The quantum harmonic oscillator expected shortfall model

Markovic, Vladimir M.; Radivojevic, Nikola; Ivanovic, Tatjana; Radisic, Slobodan; Novakovic, Nenad Article The quantum harmonic oscillator expected shortfall model Estudios de Economía Provided in Cooperation with: Department of Economics, University of Chile Suggested Citation: Markovic, Vladimir M.; Radivojevic, Nikola; Ivanovic, Tatjana; Radisic, Slobodan; Novakovic, Nenad (2023) : The quantum harmonic oscillator expected shortfall model, Estudios de Economía, ISSN 0718-5286, Universidad de Chile, Departamento de Economía, Santiago de Chile, Vol. 50, Iss. 2, pp. 233-261 This Version is available at: https://hdl.handle.net/10419/285107 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-sa/4.0/ 233 Estudios de Economía, Vol.50 - Nº 2, Diciembre 2023. Págs 233-261 The quantum harmonic oscillator expected shortfall model* El modelo de déficit esperado basado en el oscilador armónico cuántico VLADIMIR M. MARKOVIC** NIKOLA RADIVOJEVIC*** TATJANA IVANOVIC**** SLOBODAN RADISIC***** NENAD NOVAKOVIC****** Abstract This paper presents a new Expected Shortfall (ES) model based on the Quantum Harmonic Oscillator (QHO). It is used to estimate market risk in banks and other financial institutions according to Basel III standard. Predictions of the model agree with the empirical data which displays deviations from normality. Using backtesting, it is shown that the model can be reliably used to assess market risk. Key words: Expected Shortfall; market risk; Basel III standard; stock returns; S&P index. JEL Classification: G24, C22, C52, C53. * ** *** **** ***** ****** The authors would like to thank the referees and the editor of Estudios de Economía for help, advice and useful comments which have considerably improved the manuscript. Faculty of Science, University of Kragujevac, Serbia, E-mail: vmark[email protected]. Academy at applied studies Sumadia in Kragujevac, Serbia E-mail: radivojevic034@ gmail.com. Faculty of Agriculture, University of Pristina, Serbia, E-mail:tatjana.ivanovic@ pr.ac.rs. Faculty of Technical Sciences, University of Novi Sad, Serbia, E-mail: Klajvert034@ yahoo.com. Faculty of Technical Sciences, University of Novi Sad, Serbia, e-mail: novakovic. [email protected]. Received: September, 2022 Accepted: April, 2023 234 Estudios de Economía, Vol.50 - Nº 2 Resumen Este documento presenta un nuevo modelo de déficit esperado basado en el oscilador armónico cuántico para la estimación de riesgo de bancos e instituciones financieras conforme al estándar de Basilea III. Las predicciones del modelo son consitentes con los datos del mercado accionario que presentan desvíos de normalidad. Utilizando “backtesting”, se muestral que el el modelo es fiable para la evaluación del riesgo de mercado. Palabras clave: Déficit esperado; riesgo de mercado; Basilea III; retorno accionario; S&P. Clasificación JEL: G24, C22, C52, C53. 1. INTRODUCTION Back in the 1960s, Mandelbrot (1963, 1972) and Fama (1965) showed that the series of daily returns of securities have a distribution that deviates from the normal distribution and from the identical and independent distribution assumption. Fama (1965) assumed that the distribution of price change is approximately Gaussian or normal, which was confirmed by observations. It was found that extreme tails of empirical distributions are higher than those of normal distribution, and four parameter Paretian distribution was introduced to describe data. Blattnerg and Gonedes (1977) showed that returns distributions are characterized by fat tails. They considered another family of symmetric distributions that can consider fat tails. It was Student or t distribution, and authors concluded that Student model has greater descriptive validity then the normal distribution. Kan and Zhou (2017) also presented similar findings using multivariate t distribution with 7 degrees of freedom to model stock returns. They point out that due to the presence of fat tails, the assumption of normality must be rejected. Empirical evidence of non-Gaussian properties of stock market return distribution led to the development of a lot of theoretical models on this subject. From the econophysics point of view, the Brownian movement of the classical particles was used to model the stock returns in the first place (Dragulescu and Yakovenko, 2002; Roumen, 2013; Reddy and Clinton, 2016; Agustini et al. 2018). Change of the stock price return is modelled as position change of random displacement of classical Brownian particle in these papers. The main problem with this model is that lead to Gaussian-type processes (Madan and Seneta 1990). Traditional economic models were developed to 235 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic better describe the stock return distributions (Linden, 2001; Dragulescu and Yakovenko, 2002). On the other side, the real data and empirical stock return distributions show deviations from Gaussian type distributions since Probability Density Function - PDF tails decay slower than log-normal Gaussian type (Şener et. al. 2012; Zikovic and Filer, 2013; Rossignolo, et. al. 2012, 2013, Radivojevic et al. 2016b, 2017a, 2020; Doncic et al. 2022). Fat tails which include negative skewness on one side and positive excess kurtosis on the other side of the center of distribution are the most common types of deviations from Gaussian type distribution (Ahn et al. 2017). In the market models based on statistical physics, which try to make the analogy of the stock market behavior with microsystems in physics, an important role found quantum mechanics (QM), which naturally inherent statistical fluctuations via uncertainty principle (Ataullah et al. 2009). The main problem in QM is that the potential that describes the interaction of the physical system (which is used to describe market) with the environment is generally unknown. To use QM models to describe the stock market return distributions, the appropriate potential is needed (Zhang and Huang, 2010; Haijun and Guobiao, 2015; Wróblewski, 2017). The main principle is to make an analogy between some QM system, e.g. quantum particle (or systems of particles) and stock price return. In Schrodinger’s nonrelativistic QM of closed systems, the particle is described with wave functions of particle state. Physically meaning has a square of the amplitude of wave function, which should describe the PDF of stock market returns. This is the merging point of stock market returns and the QM system: there is a need for QM system with wave function, which square can describe PDF of stock market returns. Closed quantum systems with time independent potentials lead to stationary states, so some perturbation potential needs to be introduced to enable time evolution and nonstationary. It is interesting to note that stock return distributions of stable markets tend to have Gaussian properties. In general, all markets tend to reach an equilibrium state (Balvers et al. 2000), and settle to some form of Gaussian-like distributions shape (Ahn et al. 2017). Fat tails are one of the most common deviations. Stock market returns tend to settle in some equilibrium or near-equilibrium state, which can be described as a true or local minimum of the potential energy in the physics analogy. A market can be described as some sort of physical system which is in equilibrium or near equilibrium with its surrounding. Quantum mechanical systems which are isolated can be described with the Schrodinger equation in which the parameter that need to be known is its potential energy or potential. Since the potential is unknown, some reasonable guesses need to be introduced and substantiated with some real physical assumptions (Zhang and Huang, 2010). Stock markets returns in general tend to long-run equilibrium, where returns dissipate around some mean value. It implies that a 236 Estudios de Economía, Vol.50 - Nº 2 QHO can be used to describe these oscillations, which fluctuate over time, so first order perturbation theory needs to be introduced. Hence, the aim of this study is to take advantage of this opportunity. Among the first was Bachelier (1900), who described the financial assets price movement using a random walk model, and introduced the concept of Brownian motion, which is a type of random process that has played a fundamental role in the development of modern mathematical finance. From the point of view of the current paper, random processes in economics can be transformed into the form of Schrodinger equation (Wroblewski, 2017; Ahn et al, 2017; Vukovic et al, 2015), which is a fundamental equation in Quantum physics. For instance, the famous Black-Scholes equation which gives a model to pricing theory is an instance of Schrodinger equation (Vukovic et al, 2015; Contreras et al, 2010). It was shown by Vukovic et al, 2015, that starting from Black Scholes equation, using mathematical transformations, one can get to the exact form of Schrodinger equation. Phenomena that have the same or similar mathematical foundations in different disciplines, will have same or similar physical behavior. Important property of QHO is that like every bounded quantum system it has eigenstates and discrete spectrum of energies. Hence QM oscillator can be described with one of eigenstates or superposition of eigenstates. This practically means that QHO can be described as linear combinations of eigenstates. Like classical Brownian particle, QHO in ground state is described with Gaussian distribution. Since stock markets show deviations from Gaussian (negative skewness and positive excess kurtosis) classical Brownian particle is not quite suitable for describing it. On the other hand these deviations can be very well described with higher states of QHO. Eigenstates of QHO are Hermitian polynomials, which can be even or odd. Even states lead to more symmetric distributions and can contribute to the fat tail and lead to higher kurtosis. Odd states lead to distributions with a larger skewness, (Ahn et al, 2017). The paper is organized as follows: Section 1 contains the introduction. The following section gives an overview of the most significant empirical research in the area of ES models. Section 3 presents the theoretical basis of the possibility of applying QHO for predicting the movement of stock market returns. In Section 4 presented results of applying QHO. In Section 5, the backtesting results are presented, analyzed, and discussed. Section 6 summarizes the conclusions. 237 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic 2. LITERATURE REVIEW There is an abundance of papers in literature dealing with the improvements of the applicability of different market risk models according to Basel Commitment rules. All those papers can be classified into two groups. The first group consists of the papers which try to improve applicability of different ES models. In this group of papers researchers use a traditional technique for predicting behavior patterns of assets in financial markets, following known distributions. Such papers were presented by Barone-Adesi and Giannopoulos (2001), Pascual et al. (2006), Chen et al. (2011), Brandolini and Colucci (2012), (2012), Alemany et al (2012), Bee (2012), Radivojevic et al. (2016, 2017, 2020) etc. The second group includes the papers which try to improve the applicability of completely different models for prediction stock returns. Those papers are based on artificial intelligence, data mining, machine learning, and similar concepts for assessing risks to which participants in financial markets are exposed. Such papers were presented by Scaillet (2003 and 2004), Fermanian and Scaillet (2005), Atsalakis and Valavanis (2009), Thomaidis and Dounias (2012), Aguilar-Rivera et al. (2015), Cavalcante et al. (2016), Chong et al. (2017) Xing et al. (2018), Hiransha et al. (2018), Fischer and Krauss (2018), Rundo et al. (2019), Nti et al. (2019), Shah et al. (2019), Sezer et al. (2020), Doncic et al. (2022) etc. From the second group of papers, one can single out papers that focus on opportunities of applying solutions from physics. Such papers were presented by Meng et al. (2016), Agustini et al. (2018), Maruddani and Trimono (2018) etc. Inspired by a series of studies that successfully used concepts and tools from quantum mechanics to options pricing (Ye and Huang, 2008; Baaquie, 2009; Bagarello, 2009; Zhang and Huang, 2010; Pedram, 2012 and Cotfas, 2013), Agustini et al. (2018) were use Geometric Brownian Motion model for stock prices prediction. Like them, Maruddani and Trimono (2018) used multidimensional Geometric Brownian Motion model to describe stochastic process of stock price movements. However, despite of the mathematical success of quantum-mechanics models for financial instruments, only few studies have been tried to exploit quantum statistical dynamics relying on open-sys- tem concepts yet (Meng et al. 2016). The justification for applying solutions from QM can be found in empirical findings that point to the unsustainability of the efficient market hypothesis. Empirical findings such as non-Markov- ian memory (Wan and Zhang, 2008) and fat-tail deviation (Wan and Zhang, 2008, Radivojevic et al. 2020) suggest that the stock market does not satisfy the classical Brownian motion model (Ye and Huang, 2008). And Meng, et al. (2015) were among the first to point out the possibility of describing dynamical problems in the stock market using a wave function. In this context, 238 Estudios de Economía, Vol.50 - Nº 2 Meng et al. (2016) were among the first authors who presented the idea of the possibility of applying the Brownian motion quantum oscillator model. They showed that the movement of financial asset returns can be described by the Markovian Klein-Kramers equation. However, they focused only on predicting the movement of stock prices, without considering the possibility of applying the model for assessing market risk. Original idea from QM, that the more we learn of the coordinate the less we know the momentum (and vice versa) (Cohen-Tannoudji et al. 1992), can be applied to stocks because the more we know the stock price the less information we can use to estimate the trend of it (Meng et al. 2016). In QM this is the Heisenberg uncertainty principle, which states that one cannot with certainty know position of particle and its momentum or speed. This principle has its analogy in the economy. As Ye et al, 2008 stated if all the people know the price value, even though the price has deviated, it will turn back to the value swiftly and never start to fluctuate again. In this sense, precise knowledge of stock price will harden estimates of its change (see Ye et al, 2008 for details). 3. THE THEORETICAL BASIS OF THE POSSIBILITY OF APPLYING QHO FOR PREDICTING THE MOVEMENT OF STOCK MARKET RETURNSW In case of harmonic potential QM solutions of the stationary Schrodinger equation are already known and are (1)     nnn mx xm n Hmxe                 14 2 1 2 2 / ! , where Hmx n           are Hermite polynomials. Expression in equation 1 can be simplified by introducing dimensionless variable  mx  . In physics, x is the coordinate of the observed particle. This physical quantity has its analogy in the stock market model xs≡ln , which is logarithmic stock price return, where s is stock price (Meng et al, 2016). Parameter m is the mass, while ω is the angular velocity of the quantum particle, 1 05457. Js is reduced Planck constant and n=012,, ,... is positive integer number. Angular velocity is related to the energy of the quantum particle as (2) En n      1 2   . 239 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic According to Ye and Huang, 2008 and Meng et al 2015 and 2016 mass of particle correspond to inertia of the stock, energy of the particle corresponds to trading volume of the stock, while  nx  , which is called wave function, has special properties. As it was stated earlier in the text, the wave function doesn’t have physical meaning in physics, but its square of amplitude represents the probability density of finding a particle with coordinate x. Square of amplitude of the particle corresponds to the probability density distribution of the stock price (Ye and Huang, 2008 and Meng et al 2015; Meng et al 2016). General solution of stationary Schrodinger equation for the potential of the harmonic oscillator can be constructed in form of infinite expansion over Hermite polynomials or to be more precise over eigenstate wave functions given by equation 1: (3)   xc x nnn     0. Expansion coefficients have important physical implications: its square gives the probability that the system can be found n-th state. Since stock returns change over time, we need to introduce some small potential, which acts as a perturbation on our quantum system and changes states over time. If the perturbation is small, we could expect that eigen states are not changed due to the influence of small potential, and in first-order perturbation theory we could expect that the state of the system can be described as (4) xt ct xe n k nn iEt n ,       1  . Perturbation leaves eigen states unchanged, but consequently, expansion coefficients are time dependent. Let us assume that stock return distribution can be described as the probability distribution function of the QHO. Many authors use QHO to model stock return distributions (Tingting and Yu, 2017; Jaroonchokanan and Suwannay, 2018). Since the market has a tendency toward an equilibrium state (Menga et al. 2016), with some amount of fluctuations, it is quite reasonable to assume that model of QHO is mostly in the ground state, which has Gaussian shape properties, with additional impurities of excited states, which lead to the fat tails and non-Gaussian properties. Some form of the superposition state could be a real representation of the market model (Menga et al. 2016). General state function can be represented as (5)  xc tx n k nn      1, where expansion is limited to some k-th excited state. 240 Estudios de Economía, Vol.50 - Nº 2 Let us further build our model. We can take stock returns of some real markets for three years period, represented by 750 daily stock price returns. Our goal is to use the first two-year period to build the QHO model and predict stock returns of the third year. 4. MODEL ESTIMATION At start, we take the first year of stock returns to construct initial PDE of the real market. This function needs to be fitted with the square of the QHO superposition state function in the form (6)  xcx n nn      0 90 In the expansion series, let us assume the first ten members (greater members are neglectable small, which will be seen in result part of the paper). It is important to notice that the PDE of the stock return should be fitted with the square of the function in explicit form (7)                nnnn cm n He 0 9 14 2 01 2 2  / / ! In our QHO model all parameters are uncertain, i.e. mass m and angular frequency ω . These parameters should be extracted from the data of the real market. To do so, let us first introduce function (8) fx aH xe nnn x      0 92 22 ()  which will be used to fit stock market PDE. This is necessary, since it is important to find an appropriate function which can have enough degrees of freedom, so the iteration procedure of finding fitting coefficients can lead to minimal residuals. Finding the best fit is procedure to find ten coefficients and an additional κ unknown coefficient which is in correlation with all ai coefficients. Linear regression procedure of finding best parameters starting from best guess initial values of coefficients, lead to great accuracy of fit. Here, the fitting procedure was done not over the variable x, but instead, independent variables are Hermite polynomials Hx n  . This complicates the procedure a bit since function fx  need to be minimized over terms of Hermite polynomials. To perform fit over Hermite polynomials as independent variables, a least- 247 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic The perturbating potential is unknown, and possible form of time-depen- dent function that expansion coefficients should have been undetermined. This led us to use polynomial series expansion and assume that power series consisting of a few first members will be a good approximation of general function. Each expansion coefficient is now fitted with a time-dependent function in the form of (13) Ct ABtCtD tE t n    234 Fitting functions for each expanding coefficient Ct n  are presented in Fig 5 for rolling windows of the second year of stock returns. These functions are then used to calculate the expanding coefficients off the first day of the third year period of the stock returns, which is at this point an unknown variable. This enables us to write down wave function for the unknown tomorrow data, determine PDF and predict tomorrow’s outcome with a certain probability. Predictions can be extrapolated to a future period, greater than one day, but uncertainty can be large. Instead, we can wait for the empirical value for the next day and repeat the above procedure to obtain the wave function for the following day. Since empirical data in our model are known for a whole threeyear period, we iteratively repeated the procedure to predict outcomes for the whole third year. FIGURE 5 TIME EVOLUTION OF EXPANSION COEFFICIENTS FOR THE SECOND YEAR PERIOD OF STOCK RETURNS 248 Estudios de Economía, Vol.50 - Nº 2 To validate our model, predicted mean values of expected stock rerun with limits of one standard deviation are presented in Fig 6 and compared with empirical data. To calculate the standard deviation (14)    xx 22 , mean, x and mean square values,  x2need to be calculated (15)  xx dx    *, and  xx dx 22     *, where asterix refer to complex conjugation. These operations doesn’t change real functions. Since PDF distributions are roughly Gaussian shape distributions, the interval of one standard deviation covers approximately 68% of the whole range. It is expected that two-thirds of the whole data fall into the presented region in Fig 6. Newer the less only 4% of data is outside of the region. Sock return distributions for data used in the presented model have negative skewness and fat left tail and distributions are asymmetric around the mean value. This is the reason why more data are below the mean curve in Fig 6. FIGURE 6 STOCK RETURN DATA IN THIRD YEAR – DOTS; PREDICTED MEAN – FULL LINE; STANDARD DEVIATION INTERVAL AROUND MEAN – DASHED LINE; FOR THE PERIOD BETWEEN JANUARY 2ND, 1997 TO DECEMBER 30TH, 1999 249 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic A powerful tool in Quantum Mechanics are selection rules that can be derived from wave functions, determine possible transitions between quantum states and give transition probabilities. This could limit the possible outcomes and better predict tomorrow’s stock returns. For the derivation of selection rules, an exact form of perturbation potential is needed. At this point, the best-predicted value of tomorrow’s outcome can be obtained using the traditional formula (16) PP tt  1  where Pt+1 is tomorrow’s stock return, Pt is stock return at present day µ is the quantile that gives confidence interval (taken to be unit) and σ is the standard deviation calculated using the predicted wave function for tomorrow outcome. Eq (16) has great importance, since implies that tomorrow’s outcome is directly related to today’s stock return values. On Figure 7 are presented predicted outcomes given with full line, while empirical tomorrow’s outcome is plotted with gray line. It is of great importance to notice that outcomes do not falls out of the predicted interval. FIGURE 7 PREDICTED OUTCOME OF STOCK RETURN – FULL BLACK LINE; EMPIRICAL STOCK RETURN – FULL GRAY LINE; FOR THE PERIOD BETWEEN JANUARY 2ND, 1997 TO DECEMBER 30TH, 1999 250 Estudios de Economía, Vol.50 - Nº 2 6. BACKTESTING MODEL ACCORDING TO BASEL III STANDARDS The obtained data were used to estimate the Expected Shortfalls (ES) according to Basel III standards (Bank for International Settlements, 2013). More precisely, the ES are calculated for the one-day-ahead horizon for the period from January 1st, 1998, to January 1st, 1999, according to the Basel III standard, and for the period from January 1st, 2008 to January 1st, 2009, which was during an economic crisis, Figure 8. The ES estimates were made for the confidence levels of 97.5%. Since VaR does not fulfil all the characteristics of coherent risk measures, the Basel Committee has proposed fundamental changes in the regulatory treatment of financial institutions’ trading book positions (Kellner and Rösch, 2016). Among other things, the replacement of 99% VaR with the 97.5% expected shortfall (ES) for the quantification of market risk is recommended (Radivojevic et al., 2019; Doncic et al., 2022). The rest of the observations were used as the resample observations needed for the ES starting values. At the same time, to answer the question of whether the model contributes to the improvement of risk assessment, i.e., whether the model gives better results compared to traditional risk models, the performance of the model was compared with FIGURE 8 PREDICTED OUTCOME OF STOCK RETURN – FULL BLACK LINE; EMPIRICAL STOCK RETURN – FULL GRAY LINE, FOR THE CRISIS PERIOD FROM JANUARY 1ST, 2008 TO JANUARY 1ST, 2009 251 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic the performance of three ES models: GARCH models under the assumption that innovations follow the Student’s T, GED and Skewness GED distributions. Models were chosen given in the mind results of studies conducted by Radivojevic et al. (2015) and Rossignolo et al. (2013, 2012). The maximum likelihood of the estimated parameters of the GARCH models are given in Table 2. More precisely, in the first part of the table, the estimated parameters of the GARCH models for the period January 1st, 1998, to January 1st, 1999, are given, while in the second part of the Table 2 the estimated parameters of the GARCH models during the period of the economic crisis in 2008 are given. TABLE 2 THE ESTIMATES OF THE PARAMETERS OF APPROPRIATE GARCH(1,1) MODEL DURING REGULAR MARKET CONDITIONS Type of GARCH model GARCH(1,1) with Student's t GARCH(1,1) with GED GARCH(1,1) with Skewness GED 0.066 (0.004) 0.082 (0.001) 0.081 (0.003) 0.873 (0.000) 0.851 (0.000) 0.860 (0.000) 0.000 (0.023) 0.000 (0.025) 0.000 (0.027) -0.113 (0.045) η7.754 (0.000) 1.479 (0.000) 1.515 (0.000) Loglikelihood 2322.90 2318.08 2320.70 During conditions of the crisis Type of GARCH model GARCH(1,1) with Student's t GARCH(1,1) with GED GARCH(1,1) with Skewness GED 0.114 (0.000) 0.107 (0.000) 0.105 (0.000) 0.899 (0.000) 0.895 (0.000) 0.895 (0.000) 0.000 (0.226) 0.000 (0.137) 0.000 (0.133) -0.116 (0.240) η4.610 (0.000) 1.166 (0.000) 1.180 (0.000) Loglikelihood 2378.93 2381.70 2385.40 Presented models did not produce any ES breaks, which implies that the models potentially can be reliably used to assess market risk according to the requirements of the Basel III standard. However, this conclusion can only be made based on backtesting. Unlike VaR backtesting, ES backtesting is signifi- Notes: p-values are given in parentheses 252 Estudios de Economía, Vol.50 - Nº 2 cantly more complex (Doncic et al. 2022). This is the reason why the Basel III standard is not the prescribed manner of backtesting the validity of ES assessments. For that purpose, the in this paper we used two tests Berkowitz’s test (LRB) (2001) and Acerbi and Szekely’s (2014) first method (Z1). Berkowitz (2001) presented a test based on the Levy Rosenblatt transformation that can be mathematically presented as follows: (17) LR LL BMLML        20 1 22 ln ,ln,   . , where LRB is the Berkowitz’s likelihood ratio. Berkowitz’s ES back test is the test that tests a joint hypothesis of zero mean ( µ ) and unit variance ( σ 2 ) , while ( ˆ ) µ ML and ( ˆ σ ML ) are ( µ ) and () σ 2 estimates obtained using maximum likelihood. The LRB test is asymptotically distributed as χ2 with two degrees of freedom. Berkowitz’s test compares the shape of the forecasted tail of density to the observed tail. Any observations that did not fall within the tail were truncated, noting that the threshold was defined as follows: TH maxESE SE St   12 ,, . Since the Berkowitz test validity is disputed in the case of a relatively small number of exceedances, one of the authors in (Radivojevic et al. 2019) proposed the use of bootstrap simulation, where F is the unknown distribution of the estimator θ k . Thus, Berkowitz’s ES backtesting based on bootstrap simulation as presented (Radivojevic et al., 2019) was used in the paper. In fact, the estimation of the unknown density F of our ES estimates was used by repeating the simulations by the appropriate models several times1. The number of bootstrap repetitions is determined according to the Andrews and Buchinsky procedure (Andrews and Buchinsky, 1997). Determining the bootstrap repetitions number is particularly important in this case because the sample of the breaks utilized in obtaining a single ES estimate is a small fraction of the number of draws. The procedure for calculating the p-value is then continued by analogy, as previously described. The results are given in Table 3. Given the limitations of Berkowitz’s test, Acerbi and Szekely’s first method was also used to test the model’s validity. Acerbi and Szekely defined the null hypothesis: HP F t a t a 0:    for   t against the alternatives HESXES X tt 1: ,.      fortand forsomet    VaRXVa RX tt  ,.     1 According to the bootstrap method (Efron and Tibshirani, 1993), we generated multiple new samples from the data sample and calculated the value of the estimator θ k in each sample. The size of the data-sample, of the exceedances, is known as it is a direct function of the number of trials in the bootstrap simulation and the probability level used in defining the ES. We have chosen a level of error PDB equal to 10% and a confidence level equal to 95%, the initial value of bootstrap repetitions, initial excess kurtosis of the sample of ES repetitions set to zero. 253 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic fo rt   wherein Ft is the realized distribution of returns, Pt a  is the conditional distribution tail of the distribution of Pt below the quantile α . We can write this as Px Px a t a t      min ,/ .1 ES X t  ,   and Va RX t  ,   are the sample ES and VaR from the realized returns. Under the null hypothesis, the realized tail is assumed to be the same as the predicted tail of the return distribution. The alternative hypothesis rejects the ES without rejecting VaR. To test the null hypothesis, Acerbi and Szekely defined the following test statistics: (18) ZXI ES N t T tt at t 11X   (/ ) , where X denotes the vector of realized returns (X1, X2,…,XT), It – the indicator function ItRVaR R Ta    1 () that indicates the backtesting exceedance of VaR for the realized return Xt in the period t, and NI Tt T t  1 is the number of the exceedances. The simulations from the distribution under H0 were used to test for significance in the above method. More precisely, we followed the steps below: 1) simulate Xt i from Pt for all t and i = 1,2…, M; where M is a suitably large number of scenarios. 2) for every i, compute ZZX ii   ,� i.e., compute the value of Z1 using the simulations from the first step; 3) estimate the p-value as pZZx M i Mi     1 , where Z(x) is the observed value on Z1. 4) we conducted the test for a confidence level of 95%. The results of this test are shown in Table 3. ˆ) µ ML ˆ σ ML 254 Estudios de Economía, Vol.50 - Nº 2 The p-values were obtained by applying 10.000 simulations. The test was conducted for a confidence level of 95%. According to the results shown in Table 3, it can be concluded that all models successfully passed both tests. Interestingly, no cluster of ES breaks was recorded in any simulations. To answer the question of whether the model contributes to improving the risk assessment, the root mean-squared error ( RMSE RES i ii   1 255 22 255 ) was used to compare the model performances with the performances of selected models. RMSE results are given also in Table 3. Based on the RMSE results, it can be clearly seen that the model generates smaller deviations, which means smaller capital burdens for banks. Hence, it can be concluded that the model contributes to the improvement of the traditional ES model. These results were obtained under regular market conditions. In the conditions of the crisis, the results are shown in the second part of Table 3. The results, also show that the model generates better risk assessments. This means that the model contributes to the improvement of traditional models and conditions of high volatility. A comparison of models has been also performed in the context of the RMSE of the first four moments of the distribution. The results are given in Table 4 . The results show that the model produces better risk estimates in all four moments of the distribution compared to traditional models. According to Radivojevic et al. (2019), it is clear that in the case of the GARCH model, the assumption of innovations distribution is more important than model specification. In other words, the assumption that is more compatible with real conditions leads to a better-performing model. Since the Student t distribution has a higher degree of QHO GARCH(1,1) with Student's t GARCH(1,1) with GED GARCH(1,1) with Skewness GED LRB0.123 0.210 0.119 0.301 0.144 0.172 0.144 0.106 RMSE 0.038 0.052 0.046 0.042 Backtesting results during conditions of the crisis QHO GARCH(1,1) with Student's t GARCH(1,1) with GED GARCH(1,1) with Skewness GED LRB0.154 0.056 0.211 0.177 0.122 0.098 0.381 0.428 RMSE 0.075 0.156 0.107 0.082 TABLE 3 BACKTESTING RESULTS DURING REGULAR MARKET CONDITIONS Z1 Z1 255 The quantum harmonic oscillator... / Markovic, Radivojevic, Ivanovic, Radisic, Novakovic freedom parameter than the GED distribution, it was expected to better capture the kurtosis of the return’s distribution, especially in crisis conditions, which means that the GARCH(1,1)-Student t model is better equipped to handle extreme events in the data. On the other hand, the GED distribution is better suited for modeling skewness because it has a flexible shape that can be skewed in either direction. This is because the Student t distribution has a symmetric shape, whereas the GED distribution allows for skewness. In situations where the data exhibits skewness, the GARCH(1,1)-GED model may provide better estimates of the dispersion of stock returns. However, theoretical distributions are not fully able to capture empirical phenomena. As the number of extreme cases increases, different variants of Garch models produce larger deviations. They are less able to predict the probability of extreme returns occurring, as well as the magnitude of these deviations. However, it is characteristic of the used variants of the Garch model that it is not possible to make a universal conclusion about which variant is better from the aspect of smaller deviation in moments of the distribution. It is evident that their performance weakens in crisis conditions, but a general conclusion cannot be drawn about whose performance will weaken the most. On the other hand, in the case of the QHO model, the finding showed that it better captures the occurrence of extreme outliers, as well as the probability of their occurrence. This is from reasons because the quantum oscillator provides information about the current trend and momentum of the security. TABLE 4 THE RESULTS OF COMPARISON OF MODELS IN TERMS OF THE RMSE OF THE FIRST FOUR MOMENTS OF THE DISTRIBUTION QHO S&P Garch(1,1)- student t S&P Garch(1,1)- GED S&P Garch(1,1)- Skewed GED Mean 5.25E-05 1.89E-04 5.47E-05 1.30E-04 Standard Deviation 0.007 0.004 0.011 0.011 Kurtosis 1.104 1.652 1.726 2.080 Skewness 0.801 0.914 0.906 0.924 Conditions of crisis QHO S&P Garch(1,1)- student t S&P Garch(1,1)- GED S&P Garch(1,1)- Skewed GED Mean 5.37E-05 1.99E-04 5.50E-05 1.40E-04 Standard Deviation 0.011 0.011 0.011 0.011 Kurtosis 1.297 1.865 1.977 2.291 Skewness 0.920 0.921 0.947 0.929 256 Estudios de Economía, Vol.50 - Nº 2 7. CONCLUSION First order of time dependent perturbation theory is applied on QHO model of stock market returns for the market with non-Gaussian properties. Semiempirical approach is introduced, since perturbating potential is unknown, to obtain series of time dependent coefficients of QHO wave functions. Fitting procedure over Hermitian polynomial as the independent variables is introduced and enables good fit of empirical data within all second year period of stock returns. In summary, this method enables prediction of tomorrow’s outcomes of stock return. Real tomorrow’s outcomes show no fallout from predicted ranges within confidence interval of one standard deviation. In the context of meeting the model validation rules defined by the Basel III standard, the model was tested using Berkowitz’s ES backtesting based on bootstrap simulation and Acerbi and Szekely’s first method. The model provided satisfactory results. As not only the number of exceedances but also the size of the loss is relevant for the bank, it is important to allow for this criterion when comparing the models. Unfortunately, due to the scope of the work, no such comparison was made with other ES models. For the results to be comparable, for this reason, the data used by Christoffersen were taken. Model presented in the present paper uses Harmonic oscillator potential, which tends to return the position of the particle towards equilibrium state. In economic terms this implies that stock-markets have the ability of self-correc- tion of stock market returns toward equilibrium. At first glance, it seems that the model can be applied onto markets which are autocorrelated. This would be true for the classical model of the harmonic oscillator, where simple random movement around equilibrium position is described. Considering QHO, this problem is removed, since arbitrary deviation from equilibrium state can be described, even oscilations around new equilibrium state. This can be done by taking into account superposition quantum states, where displacements from equilibrium, which corresponds to pure ground state of QHO can be described with higher quantum states of the particle. Higher QM states are described with higher members of the wave function, i.e. higher orders of Hermite polynomial members in equation 3. For very unstable markets, where stock returns have a long-term tendency of increasing or decreasing, higher order polynomial members will have greater contribution, and larger values of coefficients in Table 1. From a theoretical point of view any stock return distribution can be expanded over infinite numbers of Hermite polynomial members. The only practical question is how fast convergence will occur, and after which order of member series infinite expansion can be truncated. For the crisis period, it was shown that taking into account the first 10 members, stock returns can be well modeled.