been to consider
the efficient market hypothesis.. It relies on the belief
that securities markets are extremely efficient in reflecting information about individual stocks and about the stock market
as a whole. When information arises, the news spread very
quickly and are incorporated into the prices of securities without delay. Thus, neither technical analysis, which is the study of past stock prices in an attempt to predict future prices, nor even fundamental analysis, which is the analysis of financial information such as company earnings, asset values, etc., to help investors select undervalued stocks, would enable an investor to achieve returns greater than those that could be obtained by holding a randomly selected portfolio of individual
stocks with comparable risk.\\
We derive a set of equations which are a simple model for investor behavior in a
theoretical financial market. The model incorporates the emotional aspect of investor sentiment
with memory of price history which decays exponentially in time. Within this model, the
emotional reaction of the body of investors is to buy when the recent price has been increasing
and sell when it has been decreasing. The rational motivations are based on capitalizing on the
difference between the price and intrinsic value, with the possibility of some inertia in taking
action. These two competing effects provide the basis for fluctuations and instability. \\
\newpage
\tableofcontents
\chapter{PRICE DYNAMICS IN FINANCIAL MARKETS}
\section{Efficient Market Hypothesis}
The efficient market hypothesis is associated with the idea
of a random walk,which is widely used in the finance literature to characterize a price series where all subsequent price changes represent random departures from previous prices. The logic of the random walk idea is that if any information is immediately reflected in stock prices, then tomorrow’s
price change will reflect only tomorrow’s news and will be independent of the price changes today. Thus, resulting
price changes must be unpredictable and random.\\
Strongly linked to the market efficiency hypothesis, is the
assumption of rational behavior among the traders. Rationality
of traders can be basically reassumed in two main features.
First, when they receive new information, agents update
their beliefs by evaluating the probability of hypotheses
accordingly to Bayes’ law. Second, given their beliefs, agents
make choices that are completely rational, in the sense that
they arise from an optimization process of opportune subjective utility functions.\\
\section{Behavioral Finance Theory}
By the beginning of the twenty-first century, the intellectual
dominance of the efficient market hypothesis had become
far less universal. Many financial economists and statisticians
began to believe that stock prices are at least partially predictable.
A new breed of economists emphasized psychological
and behavioral elements of stock-price determination. The
behavioral finance approach has emerged in response to the
difficulties faced by the traditional paradigm.\\
It relies in the fact that some financial phenomena can be better understood using models in which some agents are not fully rational. In some behavioral finance models, agents fail to update their beliefs correctly. In other models, agents apply Bayes’ law properly but make choices that are questionable, in the sense that they are incompatible with the optimization of suitable utility functions.\\
\section{Prospect Theory}
A strong impact in the field of behavioral finance has been
given by the introduction of the prospect theory by Kahneman
and Tversky. They present a critique of expected
utility theory as a descriptive model of decision making under
risk and develop an alternative model. Under prospect theory,
value is assigned to gains and losses rather than to final assets and probabilities are replaced by decision weights.The theory which they confirmed by experiments predicts a distinctive
fourfold pattern of risk attitudes: risk aversion for gains
of moderate to high probability and losses of low probability,
and risk seeking for gains of low probability and losses
of moderate to high probability.\\
Further development in this direction, led to the discovery that that people systematically overreacting
to unexpected and dramatic news events results in substantial weak-form inefficiencies in the stock market.\\
\section{Agent Based Modeling}
Recently, these methods have given an important contribute and provided a huge quantity of numerical simulations.The idea is to produce a big mass of artificial data and to observe how they can fit with empirical observations. This approach is now also supported by the availability of many recorded empirical data.The aim of the construction of such microscopic models of financial markets is to reproduce the observed statistical features of market movements (e.g. fat tailed return distributions, clustered volatility, cycles, crashes) by employing highly simplified models with large numbers of agents. Microscopic models of financial markets are highly idealized as compared to what they are meant to model.The relevant part of
physics that is used to build such models of financial markets
consists in methods from statistical mechanics. This attempt
by physicists to map out the statistical properties of financial
markets considered as complex systems is usually referred to
as econophysics.\\
The need to recover mathematical models which can display
such scaling properties, but also capable to deal with systems
of many interacting agents and to take into account the
effects of collective endogenous dynamics, put the question
on the choices of the most appropriate mathematical framework
to use.\\
Besides numerical simulations, it is of
paramount importance to have a rigorous mathematical theory
which permits to identify the essential features in the modelling
originating the stylized facts. The classical framework
of stochastic differential equations which played a major rule
in financial mathematics seems inadequate to describe the dynamics
of such systems of interacting agents and their emerging
collective behavior.\\
\chapter{KINETIC MODELLING FOR PRICE FORMATION}
\section{Introduction}
Kinetic theory was introduced in order
to give a statistical description of systems with many interacting
particles. Rarefied gases can be thought as a paradigm
of such complex systems, in which particles are described by
random variables which represents their physical states, like
position and velocity. A Boltzmann equation then prescribes
the time evolution for the particles density probability function.\\
The above theoretical model seems to fit very well with the necessity to prescribe how the trading agents interacting in a stock market are leaded to form their expectations and re-evaluate their choices on the basis of the influence placed on the neighbor agents’ behavior rather than the flux of news coming from some fundamental analysis or direct observations of the market dynamic.The kinetic approach reveals particularly powerful
when from some simple local interaction rules some global
features for the whole system has to be derived, but also in
the study of asymptotic regimes and universal behaviors described
by Fokker-Planck equations.\\
\section{Opinion Modeling}
The collective behavior of a system of trading agents can
be described by introducing a state variable y $\epsilon$ [-1,1] and the relative density probability function f(y) which, for each agent, represent respectively the propensity to invest and the probability to be in such a state. Positive values of y represent potential buyers, while negative values characterize potential
sellers, close to y = 0 we have undecided agents. Clearly,
\begin{equation}
\label{1}
\rho (t) = \int_{-1}^1f(y,t) dy
\end{equation}
Equation ~\ref{1} represents the number density. Moreover we define the mean investment propensity by equation ~\ref{2},
\begin{equation}
\label{2}
Y(t)=\frac{1}{\rho(t)}\int_{-1}^1f(y,t) ydy
\end{equation}
Traders are allowed to compare their strategies and to re-evaluate them on the basis of a compromise opinion dynamic. This is done by assigning simple binary interaction rules, where, if the pair (y,$y_*$) and (y',$y_*'$) represents respectively the pre-interaction and the post-interaction opinions, we have:
\begin{equation}
\label{3}
y'=(1-\alpha_1(H(y))y+\alpha_1H(y)y_*+D(y)\eta
\end{equation}
\begin{equation}
\label{4}
y_*'=(1-\alpha_1(H(y_*))y_*+\alpha_1H(y_*)y_*+D(y)\eta_*
\end{equation}
Here $\alpha_1 \epsilon$ [0,1] measures the importance the individuals place on others opinions in forming expectations about future price changes.The random variables $\eta$ and $\eta_*$ are assumed are assumed distributed accordingly to $\theta$($\eta$) with zero mean and variance $\sigma^2$ and measure individual deviations to the average behavior.
The functions H(y) and D(y) characterize respectively,
herding and diffusive behavior. Simple examples of herding
function and diffusion function are given by:
\begin{equation}
\label{5}
H(y)=a+b(1- \mid y \mid)
\end{equation}
\begin{equation}
\label{5}
D(y)=(1-y^2)^2{\gamma}
\end{equation}
With 0 $\leq$ a $+$ b $\leq$ 1, a $\geq$ 0, b $>$ 0, $\gamma>$0.
\section{Market Influence}
The traders are also influenced by the dynamics of stock
market’s price, so a coupling with the price dynamic has to
be considered. With the same kinetic setting we define the
probability density V(s,t) of a given price s at time t. The
market price S(t) is then defined as the mean value.
\begin{equation}
\label{6}
S(t) = \int_0^{\infty}V(s,t) sds
\end{equation}
Price changes are modeled as endogenous responses of the
market to imbalances between demand and supply characterized
by the mean investment propensity accordingly to the following
price adjustment.
\begin{equation}
\label{7}
s'=s+\beta\rho Y(t)s+ \eta S
\end{equation}
Where, $\beta >$0 represents the price speed evaluation and $\eta$ is a random variable with zero mean and variance $\zeta^2$ distributed accordingly to $\Psi(\eta)$.\\
\begin{figure}[H]
\centerline{\includegraphics[scale=1.5]{long.pdf}}
\caption{Loss vs. Gain}
\end{figure}
To take into account the influence of the price in the mechanism of opinion formation of traders, we introduce a normalized value function $\Theta$ = $\Theta(\frac{\frac{dS(t)}{dt}}{S(t)})$ in [-1,1] in the sense of Kahneman and Tversky that models the reaction of individuals towards potential gain and losses in the market. Thus we reformulate the binary interaction rules in the following way:\\
\begin{equation}
\label{3}
y'=(1-\alpha_1(H(y)-\alpha_2)y+\alpha_1H(y)y_*+ \alpha_2 \Theta(y)\eta
\end{equation}
\begin{equation}
\label{4}
y_*'=(1-\alpha_1(H(y_*)-\alpha_2)y_*+\alpha_1H(y_*)y_*+ \alpha_2 \Theta(y)\eta_*
\end{equation}
Here, $\alpha_1 \epsilon$ [0,1] and $\alpha_2 \epsilon$[0,1] with $\alpha_1 +\alpha_2 \leq $ 1, measure the importance the individuals place on others opinions and actual price trend in forming expectations about future price changes. This permits to introduce behavioral aspects in the market dynamic and to take into account the influence of psychology and emotivity on the behavior of the trading agents.\\
Note: Except for the particular shape of the value function,
if the mean propensity is initially (sufficiently) positive then it will continue to grow together with the price and the opposite occurs if it is initially (sufficiently) negative. The market goes towards a boom (exponential grow of the price) or a crash (exponential decay of the price).\\
\newpage
\section{Lognormal Behavior}
A set of Boltzmann equations for the evolution of the unknown
densities f(y,t) and V(s,t) can be obtained using the
standard tools of kinetic theory. Such systems are defined by the equations ~\ref{10} and ~\ref{11}.
\begin{equation}
\label{10}
\frac{\partial f}{\partial t}=Q(f,f)
\end{equation}
\begin{equation}
\label{11}
\frac{\partial V}{\partial t}=L(V)
\end{equation}
where the quadratic operator Q and the linear operator L can
be conveniently defined in weak form by equation ~\ref{12} and ~\ref{13}.
\begin{equation}
\label{12}
\int_{-1}^1 Q(f,f)\Phi(y)dy=\int_{[-1,1]^2}\int_{R^2}B(y,y_*)f(y)f(y_*)(\Phi(y)-\Phi(y')) d\eta d\eta_* dy dy_*
\end{equation}
\begin{equation}
\label{13}
\int_0^{\infty}L(V)\Phi(s)ds=\int_0^{\infty}\int_R b(s)V(s)(\Phi(s')-\Phi(s)) d\eta ds
\end{equation}
In the above equations $\Phi$ is a test function and the transition rates have the form given by the equations ~\ref{14} and ~\ref{15}.
\begin{equation}
\label{14}
B(y,y_*)=\Theta(\eta)\Theta(\eta_*)\chi(\mid y' \mid \leq 1)\chi(\mid y_*' \mid \leq 1)
\end{equation}
\begin{equation}
\label{15}
b(S)=\Psi(\eta)\chi(s'\geq 0)
\end{equation}
With $\chi(.)$ representing the indicator function.
A simplified Fokker-Planck model which preserves the
main features of the original Boltzmann model is obtained under
a suitable scaling of the system. In such scaling all agents
interact simultaneously with very small variations of their investment propensity.This allows us to recover the following Fokker-Plank system:\\
\begin{equation}
\label{16}
\frac{\partial f}{\partial t}\frac{\partial [(\rho \alpha_1 H(y)(Y-y)+\rho \alpha_2(\Theta-y))f]}{\partial y}=\frac{\sigma^2 \rho}{2}\frac{\partial^2 [D^2(y)f]}{\partial^2 y^2}
\end{equation}
\begin{equation}
\label{17}
\frac{\partial V}{\partial t}+\frac{\partial [\beta \rho YsV]}{\partial s}=\frac{\zeta^2}{2}\frac{\partial^2 s^2V}{\partial^2 s^2}
\end{equation}
where we kept the original notations for all the scaled quantities.\\
The above equation for the price admits the self similar lognormal solution.\\
\begin{equation}
\label{18}
V(s,t)=\frac{1}{s(2log(Z(t)^2)\pi)^{\frac{1}{2}}}exp(\frac{-log(sZ(t))^2}{2log(Z(t)^2)})
\end{equation}
Where, $Z(t)= \sqrt{E(t)}/S(t)$, and E(t) satisfies the differential equation given by equation ~\ref{19}.
\begin{equation}
\label{19}
\frac{dE}{dt}=(2\beta Y + \zeta^2)E
\end{equation}
\newpage
\section{A Different Strategy}
We consider now in the stock market the presence of traders
who deviate their strategy from the mass. We introduce trading
agents who rely in a fundamental value for the traded security.
They are buyer while the price is below the fundamental
value and seller while the price is above. Expected gains or
losses are then evaluated from deviations of the actual market
price and just realized only wether or not the price will revert
towards the fundamental value. Such agents are not influenced
by other agents’ opinions.\\
The microscopic interactions rules for the price formation
now reads.\\
\begin{equation}
\label{20}
s'=s+\beta(\rho Y(t)s+\rho_F\gamma(S_F-s))+\eta_s
\end{equation}
Where, $S_F$ represents the fundamental value of price, $\rho_F$ is the number density for such trading agents performing a different strategy, while $\gamma$ is the reaction strength to the deviations from fundamental value. If we are interested in steady
states we can ignore the possibility of a strategy exchange between
traders and the resulting kinetic system has the same
structure (~\ref{16} and ~\ref{17}).\\
\chapter{MARKET FLUCTUATION PSYCHOLOGY}
\section{Introduction}
Large price fluctuations in financial markets are often far in excess of the changes in intrinsic
value. These fluctuations are generally attributed to the psychological reactions and overreactions
to rapid but smaller changes in fundamental value. In this paper, we model the
behavior of a theoretical financial market based on the psychological motivation of a body
of investors. \\
The modeling of a financial system with a large number of decision makers is analogous
to modeling a physical system consisting of many degrees of freedom. The alternatives can
be grouped into at least two categories:\\
\begin{enumerate}
\item The states in which the investor is in, for example, he is in a buying mode or a selling mode is determined by giving individual probabilistic weighing. Within this model, the
emotional reaction of the body of investors is to buy when the recent price has been increasing
and sell when it has been decreasing.This is
analogous to a statistical mechanical approach to thermodynamics.
\item In this model one considers the total number of investors in each state and the total rate of transition of these investors from one state to another just like the transfer of molecules from one state to another. This is analogous
to chemical kinetics or Boltzmann dynamics for a gas.
\end{enumerate}
The intent of this section is that to show that such price fluctuations can be modeled using an equation which relates the current price change to an integral, $\zeta(t)$ involving past price and price derivative history. Thus it is not only the current prices but the past prices and the past price derivatives that help in determining the future prices and predict the market behavior.
\section{The Model}
The price of the stock commodity is determined by the behavior of the share holders who might buy, sell or hold the stock. It is assumed that the share held by the investors is small compared to the total value in the market. An investor has four options (A,B,C,D). A- sell the stock, B- hold the stock, C- sell the cash, D- hold the cash. Each of these options of the investor defines the state in which he is. There may be several transfer of investors from one state to another just like the transfer of molecules from one energy state to another in the chemical kinetics -
that is the laws of mass action. For example, transitions occur from B to A, from A to D, from D
to C, and from C to B. The corresponding rate equations are then :
\begin{equation}
\label{31}
\frac{dA}{dt}=-k_1A+k_4B
\end{equation}
\begin{equation}
\label{30}
\frac{dB}{dt}=k_3C-k_4B
\end{equation}
\begin{equation}
\label{29}
\frac{dC}{dt}=-k_3C+k_2D
\end{equation}
\begin{equation}
\label{28}
\frac{dC}{dt}=k_1A-k_2D
\end{equation}
Where the $k_i$ will depend on the history of the price and its derivative. Note that summing
the four equations implies that A + B + C + D is constant. \\
The crucial assumptions on mass psychology of investors are embedded in the rate co-efficients $k_i$ which have two factors. The first factor is the “emotional” aspect of the investor urges him to sell the stock when the prices are falling. It is assumed that each investor has some idea about the rise and fall of the stock in the past and the price history derivatives and remembers it through an exponentially failing memory. Faster the memory fails, more the emphasis on the current changes in price.We thus have a product of the fractional change in price, i.e., $P^{-1}\frac{dP}{dt}$, with a
natural function for decay; the exponential, integrated over all past times. This yields the
first integral of ~\ref{33} below. The second factor is the “rational” aspect is based on the motivation to buy when
the price is below the realistic or intrinsic value, $P_a(t)$ of the shock.Here, there is also a
decaying exponential term due to intellectual inertia, or lag time between actual changes
and investor action, multiplied by the fractional discount. \\
Hence we define the “investor sentiment” or tendency to buy as :\\
\begin{equation}
\label{33}
\zeta(t)=q_1c_1\int_{-\infty}^te^{-c_1(t-\tau)}\frac{1}{P(\tau)}\frac{dP(\tau)}{d\tau}d\tau+q_2c_2\int_{-\infty}^te^{-c_2(t-\tau)}[\frac{P_a(\tau)-P(\tau)}{P_a(\tau)}]d\tau
\end{equation}
Here, $q_1$ is the contribution to the "emotional" aspect, $q_2$ is the introduction to the intellectual aspect, $c_1^{-1}$ is the "memory length", and $c_2^{-1}$ is the "intellectual inertia".e.g., a
large $q_1$ and $c_1$ means means that a recent sharp price increase will improve investor sentiment.
Choosing an appropriate monotone positive function of $\zeta$ as for instance, $\frac{1}{2}+\frac{1}{2} tanh(\zeta(t))$, one may write:\\
\begin{equation}
\label{38}
k_2=\alpha[\frac{1}{2}+\frac{1}{2} tanh(\zeta(t))]
\end{equation}
\begin{equation}
\label{34}
k_4=\beta[\frac{1}{2}-\frac{1}{2} tanh(\zeta(t))]
\end{equation}
Note that since $k_2$ is is the rate at which investors decide to buy stock while $k_4$ is the rate at
which they decide to sell stock, we have assumed the relation $\frac{k_2}{\alpha}+\frac{k_4}{\beta}=1$, where $\alpha$ and $\beta$ are constant amplitudes .Due to simplifications, it is not necessary
to derive $k_1$ and $k_3$.
Next, we discuss the equation for the price, P. The relative price change should be given
by a function, f, of buyers and sellers. That is:\\
\begin{equation}
\label{37}
\frac{dP}{dt}=Pf(C/A)
\end{equation}
where f is an increasing function such that f(1) = 0. For example, f(z) = $\delta log(x)$ is acceptable.\\
The assumptions inherent in ~\ref{33}, ~\ref{38}, and ~\ref{34} lead to a preliminary interpretation for price
fluctuations. Further the concept of "panic buying and selling" has been modeled using the kinetic theory. The price fluctuations are a result of a difference between the perceived and the expected prices. The emotional tendency of the stock holder tends to serve as a destabilizing factor for this gap rather than bridging it. Suppose there is a price fluctuation with the price increasing a bit, due to behavior of stock holders $zeta(t)$ further increases and the same analogy can be applied to a slight decrease in price. On
the other hand, the latter part of ~\ref{33} is a stabilizing factor in that a deviation from
actual value ($P_a(t)$) results in a change in investor sentiment which is in the direction of
restoring equilibrium. This process is usually called “bargain hunting” or “profit taking”.
Consequently, there is a basic competition between these stabilizing and destabilizing forces.
Within our model, the result of this competition decides whether one will observe abrupt
reversals in price (and fraction of buyers) in the absence of dramatic changes in fundamental
value of the underlying securities. \\
\section{Simplification of The Model}
The equations ~\ref{31},~\ref{30},~\ref{29}, and ~\ref{28} can be reduced to a single equation with the additional assumption
that the transition from seller to sitter is very rapid. This is realistic since buy and sell
orders are usually executed almost immediately. More precisely, we assume that :\\
\begin{equation}
\label{40}
k_1=\frac{\bar{k_1}}{\epsilon}
\end{equation}
\begin{equation}
\label{41}
k_3=\frac{\bar{k_3}}{\epsilon}
\end{equation}
Where, 0$\leq \epsilon \ll$ 1 and $\bar{k_1}$ and $\bar{k_3}$ are O(1). Then letting $\epsilon \longrightarrow$ 0, we see that A $\longrightarrow$0, C$\longrightarrow$0, $k_1$A $\longrightarrow k_4$B, and $k_3$C$\longrightarrow k_2$D, so that ~\ref{31} reduces to :\\
\begin{equation}
\label{42}
\frac{dD}{dt}=k_4B-k_2D=-\frac{dB}{dt}
\end{equation}
From this, it is clear that D = 1 - B and therefore that:\\
\begin{equation}
\label{43}
\frac{dB}{dt}=k_2(1-B)-k_4B
\end{equation}
\begin{equation}
\label{44}
\frac{C}{A}=\frac{k_2\bar{k_1}}{k_4\bar{k_3}}\frac{1-B}{B}
\end{equation}
which reduces the price equation (~\ref{37}) to one only involving B, namely:\\
\begin{equation}
\label{45}
\frac{dP}{dt}=Pf(\frac{k_2\bar{k_1}}{k_4\bar{k_3}}\frac{1-B}{B})
\end{equation}
In togetherness, the equations ~\ref{43} and ~\ref{45} coupled along with equation ~\ref{33},~\ref{38}, and ~\ref{34} constitute the
model based on these assumptions.\\
Unlike many mathematical models of markets, a key feature of our model is the absence
of explicit probabilistic concepts. The central assumption to our model is the use of a realistic price value of stock
,$P_a(t)$.\\
\chapter{CONCLUSION}
CONCLUSION FOR CHAPTER 3: We have therefore seen that the mathematical model based on kinetic theory that we have used not only models the price fluctuations based on the current prices but also takes into account the past prices and the price history derivatives. Due weight-age has been given to customer motivation and behavior. The model does not use any explicit probabilistic concept but revolves around the concept of assigning a realistic value to stocks which forms a central concept in the stock market.
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