# Modulational Instability of the Coupled Nonlinear volatility and option
price model

C. Gaafele<EMAIL_ADDRESS>Department of Physics
and Astronomy, Botswana International University of Science and Technology,
Private Mail Bag 16 Palapye, Botswana  Edmond B. Madimabe
<EMAIL_ADDRESS>Department of Physics and Astronomy,
Botswana International University of Science and Technology, Private Mail Bag
16 Palapye, Botswana  K. Ndebele<EMAIL_ADDRESS>Department of Physics and Astronomy, Botswana International University of
Science and Technology, Private Mail Bag 16 Palapye, Botswana  P. Otlaadisa
<EMAIL_ADDRESS>Department of Physics and Astronomy, Botswana
International University of Science and Technology, Private Mail Bag 16
Palapye, Botswana  B. Mozola<EMAIL_ADDRESS>Department of Physics and Astronomy, Botswana International University of
Science and Technology, Private Mail Bag 16 Palapye, Botswana  T. Matabana
<EMAIL_ADDRESS>Department of Physics and Astronomy,
Botswana International University of Science and Technology, Private Mail Bag
16 Palapye, Botswana  K. Seamolo<EMAIL_ADDRESS>Department of Physics and Astronomy, Botswana International University of
Science and Technology, Private Mail Bag 16 Palapye, Botswana  P. Pilane
<EMAIL_ADDRESS>Department of Physics and Astronomy,
Botswana International University of Science and Technology, Private Mail Bag
16 Palapye, Botswana

(May 2024)

###### Abstract

We study the Coupled Nonlinear volatility and option price model via both
Modulational instability analysis and direct simulations. Since the coupling
term for both the volatility and the option price equation is the same, the MI
results are dependent on it, and the stability of the volatility exists for
the same condition as that of the price. The numerical simulations are done to
comfirm the conditions of MI.

##  I Introduction

Nowadays, the field of economics and finance is extensively researched
globally. Experts and researchers use various technological and scientific
tools to develop sophisticated products aimed at optimizing daily life
situations. As users engage with these devices, they strive to maximize
benefits and achieve optimal outcomes. Traditionally, users rely on their
existing knowledge to assess product values, leading to various challenges for
buyers, sellers, online platforms, and banks both locally and internationally.
These issues are systematically explored using scientific principles,
resulting in innovative approaches that highlight the importance of monitoring
financial markets. The intricate modeling of global financial markets
generates comprehensive information systems, particularly dynamic systems that
enable detailed analysis of production processes. Observing the transmission
of products from producers to consumers through various channels like
highways, air travel, and shipping is also crucial. Initial steps in this
process involve formulating mathematical models, whether complex or real-
valued, incorporating wave functions. Numerous models have been developed to
extract wave distributions, helping to understand current trends and future
directions. Soliton theory is often used due to its ability to provide precise
information about periodic, singular, dark, bright, complex, and traveling
dynamics. These insights are essential for comprehending, predicting,
controlling, and planning for complex production behaviors. Notably,
researchers like Sharp et al. have investigated stochastic differential
equations (SDE) in finance sharp1990stochastic , while Irving and colleagues
have explored mixed linear-nonlinear coupled differential equations using
multivariate discrete time series sequences irving1997determining . Jin et al.
have demonstrated optimal consumption and portfolio strategies in continuous-
time finance models jin1997existence , and Ganesh et al. have analyzed
fractional order impulsive stochastic differential equations for
controllability priya2016controllability . Additionally, various economic
models, including robust economic models and national economy models, have
been proposed and analyzed by researchers such as Decardi-Nelson
decardi2021robust  and Adomian adomian1984modeling , employing decomposition
methods and numerical integration techniques. Advancements in computer
technology have also led to studies on modeling large power plants
maffezoni1984computer .

The classic Black-Scholes (BS) model stands as a pivotal achievement in
financial mathematics, capturing the evolution of market prices for financial
assets over time, such as stock options black1973pricing . This model operates
under certain assumptions, where variables like asset price $s$, drift
parameter $\mu$, and volatility $\sigma$ are treated as constants.
Additionally, it assumes frictionless, arbitrage-free, and efficient markets
gonzalez2016nonlinear , edeki2016modified . However, to address the
limitations posed by these assumptions, various modifications have been
proposed, including stochastic interest models, jump-diffusion models,
stochastic volatility models, and models incorporating transaction costs.
Consequently, the Ivancevic option pricing model (IOPM), an alternative to the
Black-Scholes equation, has garnered increasing attention in recent years due
to its effectiveness in estimating option values chen2022soliton ,
ali2023physical , elmandouh2022integrability , chen2021dark ,
gonzalez2017solving . The price function $s=s(t)$ for $0\leq t\leq T$ and
follows the geometric brownian motion (BM) $ds=s(\mu dt+\sigma dW_{t})$, where
$\mu$ is the drift rate, $W_{t}\equiv W(t)$ is the Weiner process. The BS
model is given by

$\frac{\partial\psi}{\partial t}+rs\frac{\partial\psi}{\partial
s}+\frac{1}{2}\sigma^{2}s^{2}\frac{\partial^{2}\psi}{\partial s^{2}}-r\psi=0.$
(1)

Even though there exists some versions of the BS model for the american stock
regulations, the above BS model works for the european stock regulations,
where the option is exercised at $T$. Recently, Vukovic
vukovic2015interconnectedness  established a connection between two equations:
the Schrödinger equation (SE) and the Black-Scholes (BS) model equation, using
principles from quantum physics, particularly the Hamiltonian operator. It was
further noted that the BS equation could be derived from the SE by employing
tools derived from quantum mechanics contreras2010quantum . While the SE is a
complex-valued equation, the BS model represents a real-valued equation
governing the price function. Equation (1) of the BS model can be extended to
one-dimensional option models associated with $\psi$ and $s$. As discussed in
voit2003statistical , one can determine the probability density function (PDF)
by solving the Fokker–Planck equation using classical Kolmogorov probability
methods, instead of relying on the option value obtained from the BS equation.

Hence, in the contemporary era, delving into the intricate aspects of economic
and financial issues through mathematical modeling stands as a highly explored
domain owing to its broad utilization within nonlinear science. These
mathematical models typically manifest as nonlinear partial differential
equations (NPDEs). Among the extensively researched NPDEs lies in the
Ivancevic option pricing model. Ivancevic ivancevic2010adaptive  utilized
quantum-probabilistic principles to derive a PDF equivalent for the value of a
stock option, by constructing a complex-valued function. This approach led to
the proposal of a nonlinear model cont2001empirical . Subsequently, this model
was rebranded as the Ivancevic Option Pricing Model (IOPM), outlined as
follows

$i\frac{\partial\psi}{\partial
t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}\psi}{\partial
s^{2}}+\beta|\psi|^{2}\psi=0,$ (2)

where $\psi(s,t)$ represents the option price function at time $t$, $\sigma$
is the volatility which is constant for this model and the adaptive market
potential given by $\beta$ which is the Landau coefficient representing the
adaptive market potential. In its simplest from it is the same as the interest
rate $r$ but in this case it depends on the set of adjustable parameters
$\\{{w_{i}}\\}$ called the synaptic weight vector components. The adaptive
market potential can be related to the market temperature which obeys the
Maxwell-Boltzman distribution kleinert2009path . Additional relevant studies
have been referenced here to enhance the comprehension of the current
analysis. Edeki et al. edeki2017analytical  and Gonzalez-Gaxiola and Ruiz de
Chavez gonz2015solving  examined the Ivancevic option pricing model employing
analytical approaches, specifically the projected differential transformation
method and a hybrid technique combining the Adomian decomposition method with
the Elsaki transform, respectively. Chen et al. chen2019new  introduced a
novel operator splitting method for resolving the fractional Black-Scholes
model concerning American options.

The dynamics of spatial pattern formation in nonlinear media holds
significance across various physical phenomena MI1 , MI2 , MI3 , MI4 , MI5 ,
MI6 , MI7 , MI8 , MI9 , MI10 . Modulational instability (MI) serves as a
fundamental mechanism for elucidating pattern formation within a uniform
medium. MI arises when a constant-wave background becomes unstable to induced
sinusoidal modulations due to the combined influences of nonlinearity and
diffraction or dispersion in a spatially nonlinear field. This instability
leads to the fragmentation of a uniform medium into pulsed "solitary waves"
MI2 . Since the Ivancevic option pricing model is a form of a Nonlinear
Schrodinger equation (NLS) it can be subjected to this analysis. The MI of a
generic NLS excists for a positive product of the interaction and dispersion
term, that is they should have the same sign, but since the Ivancevic option
pricing model’s dispersion term is always positive, instability occurs for
positive values of the adaptive market potential (hot adaptive market
temperatures). MI for the Ivancevic option pricing model was first performed
by chen2022soliton . MI has been used to study several branches of physics,
such as biophysics BP1 , BP2 , BP3 , BP4 , BP5 , hydrodynamics hydro1 , hydro2
, optical fibers opti1 , opti2 , opti3 , opti4 , Bose-einstein condensates
BEC1 , BEC2 , BEC3 , BEC4 , BEC5 , BEC6 , plasma physics pp1 , pp2 , pp3 , pp4
, pp5 , pp6 , electrical transmission lines et1 , et2 , et3 , metamaterials
mm1 , mm2 , mm3 , mm4 , mm5 , mm6 , mm7 , and finally quantum finance
chen2022soliton  amongst many others.

The remaining section are as follows: Section II will focus on the time-
fractional Ivancevic option pricing model and some basic fractional calculus.
Section III will focus on the linear stability analysis of the model deriving
the Bogoliubov dispersion relation of the model. Section 8 gives the numerical
simulations of the system and the paper is concluded by Section V.

##  II The Model

We investigate the volatility and option price coupled Ivancevic model given
by

$\displaystyle i\frac{\partial\sigma}{\partial
t}+\frac{1}{2}\frac{\partial^{2}\sigma}{\partial
s^{2}}+\beta(|\sigma|^{2}+|\psi|^{2})\sigma=0,$ (3) $\displaystyle
i\frac{\partial\psi}{\partial t}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial
s^{2}}+\beta(|\sigma|^{2}+|\psi|^{2})\psi=0,$ (4)

where $\sigma\equiv\sigma(s,t)$ is the volatility, $\psi\equiv\psi(s,t)$ is
the option price and

$\displaystyle\beta(r,\omega)$ $\displaystyle=r\sum_{i=1}^{N}\omega_{i}g_{i},$
(5)

for $\dot{\omega_{i}}=-\omega_{i}+c|\sigma|g_{i}|\psi|$ivancevic2010adaptive ,
is the adaptive market potential. The $w-$ODE is the coupling based continuous
Hebbian learning, where $c$ is the learning rate. The dot product of the
synaptic weight vector $w_{i}$ and the Gaussian kernel vector $g_{i}$ are
proportional to the adaptive market-heat-potential. The product can be
negative, positive or zero depending on the value of the synaptic weight
vector, since the Gaussian kernel vector can neither be zero nor negative. The
weight can be dynamically adjust based on various factors, including
historical data, market conditions, and external events, affecting the overall
behavior and dynamics of the market.

The bidirectional associative memory model Eq.(3) - (4) effectively engages in
quantum neural computation, offering a spatial and temporal quantum extension
of Kosko’s bidirectional associative memory (BAM) neural network
familykosko1988bidirectional . Furthermore, the characteristics of shock waves
and solitary waves in the coupled NLS equations might capture phenomena akin
to those observed in financial markets, such as the propagation of
volatility/prices, and the interaction of shock and solitary waves.

The coupled NLS-system, when $w-$learning is absent (i.e., for a constant
$\beta=r$, representing the interest rate), essentially defines the well-
established Manakov system, as established by S. Manakov in
1973manakov1974complete . This system is proven to be completely integrable
due to the existence of an infinite number of involutive integrals of motion.
It features solutions characterized as ’bright’ and ’dark’ solitons. The
Manakov system finds application in describing the interaction between wave
packets in dispersive conservative media, as well as the interaction between
orthogonally polarized components in nonlinear optical fibers, as seen in
various studies and references.

##  III Modulational Instability

We perform Modulational Instability on the coupled volatility and option price
model. The stationary solution for both the volatility and the option price

$\displaystyle\sigma$ $\displaystyle=\sqrt{n_{\sigma}}e^{-ia_{0}t},$ (6)
$\displaystyle\psi$ $\displaystyle=\sqrt{n}e^{-ia_{0}t},$ (7)

where $a_{0}=-\beta{(n_{\sigma}+n)}$ and $n,n_{\sigma}$ are the price and
volatility density, respectively. We perturb solution

$\displaystyle\sigma=(\sqrt{n_{\sigma}}+\eta)e^{-ia_{0}t},$ (8)

$\displaystyle\psi$ $\displaystyle=(\sqrt{n}+\chi)e^{-ia_{0}t}.$ (9)

Substituting Eqs.(8) and (9) into Eq.(3) and Eq.(4) respectively gives the
linearized equation for the perturbations $\eta$ and $\chi$

$\displaystyle i\frac{\partial\eta}{\partial t}$
$\displaystyle=-\frac{1}{2}\frac{\partial^{2}\eta}{\partial s^{2}}-\beta
n_{\sigma}(\eta+\eta^{*})-\beta\sqrt{n_{\sigma}n}(\chi+\chi^{*}),$ (10)
$\displaystyle i\frac{\partial\chi}{\partial t}$
$\displaystyle=-\frac{1}{2}\frac{\partial^{2}\chi}{\partial s^{2}}-\beta
n(\chi+\chi^{*})-\beta\sqrt{nn_{\sigma}}(\eta+\eta^{*}).$ (11)

We assume the ansatz $\eta=\alpha_{1}e^{i(bs-at)}+\alpha_{2}e^{-i(bs-at)}$ and
$\chi=\alpha_{3}e^{i(bs-at)}+\alpha_{4}e^{-i(bs-at)}$, for real amplitudes
$\alpha_{i}$,$i=1,2,3,4$, wavenumber $s$ and frequency $a$. Substituting the
ansatz and separating the $e^{i(bs-at)}$ and the $e^{-i(bs-at)}$ terms, we get
a system of equations $M\vec{\alpha}=\vec{0}$, where $\vec{\alpha}$ is a
vector, $(\alpha_{1},\alpha{2},\alpha_{3},\alpha_{4})^{T}$ $M$ is a matrix

$M=\begin{pmatrix}(a+\beta n_{\sigma}-\frac{b^{2}}{2})&\beta
n_{\sigma}&\beta\sqrt{n_{\sigma}n}&\beta\sqrt{n_{\sigma}n}\\\ -\beta
n_{\sigma}&(a-\beta
n_{\sigma}+\frac{b^{2}}{2})&-\beta\sqrt{n_{\sigma}n}&-\beta\sqrt{n_{\sigma}n}\\\
\beta\sqrt{nn_{\sigma}}&\beta\sqrt{nn_{\sigma}}&(a+\beta
n-\frac{b^{2}}{2})&\beta n\\\
-\beta\sqrt{nn_{\sigma}}&-\beta\sqrt{nn_{\sigma}}&-\beta n&(a-\beta
n+\frac{b^{2}}{2})\end{pmatrix}.$ (12)

The nontrivial solution of the above matrix are given by equating it’s
determinant to zero, then solving for the fraquency $a$, to get the dispersion
relation

$\displaystyle a_{1,2}=\sqrt{-T_{2}\pm\sqrt{T_{2}^{2}-T_{0}}},$ (13)

$\displaystyle a_{3,4}=-\sqrt{-T_{2}\pm\sqrt{T_{2}^{2}-T_{0}}},$ (14)

where

$\displaystyle\begin{split}T_{0}=\frac{b^{8}}{16}-\beta\frac{b^{6}}{4}\left(n+n_{\sigma}\right),\end{split}$
(15)

$\displaystyle\begin{split}T_{2}=\beta\frac{b^{2}}{2}\left(n+n_{\sigma}\right)-\frac{b^{4}}{4}.\end{split}$
(16)

The above equations, Eq.(13) and Eq.(14) can be negative or positive, real or
complex. Instability occurs when $a_{i,j}$ is imaginary, that is when the
solution of the model are solitons or solitary waves, but for real $a_{i,j}$,
the model assumes plane wave solutions. Hence $\xi=Im(a)$ is called the MI
gain of the system shown in figure 1, depending on the wavelength and the
adaptive market potential.

The only coupling term in the system of equations is the adaptive market
potential.That means that instability conditions are the same for both the
volatility and the option price (miscible)tamilthiruvalluvar2019impact .
Figure 1 and 2 shown the conditions for Instability. Even though the simple
Ivancevic option pricing model enterpreted as a simple Nonlinear Schrodinger
equation, instability occurs for $\beta>0$ dauxois2006physics
,rapti2003modulational  the coupled volatility-option price Ivancevic model is
unstable for hot market temperatures $\beta>0$ and also cold market
temperatures $\beta<0$ depending on the values for $beta$ and $b$. Figure 2
shows the MI gain for different selected values of the wavenumber $b$ which is
the same for the negative side of the same numbers since the MI gain plot is
symetric with respect to the wavenumber axis.

Figure 2, shows the MI gain for different values of $b$. For $b=0.78$ the
system is unstable for positive $\beta$, for $b=0.28$ and $b=0.18$ the system
is unstable for positive $\beta$.

Figure 1: The panels show the variation of the MI gain in the $(b-\beta)$
plane. Figure 2: The Figure shows the distribution of MI gain for different
values of $b$ (the wavenumber).

##  IV Numerical Experiment

We perform a runge-Kutta method on the coupled volatility option price model.
Time in numerical simulations is discrete and increases in timesteps
$t_{n}+\Delta t$, where at $t_{0}$ the assumed ansatz is given by
$\psi(s,0)=\sigma(s,0)=\sqrt{(}n_{0})+\epsilon cos(bs)$, where
$\epsilon=0.01$,$n_{0}=n=n_{\sigma}=10$. In the code we used $\Delta
s=\frac{L}{N}$, where $L=64\pi$, $N=2^{10}$ and time step is
$\Delta=\frac{1}{4000}$.

We can see that in this environment the condition give way to solitons, even
though we started with a modulated initial condition. In figure the figures
below the wave for $\sigma(s,t)$ and $\psi(s,t)$ are identical, which is the
result of having the same coupling term. In the figures below we observe
stable waves at $\beta=0.1$ for the $b=0.78$ case, but for $\beta=0.1$ we
observe solitons for the $b=0.28$ and the $b=0.18$ case. Otherwise we observe
solitons for $\beta=-0.15$ for the $b=0.78$ case, whereas we observe
planewaves for $\beta=-0.15$ in the $b=0.28$ and the $b=0.18$ cases.

Figure 3: The Figure shows the temporal development of the CW wave of stock
prices $s$ for $b=0.78$, $\beta=-0.15$.

Figure 4: The Figure shows the temporal development of the CW wave of stock
prices $s$ for $b=0.78$, $\beta=0.1$.

Figure 5: The Figure shows the temporal development of the CW wave of stock
prices $s$ for $b=0.28$, $\beta=-0.15$.

Figure 6: The Figure shows the temporal development of the CW wave of stock
prices $s$ for $b=0.28$, $\beta=0.1$.

Figure 7: The Figure shows the temporal development of the CW wave of stock
prices $s$ for $b=0.18$, $\beta=-0.15$.

Figure 8: The Figure shows the temporal development of the CW wave of stock
prices $s$ for $b=0.18$, $\beta=0.1$.

##  V Conclusion

we have studied the modulational instability of the coupled volatility option
price Ivancevic model both analytically and numerically. Depending on the
value of the wavelength of the pertubations, stability may occur for cold
$\beta<0$ or hot $\beta>0$ market temperatures and also direct numerical
simulations shows that the solutions are the same for both the option price
and the volatility, proving the miscibility of the coupled equation.

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