Calibration of Quantum harmonic oscillator as a stock return distribution model on the Index of NSEI
Atman Bhatt ^{1} , Ravi Gor ^{2}
^{1}^{ }Research
Scholar, Department of Applied Mathematical Science, Actuarial Science and Analytics,
Gujarat University, Ahmedabad, India
^{2}^{ }Department
of Applied Mathematical Science, Actuarial Science and Analytics, Gujarat
University, Ahmedabad, India

ABSTRACT 

Stock returns
have a mixed distribution, which describes Gaussian and nonGaussian
characteristics of the stock return distribution, according to the solution
of the Schrodinger equation for the quantum harmonic oscillator. As a model
for the market force, A quantum harmonic oscillator which uses a stock return
from shortrun oscillations to longrun equilibrium will be suggested. We
will calculate fitting errors and goodness of fit statistics by analysing the AllShare Index of the National Stock
Exchange of India. 

Received 15 October 2022 Accepted 15 November 2022 Published 01 December 2022 Corresponding Author Atman
Bhatt, bhattatman31794@gmail.com DOI 10.29121/IJOEST.v6.i6.2022.423 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or notforprofit sectors. Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CCBY, authors retain the copyright, allowing anyone to download,
reuse, reprint, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. 

Keywords: Quantum
Harmonic Oscillator, Gaussian Distribution, NonGaussian Properties, EigenState, EigenEnergy, Angular Frequency, Schrodinger
Equation, Stock Return Distribution 
1. INTRODUCTION
Standard stochastic process models do not match the results of empirical evidence. Instead of standard financial models, this paper calibrates stock distribution using quantum harmonic oscillator model because the stochastic dynamics of stock return can be studied using quantum models, which can also be used to define its statistical characteristics. The inclusion of market conditions gives quantum models an edge over conventional stochastic models. Quantum harmonic oscillator is the quantum counterpart of the classical harmonic oscillator. It is one of the most crucial model systems in quantum mechanics because it may frequently be represented as a harmonic potential at a stable equilibrium point. It is also one of the few quantummechanical systems that has a precise, analytical solution and the market force that pulls a stock return from shortterm oscillations to the longterm equilibrium can also be captured by the quantum harmonic oscillator with ease.
2. Literature Review
Grabert et al. (1988) proposed generalisation of the FeynmanVernon impact functional by applying functional integral methods to the quantum mechanical dynamics of a particle connected to a heat bath. The time evolution of equilibrium correlation functions and nonfactorizing beginning states is described by the extended theory. Exact solvable models shed light on the theory.
Ye and Huang (2008) suggested a nonclassical oscillator model based on Quantum Mechanics. He studied fluctuations of stock markets. Since the same stock has a price range rather than a set price at different times, it is expected that the value might be a packet of trends that determines the possibilities of each price.
Meng et al. (2015) use a quantum spatialperiodic harmonic model to examine the stock market behaviours of equities in daily pricelimited stock markets. The effectiveness of price limits is reconsidered, and quantum model is employed to study several observable characteristics of China's pricelimited stock markets.
Meng et al. (2016) built an econophysical ‘outline for the stock market using the
physical ideas and mathematical constructions of quantum mechanics. Using this
framework, he analogously mapped a large number of
individual stocks into a reservoir made up of numerous quantum harmonic
oscillators, and their stock index into a typical quantum open system, or
quantum Brownian particle.
Ahn et al. (2018) developed a quantum harmonic oscillator as a model for the market force that pulls a stock return from shortrun oscillations to the longrun equilibrium. Additionally, using analogies, they established an economic justification for physics notions like the eigenstate, eigenenergy, and angular frequency, which clarifies the connection between the literature on finance and econophysics.
JeknićDugić et al. (2018) pursued the quantummechanical challenge to the efficient market hypothesis for the stock market by employing the quantum Brownian motion model. He also introduced the external harmonic field for the Brownian particle and use the quantum CaldeiraLeggett master equation as a potential phenomenological model for the stock market price fluctuations.
Lee et al. (2020) examined the weakform efficient market hypothesis of the crude palm oil market by adopting the quantum harmonic oscillator. This method permits Lee to analyse market efficiency by approximating one constraint: the probability of finding the market in a ground state where conclusion established that the crude palm oil market is more efficient than the West Texas Intermediate crude oil market.
Orrell (2020) addressed issues regarding intrinsically uncertain demand by consuming a quantum context to model supply and demand as, not a cross, but a probabilistic wave, with an allied entropic force. The approach is used to derive from first principles a technique for modelling asset price changes using a quantum harmonic oscillator, that has been previously used and empirically tested in quantum finance. The method is established for a simple system and claims in other areas of economics are discussed.
Bhatt and Gor (2022) showcased an interesting structure of Risk Neutral system. They also examine single step and multistep quantum binomial option pricing model. This approach elaborates circuit proposed by A. Meyer. Bhatt and Gor (2022) review applications of quantum harmonic oscillator model in financial mathematics and also discussed about different applications of quantum harmonic oscillator and its characteristics.
3. Data Collection
To calibrate quantum harmonic oscillator model; daily, weekly, and monthly dataset is implemented from Nifty Index of India, spanning 1^{st} January 2021 to 1^{st} January 2022 from yahoofinance.
4. Methodology
Firstly, stock returns for different holding periods are calculated and also some insights about the data were found with the help of statistical software Jamovi.
4.1. Summary statistics of stock
returns for different holding periods
=1,5 and 20 holding periods Table 1
Table 1
Table 1 Data Exploration 

Summary
Statistics of Stock Returns for Different Holding Periods 

Holding
Period 
1 
5 
20 
Number
of Observation 
248 
53 
12 
Mean 
0.101 
0.0956 
0.110 
Median 
0.134 
0.0669 
0.0545 
Standard
deviation 
1.09 
0.452 
0.190 
Variance 
1.18 
0.205 
0.0359 
Minimum 
4.21 
1.19 
0.217 
Maximum 
5.08 
1.65 
0.456 
Skewness 
0.248 
0.404 
0.437 
Std.
error skewness 
0.155 
0.327 
0.637 
Kurtosis 
2.76 
2.57 
0.0349 
Std.
error kurtosis 
0.308 
0.644 
1.23 
4.2. Parameters of Quantum Harmonic
Oscillator
Wiener process is considered to introduce the probability distribution function which is obtained by the FokkerPlanck Equation. Diffusion coefficient is considered as half of the value of the variance which is used to solve the FP Equation and the mass of the Schrodinger equation.
This whole process is supposed to be a timeindependent process. Stationary density function is found with the help of timeindependent potential and normalization constant.
This all observations are done with fixing value of speed for mean reversion which is equal to. After that angular frequency is calculated which gives insights about from which point the price for a security begins to increase. Five different states are defined considering different five eigenvalues which are derived by the obtained solution of the FP equation.
Probabilities of respective eigenvalues are obtained with the normalization factor which shows the probability of a stock return residing theeigenstate.
Probability density function of the calibrated model is treated as a product of the eigenvalues and its density function. Same process is followed for different holding periods (i.e., 1, 5 and 20 Days).
Calculated values for the abovementioned parameters of quantum harmonic oscillator and its probability density function are given below. Table 2, Table 3, Table 4.
Table 2
Table 2 Daily Observations 

Holding Period 
1 

Mass 


Number of observations 
248 

angular frequency 


Factor 
0.0054 

Diffusion Coefficient 
0.59 

Harmonic Potential 
0.00009077 

Planck Constant 


State 
0 
1 
2 
3 
4 
Energy 
0.005431 
0.016294 
0.027157 
0.03802 
0.048883 
Hermite Polynomial 
1 
0.06106 
1.98509 
0.7309 
11.82126 
Amplitude 
10.42247 
18.05224 
23.30534 
27.57525 
31.2674 
Eigen Function 
0.232652 
0.010045 
0.11546 
0.02454 
0.140348 
Eigen Value 
0.561134 
0.67982 
0.434081 
0.207416 
0.082253 
Probability 
0.310144 
0.455218 
0.185598 
0.042375 
0.006664 
Density Function 
0.999906 
0.061054 
1.9849 
0.73083 
11.82015 
Probability Density Function 
0.561081 
0.041506 
0.86161 
0.15159 
0.972242 
Table 3
Table 3 Weekly Observations 

Holding Period 
5 

Mass 


Number of observations 
53 

angular frequency 


Factor 
0.0023 

Diffusion Coefficient 
0.1025 

Harmonic Potential 
0.002263846 

Planck Constant 


State 
0 
1 
2 
3 
4 
Energy 
0.0022638 
0.006792 
0.011319 
0.01585 
0.02037 
Hermite Polynomial 
1 
0.028415 
1.999192 
1.147177 
11.99031 
Amplitude 
6.728813 
11.65465 
15.046083 
17.80277 
20.18644 
Eigen Function 
0.289534 
0.00582 
0.204649 
0.047941 
0.17716 
Eigen Value 
0.55784 
0.66792 
0.421490 
0.19904 
0.07801 
Probability 
0.317324 
0.45492 
0.181157 
0.040399 
0.006205 
Density Function 
0.999798 
0.02841 
1.998789 
1.146946 
11.9879 
Probability Density Function 
0.557727 
0.01898 
0.842466 
0.228289 
0.93515 
Table 4
Table 4 Monthly Observations 

Holding
Period 
12 

Mass 


Number
of observations 
12 

angular frequency 


Factor 
0.0009474 

Diffusion
Coefficient 
0.01795 

Harmonic
Potential 
0.000011463 

Planck
Constant 


State 
0 
1 
2 
3 
4 
Energy

0.00095 
0.00284 
0.00474 
0.00663 
0.00853 
Hermite
Polynomial 
1.00000 
0.05054 
1.99745 
0.60637 
11.87739 
Amplitude 
4.35285 
7.53936 
9.73327 
11.51656 
13.05855 
Eigen
Function 
0.35990 
0.01286 
0.25417 
0.03150 
0.21814 
Eigen
Value 
0.55781 
0.66782 
0.42138 
0.17203 
0.06082 
Probability 
0.32144 
0.46073 
0.18344 
0.03057 
0.00382 
Density
Function 
0.99936 
0.05051 
1.99617 
0.60598 
11.86981 
Probability
Density Function 
0.55746 
0.03373 
0.84116 
0.10425 
0.72194 
4.3. Using Goodness of fit test for
Parameter estimation of Quantum Harmonic Oscillator
The Cramer–von Mises criterion is a criterion used for adjudicating the goodness of fit of a cumulative distribution function. For goodness of fit test, mean and standard deviation are used to find the value of ttest where mean is summation of products of eigenvalues and its probabilities respectively.
Pvalues for the given holding periods are zeroes which indicates the curve fitting for the historical data taken. Table 5
Table 5
Table 5 Cramer Von Mises Test 

Number of observations 
248 
53 
12 
Holding Period 
1 
5 
20 
Mean 
0.101 
0.0956 
0.110 
Standard deviation 
1.09 
0.452 
0.190 
Quantum Harmonic Oscillator 

Mean 
0.114680 
0.113148 
0.113955 
Variance 
0.016717 
0.016385 
0.016973 
Standard deviation 
0.129294 
0.128003 
0.130280 
PDF 
0.561635 
0.441095 
0.367726 
Cramer Von Mises Goodness of fit Test 

T 
12.7637 
2.11028 
4.93087 
Standard deviation 
0.129294 
0.128003 
0.130280 
zstat 
85.6215 
14.1562 
33.0773 
To compare the
fitting results of the Cramer Von Mises goodness of fit statistic, following
ways are calculated: First, to determine the 5th, 10th, and up to 100th
percentiles of returns. The actual number of
empirical returns were evaluated, falling between the 5 and
the 5 percentiles and evaluated by:
where
is mean return
falling between the fixed percentiles. Here degree of freedom for the
respective model is 14 and in the case of quantum harmonic oscillator all the
three pvalues of daily, weekly, and monthly data are observed to be greater
than 0.05 which is calculated by the found value of zstat. So, the null
hypothesis that the data comes from the distribution of the quantum harmonic
oscillator model cannot be rejected.
4.4. Graphical representation of Goodness of fit Test
Graphical representations of the goodness of fit test are constructed with MS Excel where annual logreturn is used to find respective bins and data. It is distributed normally with the mean and standard deviation derived from quantum harmonic oscillator model. Figure 1, Figure 2, Figure 3.
Figure 1
Figure 1 Daily Observations 
Figure 2
Figure 2 Weekly Observations 
Figure 3
Figure 3 Monthly Observations 
5. Conclusion
The results derived from the quantum harmonic oscillator model indicates a modest but distinct possibility of observing the particle in the traditionally forbidden zone. Quantum harmonic oscillator gives many insights about stock listing of NIFTY50 like diffusion coefficient, Amplitude, Harmonic potential which actually helps to find the Eigen values and its probabilities. Mean value of the quantum harmonic oscillator is nearer to the mean value of the stock return. Prior is the reason for which QHO gives identical values for CramerVon Mises Goodness of fit Test and it can be observed from the graphs of different holding periods.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
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