Development of Instrumental Approaches to Forecasting the Volatility of the Return of Financial Assets Текст научной статьи по специальности «Экономика и бизнес»

Research article

DOI: https://doi.org/10.48554/SDEE.2023.2.1

Development of Instrumental Approaches to Forecasting the Volatility of the

Return of Financial Assets

John Guyomey1*, Andrey Zaitsev2

1 Tamale Technical University, Tamale, Ghana, [email protected]

2 Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russian Federation, [email protected] * Corresponding author: [email protected]

Abstract

Measurement and forecasting of volatility and income correlation are achieved by non-parametric methods using high-frequency price data. Due to accurate calculations of conditional volatility and correlation forecasting, it is possible to correctly identify financial derivatives and make risk decisions and relative asset allocation decisions. This article systematises the methods for modelling the volatility of financial asset returns, considers the theoretical foundations of the generalised autoregressive conditional heteroscedasticity model, and predicts and analyses the volatility of US stock indices and stocks using high-frequency volatility estimates (realised volatility indicators). The stock indices studied are the Dow Jones Industrial Average (DJI), Standard and Poor's 500 (SP500), and the Nasdaq Composite Index (NASDAQCOMP). Stocks analysed include stocks in Microsoft, Bank of America, and Coca-Cola. The results of the study support conclusions regarding the effectiveness of volatility estimators within two Bank of America volatility forecasting models, the superiority of the HAR-RV model for trading options in a specific market, and the best model for Microsoft. Thus, systematic analysis of news information is useful for predicting the volatility of returns on financial assets, but its effectiveness depends on the individual company. Future studies should explore the usefulness of the systematic analysis of news information in predicting the volatility of returns on financial assets in other markets and for other asset classes.

Keywords: volatility, high-frequency volatility estimates, modelling and forecasting methods, yield volatility, financial assets, stock indices, US stocks, information environment

Citation: Guyomey, J., Zaytsev, A., 2023. Development of Instrumental Approaches to Forecasting the Volatility of the Return of Financial Assets. Sustainable Development and Engineering Economics 2, 1. https://doi.org/10.48554/SDEE.2023.2.1

This work is licensed under a CC BY-NC 4.0

© Guyomey, J., Zaytsev, A., 2023. Published by Peter the Great St. Petersburg Polytechnic University

8

Economics of engineering decisions as a part of sustainable development

Научная статья УДК 519.862.6

DOI: https://doi.org/10.48554/SDEE.2023.2.1

Развитие Инструментальных Подходов к Прогнозированию Волатильности

Доходности Финансовых Активов

Джон Гайомей1*, Андрей Зайцев2

1 Технический университет Тамале, Тамале, Гана, [email protected]

2 Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Российская Федерация, [email protected]

* Автор, ответственный за переписку: [email protected]

Аннотация

Измерение и прогнозирование волатильности и корреляции доходности активов осуществляется непараметрическими методами, для которых используются высокочастотные ценовые данные. Благодаря точным моделям прогнозирования условной волатильности и корреляции возможно корректное определение производных финансовых инструментов, управление рисками и принятие решений относительно распределения активов. В данной статье проведена систематизация методов моделирования волатильности доходности финансовых активов, рассмотрены теоретические основы общей модели GARCH, а также спрогнозированы и проанализировали волатильность фондовых индексов и акций США при помощи высокочастотных оценок волатильности (показатели реализованной волатильности). Изучаемыми фондовыми индексами являются индексы Dow Jones Industrial Average (DJI), Standard and Poor's 500 (SP500) и Nasdaq Composite Index (NASDAQCOMP). Акции, с другой стороны, включают акции Microsoft, Bank of America и Coca-Cola. Результатами исследования стали выводы касательно эффективности оценщиков волатильности в рамках двух моделей прогнозирования волатильности акций Bank of America, превосходство HAR-RV модели для торговли опционов определенного рынка, найден наилучшая модель для акций Microsoft. В свете приведенных выше результатов был сделан вывод о том, что систематический анализ новостной информации полезен для прогнозирования волатильности доходности финансовых активов, однако его эффективности зависит от конкретной компании. Было рекомендовано, чтобы в будущих исследованиях изучалась полезность систематического анализа новостной информации для прогнозирования волатильности доходности финансовых активов на других рынках и для других классов активов.

Ключевые слова: волатильность, высокочастотные оценки волатильности, методы моделирования и прогнозирования, волатильность доходности, финансовые активы, фондовые индексы, акции США, информационная среда.

Цитирование: Гайомей, Д., Зайцев, А., 2023. Развитие Инструментальных Подходов к Прогнозированию Волатильности Доходности Финансовых Активов. Sustainable Development and Engineering Economics 2, 1. https://doi.org/10.48554/SDEE.2023.2.1

Эта работа распространяется под лицензией CC BY-NC 4.0

© Гайомей, Д., Зайцев, А., 2023. Издатель: Санкт-Петербургский политехнический университет Петра Великого

Экономика инженерных решений как часть устойчивого развития

9

1. Introduction

The trend towards measuring and predicting volatility and correlation of asset returns using non-parametric methods has two directions for study: continued research and development of methods for using volatility information in high-frequency data, and modelling and forecasting volatility in a multidimensional environment that is relevant to practical financial economics. The realised volatility approach, which is the most obvious nonparametric measure of volatility, deals with both directions. The benefit is the lack of modelling of intraday observations of returns, which allows for the accurate measurement and prediction of volatility in multidimensional environments. The inherent problem with modelling and forecasting conditional volatility turns is that volatility is unobservable—what is found— modelling what should be indirect. The problem of lagging returns volatility is frequently addressed by inferring volatility based on characteristic assumption parameters, using, for example, autoregressive conditional heteroscedasticity-generalised autoregressive conditional heteroscedasticity "ARCH-GARCH" or a stochastic volatility model. Therefore, it is necessary to develop alternative approaches that would allow for expanding the available price indicators with additional data available for prompt receipt and research. Such data are indicators of the news, information, and digital components that have an impact on financial market participants (Kulakov, 2004a; Kulakov, 2004b).

The main aim of this study is to develop instrumental approaches to forecasting the volatility of return on financial assets. In common parlance, volatility refers to fluctuations observed in a phenomenon over time. This is the change in the results of an uncertain variable, such as the return on assets. Volatility is a statistical measure of the degree to which a trading price changes over a given period. In terms of returns on financial assets, volatility is the standard deviation of returns on investments on an annualised basis. Alexander (2008) defined volatility as "an annual measure of dispersion in a stochastic process that is used to model logarithmic returns". Andersen et al. (2010) classified approaches to the empirical quantification of volatility into two categories: procedures based on the evaluation of parametric models (parametric measurement and volatility modelling), and direct non-parametric measurements (non-parametric measurement and volatility modelling).

Table 1 summarises the various methods for empirically quantifying return volatility within parametric and non-parametric approaches.

Table 1. Approaches to measuring and modelling the volatility of returns on financial assets

Source: compiled by the authors

Parametric measurement and volatility modelling Non-parametric measurement and volatility modelling

Discrete-time parametric volatility models Continuous time parametric volatility models Implied volatility based on a parametric model Measures of instantaneous volatility, h ^ 0: 1. ARCH filters and smoothers (ARCH filters and smoothers are used to measure instantaneous volatility; filters only use information up to time a] .t = t while smoothers are based on t > t. Implemented measures of volatility, h > 0 1. The implemented volatility methods directly measure conditional volatility over fixed time intervals. 2. They can be classified based on whether the notional volatility measurement uses only the price data contained within the [t-h,t] interval itself, or whether filtering/smoothing techniques are used to also include returns outside [t-h,t]. 3. The most obvious non-parametric measure of volatility is the "ex-post return squared spanning the time interval [t-h,t], that is, measures of realised volatility".

Methods for parametric measurement of volatility

Volatility models in continuous time Discrete Time Models

Continuous diffusion along the sampling path 1. Invariant diffusion in time 2. Ornstein-Uhlenbeck (OU) and Cox-In-gersoll-Ross (1985) processes (CIR) 3. Square root volatility model Hopping diffusions and processes controlled by the levy ARCH-GARCH models 1. ARCH (m) (Engle, 1986) 2. GARCH (1,1) (Engle, 1986) 3. GARCH (p,q) 4. Symmetric normal GARCH Model 5. Asymmetric GARCH models 5.1 GJR-GARCH (Glosten, 1993) 5.2 Exponential GARCH (E-GARCH) (Nelson, 1991) 5.3 Asymmetric power ARCH model ("apARCH") (Ding, 1993) 5.4 Component sGARCH model ('csGARCH') (Lee and Engle, 1999) 5.5 GARCH family model ("fGARCH") (Hentschel, 1995) 6. Abnormal GARCH models 6.1 Student tGARCH (Bollerslev, 1987) 6.2 Normal mixture GARCH 6.3 Markov switching GARCH (Hamilton and Susmel, 1994) 7. Multivariate GARCH models 7.1 The VECH model presented by Bollerslev et al. (1988) 7.2 Baba-Engle-Craft-Kroner (BEKK) model formalised by Engle and Kroner (1995) 7.3 Factorial and orthogonal GARCH models 7.4 The class of constant conditional correlation (CCC) models proposed by Bollerslev (1990) 7.5 Dynamic conditional correlation (DCC) proposed by Engle III and Sheppard (2001) Stochastic volatility models 1. Autoregressive volatility model, or SARV(p) model: 1.1 Lognormal stochastic autoregressive volatility model 1.2 Stochastic autoregressive square root model or SR-SARV

Approaches to Modeling Financial Asset Return Volatility Using High-Frequency Data

Method/Evaluator Description

Realised variance or realised volatility (rRVar) (Andersen, 2003) This estimator calculates daily realised variance or realised volatility (RV)

Realised covariances using subsample averaging (rAVGCov) It calculates realised variances by averaging RV over partially overlapping grids (Zhang et al., 2005)

Modulated realised covariance (rMRCov) The modulated realised covariance computes a univariate or multivariate pre-averaged estimator by Hautsch and Podolskij (Hautsch and Podolskij, 2013).

Two-time scale of covariance estimation (rTSCov) It calculates the covariance matrix on a two-time scale (Zhang et al., 2005; Zhang, 2011).

Reliable estimation of covariance on a two-time scale (rRTSCov) It calculates the robust two-time covariance matrix (Boudt and Zhang, 2015)

Implemented kernel estimator (rKernel-Cov) It calculates the realised covariance using the kernel estimator

Realised two power covariance (rBPCov) It calculates the realised BiPower covariance (rBPCov) (Barndorff-Nielsen and Shephard, 2004)

Minimum realised variance (rMinRVar) It calculates rMinRVar (Andersen et al., 2012)

Median realised variance (rMedRVar,) It calculates rMinRVar (Andersen et al., 2012)

Threshold Covariance (rThresholdCov) It calculates the threshold covariance matrix (Mancini and Gobbi, 2012)

Hayashi-Yoshida Covariance (rHYCov) It calculates the Hayashi-Yoshida covariance estimate (Hayashi and Yoshida, 2005)

Realised appearance weighted covariance (rOWCov) It calculates the realised distance weighted covariance (rOWCov) (Boudt and Zhang, 2015)

Realised semi-dispersion of high-frequency reciprocal series (rSVar) It calculates the realised semivariances (Barndorff-Nielsen et al., 2008)

Realised range based variance (RRV) It computes a range-based realised estimator (Christensen and Podolskij, 2007)

Quantile-based realised variance (QRV) It calculates the quantile realised variance (Christen et al., 2010)

Estimating duration based realised variance (DRV) in Andersen, Dobrev, and Schaumburg (2009) It calculates the long-term realised variance (Andersen et al., 2009)

Let a2n and an be the variance and volatility of the market variable on day n, respectively. Let S denote the value of the variable at the end of the i-th day. For an, there will be dispersion and volatility will rush to day n respectively. Let S denote the value of the variable at the end of the i-th day. Let u. be the continuously added variable returns during the i-th day:

U = lnf-(1) (1)

An objective estimate of the daily variance, a2 , based on the most recent observations at ut is:

a2 = —V m U - u )2 (2)

Let's make the following changes to the above formula for estimating the variance, a2n ,:

- u is defined as the percentage change in the market variable between the end of day i-1 and the end of day i, so that

S - S 1

u (3)

- U = 0;

- m-1 is replaced by m.

The above changes simplify the original variance formula to:

a2n = m E>„2-i (4)

m i=1

However, there is still a problem with this. The problem with the above simplified formula for calculating the variance is that it gives a high weight density u2-1,un2-2...., u2n-m .,,, (i.e. profitability). Our goal is to conditionally estimate volatility, and predictively assign large amounts of losses. The model that adopts this is:

al=V lauli (5)

The variable ai is the amount of weight assigned to the observation i days ago. a are positive and must sum to one, that is:

y " a. = 1 (6)

/—a=i •

An extension of the idea in the above weighting schemes is to assume that there is a long-term average variance and that some weight should be given to it. The result is a model that takes the following form:

* =yVL + S m atu2ni (7)

n t L =1 i n—i

where VL is the unconditional variance and Y is the weight given to VL . The sum of the weights must be equal to one, so we have

Y + S ^ =1 (8)

The above model is called the ARCH(m) model. It was introduced by Engle (1982). It estimates the variance based on the long-term mean variance, VL , and m observations. If we w =yVL, the ARCH(m) model becomes

a2 =w +

S m au2 . (9)

/ j i=i i n—i

A generalisation of the above model is the GARCH model. The GARCH model generalises En-gle's (1982) ARCH model. In GARCH (1,1), *2n is calculated based on the following equation:

a^ =AVl +auh + 0*11 (10)

where X is the weight assigned to VL , a is the weight assigned to un2—1, and P is the weight assigned to *l— t. The sum of the weights must be equal to one, so it follows that:

Y+a+0= 1 (11) If we set w =yVL, GARCH(1,1) becomes:

*2 =w+au2—1 +0*1, (12)

The given estimated values w,a and P, y can be calculated as:

Y = 1 — a—0 (13)

The long-term variance VL will then be given as:

VL=W (14) Y

To ensure a stable GARCH (1,1) process, it is required that:

a+0< 1 (15)

"(1, 1)" in GARCH (1,1) indicates that <j2n is calculated based on the most recent observation u2 and the most recent variance estimate, that is, is calculated ul—l and a2^. The general GARCH (p, q) model computes o_nA2 using the most recent p observations on uA2 and the most recent q variance estimates (Hull, 2018). The main difference between parametric and non-parametric approaches is related to the choice of the time interval to which the measure of volatility refers, for example, a discrete interval, where h > 0, or a point in time (instantaneous) measure, where h ^ 0.

Considering the ARCH-GARCH models, there are times when volatility is unusually high, and there are times when volatility is unusually low. There is extensive empirical evidence for the clustering of volatility in financial markets, dating back to Mandelbrot (1963). Volatility clustering has significant implications for option pricing, hedging, and risk measurement (Andersen et al., 2006.; Kornikov et al., 2002). A big shock to markets leads to changes in volatility and further increases the likelihood of another big shock. This must be considered when pricing options and assessing portfolio risks (Egorova, 2002; Senko, 2001).

GARCH models have been introduced to capture the clustering of returns volatility (Chekulaev et al., 2004). Given that GARCH models reflect volatility clustering, their predictions are not equal to the current estimate. GARCH volatility can be above or below average in the short term, but as the forecast horizon increases, GARCH volatility forecasts converge towards long-term volatility. The advantage of GARCH is that it calculates short- and medium-term volatility forecasts based on a robust econometric model. GARCH models have been very successful in predicting notional return volatility. They have been finalised and expanded to include additional features. Traditional GARCH models have been extended by incorporating implemented measures into the GARCH equation to enhance their predictive capabilities. Important works in this direction include Engle and Gallo (2006), Hansen (2012), Shepard and Sheppard (2010).

2. Materials and Methods

We estimated, predicted, and analysed the volatility of US stock indices and stocks using existing high-frequency volatility estimates (realised volatility measures). The stock indices studied are the Dow Jones Industrial Average (DJI), Standard and Poor's 500 (SP500), and the Nasdaq Composite Index (NASDAQCOMP). Stocks include stocks in Microsoft, Bank of America, and Coca-Cola. We also valued call and put options on Bank of America, Coca-Cola, and Microsoft using various high-frequency volatility forecasts. The study used both primary and secondary data sources. Daily and intraday data on the prices of the studied stocks and indices, as well as data on option contracts, were obtained from official foreign sources that are participants in the US financial market. Secondary data were obtained based on an in-depth study of scientific periodicals, as well as reference books, monographs, and textbooks in the subject area of research.

To evaluate and predict the volatility of the above stocks and stock indices, we downloaded historical closing price data (data for 5 minutes) for these assets from Finam (Broker Finam, 2022)1. The sample data covered the period 2020.02.06-2021.09.02. We also downloaded option chains for Microsoft (MSFT), Bank of America (BAC), and Coca-Cola (KO) at the end of each trading day from Yahoo Finance2. Option chains covered the period from 2021.08.02 to 2021.09.02. To estimate the volatility of stocks underlying option contracts, this study used the recently proposed high-frequency volatility estimates available in the volatility literature. Table 1 provides data to describe these volatility estimates. To keep tabular presentations simple, we have abbreviated the above high-frequency volatility estimates as follows: RV, AV, MRC, TS, RTS, Epa, Par, mTH, BP, MiRV, MeRV, Thr, HY, OW, SV.do, SV.up, RRV, QRV, and mQRV.

To predict realised volatility, this study used Corsi's (2009) heterogeneous autoregressive model of realised variance (HAR-RV model). The dynamics of the HAR-RV model are represented by:

RV+d = c + 0( d ) RV( d ) + 0( w) RV(w) + 0(m ) RV(m) + e+ (16)

RVtid) is the realised volatility for day t.

RV(w) is the average realised volatility for the last week (last 5 trading days), calculated as follows:

Rv!^5(RV!d>+Rv—>+Rv—l+Rv!i+Rv!i) (17)

RVt{m) is the average realised volatility for the last month (last 22 trading days), calculated as follows:

RV} m)= 22 (RV,")+ RVH+... + RV^ + RVt« ) (18)

The HAR-RV model was estimated using the ordinary least squares method, assuming that at

ormation about financial instruments IIRT,- https

14

time t the conditional mean value s_(t+1d)A((d)) is zero. For our study, we used a HAR-RV model with the following characteristics: Type, HAR; Lags, 1 5 22; Window Type, "rolling"; and Maximum lags, 22.

We used a 5-minute series of returns (30,808 5-returns) to estimate and predict stock volatility. We were of the opinion that almost all information from high-frequency data is contained in 5-minute data, so we decided to estimate and predict stock volatility using a 5-minute sampling rate. To value option contracts, this study used the Black-Scholes-Merton formulas for the prices of European call and put options in determining the price of option contracts.

3. Results and Discussion

The most popular stock indices for tracking the dynamics of the US stock market are the Dow Jones Industrial Average (DJIA), the Standard and Poor's 500 (S&P500), and the Nasdaq Composite (NASDAQCOM). We estimated the realised volatility of these stock indices for the period 2020.01.062021.10.15 using high-frequency volatility estimates. The figures below show volatility estimates obtained for the period under review (Figure 1) (Gayomey, 2022).

Figure 1. DJI daily realised volatility, NASDAQCOMP Daily Realised Volatility, Daily Realised Volatility of SP500

The realised volatility of all three stock indices shows that the US stock market was volatile towards the end of the first quarter of 2020 (volatility of 10% and above). The figure also shows that from the beginning of the second quarter of 2020 until the end of the evaluation period (2021.10.15), realised volatility has consistently been below 5%, indicating a stable price level in the market. Estimates, however, show that volatility changes over time. Volatility clustering was also observed. We then projected volatility for the Dow Jones Industrial Average, S&P 500, and Nasdaq Composite for the next 30 days (2021.10.18-2021.11.26) using high-frequency volatility estimates. We also compared the volatility forecasts with the CBOE Volatility Index (VIX) for the next 30 days (2021.10.18-2021.11.26) to assess the accuracy of the forecast. The CBOE Volatility Index (VIX) is a popular measure of expected stock market volatility based on S&P 500 index options. It is based on the prices of options on the SPX index with the nearest expiration dates and thus gives a 30-day forecast of volatility in advance (Gayomey,

Sustain. Dev. Eng. Econ. 2023, 2, 1. https://doi.org/10.48554/SDEE.2023.2.1 15

2022a).

Figures 2-4 show the expected annual change in the Dow Jones Industrial Average, Nasdaq Composite, and S&P500 over the next 30 days: 2021.10.16-2021.11.26, according to the realised volatility approach.

Expected annual change in the DJIA over the next 30 days: 2021.10.16-2021.11.26

Implemented measures Volatility forecast CBOE Volatility Index Forecast Loss

RV 16.2897 16.3000 0.0103

AV 12.3653 16.3000 3.9347

MRC 8.8899 16.3000 7.4101

TS 10.7024 16.3000 5.5976

RTS 9.4100 16.3000 6.8900

Epa 14.8305 16.3000 1.4695

Par 14.8305 16.3000 1.4695

mTH 14.8305 16.3000 1.4695

BP 12.4816 16.3000 3.8184

MiRV 11.5836 16.3000 4.7164

MeRV 11.2091 16.3000 5.0909

Thr 10.2701 16.3000 6.0299

HY 10.9682 16.3000 5.3318

OW 8.7731 16.3000 7.5269

SV.do 9.9558 16.3000 6.3442

SV.up 12.4042 16.3000 3.8958

RRV 13.7790 16.3000 2.5210

QRV 12.9370 16.3000 3.3630

mQRV 12.7933 16.3000 3.5067

Note: The highlighted cell indicates high-frequency volatility estimators whose volatility forecast was close to that of the CBOE Volatility Index (VIX).)

Figure 2. Expected annual change in the DJIA over the next 30 days: 2021.10.16-2021.11.26

Expected annual change in the NASDAQ COMP over the next 30 days: 2021.10.16-2021.11.26

Implemented measures Volatility forecast CBOE Volatility Index Forecast Loss

RV 16.2312 16.3000 0.0688

AV 22.6563 16.3000 -6.3563

MRC 11.5679 16.3000 4.7321

TS 12.3602 16.3000 3.9398

RTS 10.9333 16.3000 5.3667

Epa 16.9672 16.3000 -0.6672

Par 16.9672 16.3000 -0.6672

mTH 16.9672 16.3000 -0.6672

BP 11.1340 16.3000 5.1660

MiRV 10.5268 16.3000 5.7732

MeRV 10.4024 16.3000 5.8976

Thr 9.6806 16.3000 6.6194

HY 11.0299 16.3000 5.2701

OW 8.7697 16.3000 7.5303

SV.do 10.3777 16.3000 5.9223

SV.up 12.1616 16.3000 4.1384

RRV 13.4633 16.3000 2.8367

QRV 10.3678 16.3000 5.9322

mQRV 11.8126 16.3000 4.4874

Note: The highlighted cell indicates high-frequency volatility estimators whose volatility forecast was close to that of the CBOE Volatility Index (VIX).)

Figure 3. Expected annual change in the NASDAQ COMP over the next 30 days: 2021.10.162021.11.26

Expected annual change in the SP500A index over the next 30 days: 2021.10.16-2021.11.26

Implemented measures Volatility forecast CBOE Volatility Index Forecast Loss

RV 14.8594 16.3000 1.4406

AV 20.4546 16.3000 -4.1546

MRC 9.2252 16.3000 7.0748

TS 10.2678 16.3000 6.0322

RTS 9.0695 16.3000 7.2305

Epa 15.2176 16.3000 1.0824

Par 15.2176 16.3000 1.0824

mTH 15.2176 16.3000 1.0824

BP 9.9485 16.3000 6.3515

MiRV 9.5236 16.3000 6.7764

MeRV 9.3557 16.3000 6.9443

Thr 8.6203 16.3000 7.6797

HY 9.5312 16.3000 6.7688

OW 7.9498 16.3000 8.3502

SV.do 8.9054 16.3000 7.3946

SV.up 11.4562 16.3000 4.8438

RRV 11.6070 16.3000 4.6930

QRV 8.6699 16.3000 7.6301

mQRV 10.1537 16.3000 6.1463

Note: The highlighted cell indicates high-frequency volatility estimators whose volatility forecast was close to that of the CBOE Volatility Index (VIX).)

Figure 4. Expected annual change in the SP500A index over the next 30 days: 2021.10.16-2021.11.26

According to Figures 2-4, for the DJI and NASDAQCOMP indices, the realised volatility (RV) estimate had the lowest forecast error compared to the CBOE Volatility Index, while for the SP500 index, the realised core estimates (Epa, Par, and MTH) performed the best. The results in the tables also show that the projections for the realised volatility and the realised core estimates for the three indices are, in most cases, very close to the forecast for the CBOE volatility index; that is, the forecast of these two estimates is very similar to the CBOE Volatility Index forecast. The tables also show that more high-frequency volatility estimators provide lower volatility forecasts for the US stock market during the period under review. Consider specific examples (Gayomey, 2022b).

Estimating and predicting the volatility of Bank of America, Coca-Cola, and Microsoft stocks

We predicted and analysed the annual volatility of these stocks. The figures below show the implemented volatility estimates for Bank of America, Coca-Cola, and Microsoft for the period 2020.01.062021.10.15 (fig. 5).

o >

"O

10.104> I

JlfciJUykiim

if1

0.00 -

2020-01 2020-07 2021-01 2021-07 Estimation Period

Daily realized volatility for Bank of

AV BP Epa HY

MeRV

MIRV

mQRV

MRC

mTH

ow

Par QRV RRV RTS

Figure 5. Daily realised volatility for Bank of America, for Coca-Cola, and Microsoft

The charts above show the realised volatility, confirming the volatility situation in the US stock market at the end of the first quarter of 2020, which remains relatively stable during the remainder of the forecast period. Of all three stocks, Microsoft shares experienced the least volatility during the remainder of the forecast period (Gayomey, 2022a).

Figure 6 shows the annual volatility forecast for Bank of America, Coca-Cola, and Microsoft.

Annual Volatility Forecast for BAC, KO and MSFT: Realized Volatility Metrics

Implemented measures BAC KO MSFT

RV 0.2922 0.1802 0.2145

AV 0.2327 0.1454 0.1816

MRC 0.1955 0.1045 0.1385

TS 0.2213 0.1352 0.1809

RTS 0.2069 0.1174 0.1516

Epa 0.2945 0.1689 0.2094

Par 0.2945 0.1689 0.2094

mTH 0.2945 0.1689 0.2094

BP 0.2377 0.1567 0.1833

MiRV 0.2372 0.1546 0.1828

MeRV 0.2294 0.1474 0.1788

Thr 0.2194 0.1309 0.1667

HY 0.2245 0.1432 0.1814

OW 0.2060 0.1147 0.1517

SV.do 0.2030 0.1335 0.1653

SV.up 0.2060 0.1248 0.1453

RRV 0.2608 0.1708 0.2227

QRV 0.1961 0.1438 0.1762

mQRV 0.2397 0.1639 0.1939

Note: Cells highlighted in blue indicate the minimum volatility forecast, while cells highlighted in pink indicate the maximum volatility forecast

Figure 6. Annual volatility forecast for BAC, KO, and MSFT: Realised volatility metrics

Figure 6 shows that the minimum volatility prediction for all three stocks is given by a modulated realised covariance (MRC) estimate. The table also shows that the maximum volatility forecast for Bank of America, Coca-Cola, and Microsoft was derived using realised variance (RV), realised core (Epa, Par, and mTH), and realised range (RRV) volatility estimates, respectively. Table 5 further confirms that all volatility estimates project high volatility for Bank of America (annualised volatility forecast of at least 19% and a maximum of 29%). It can also be seen that Coca-Cola's year-on-year volatility forecast across all volatility estimates is less than 19%, with a minimum volatility forecast of 10%. For Microsoft's stock, the year-on-year volatility forecast ranges from 13.8% to 22%. The forecast above clearly shows that Bank of America is expected to be more volatile over the coming year compared to other stocks, while Coca-Cola is expected to remain more stable over the same period (Gayomey, 2022b).

4. Conclusion

The purpose of this article was to develop methods for modelling and forecasting the volatility of financial asset returns based on the assessment of high-frequency data and the dynamics of the information environment. To this end, we reviewed and systematised theoretical and methodological approaches to measuring and forecasting the volatility of return on financial assets. We also estimated, predicted, and analysed the volatility of stock indices and US stocks using existing high-frequency volatility estimates (realised volatility measures). The stock indices studied are the Dow Jones Industrial Average (DJI), Standard and Poor's 500 (SP500), and the Nasdaq Composite Index (NASDAQCOMP). Stocks, on the other hand, include stocks in Microsoft, Bank of America, and Coca-Cola. We also valued call and put options on Microsoft, Bank of America, and Coca-Cola using various high-frequency volatility forecasts.

The results of the analysis and systematisation of theoretical and methodological approaches to measuring and forecasting the volatility of the profitability of financial assets have shown that the existing approaches and methods for assessing and forecasting the volatility of the profitability of financial assets have some limitations that make the use of these methods for predicting volatility unsatisfactory. In particular, within the framework of the GARCH model and stochastic volatility, the following are salient:

- Volatility is usually derived from daily returns squared, which are objective but noisy estimates of daily conditional volatility.

- High-frequency data are rarely used.

- Evaluation of GARCH and stochastic volatility models is difficult.

- Evaluation of these models often gives unsatisfactory results. Forecasts are inaccurate.

- Standardised returns usually have fat tails, which leads to the search for suitable error distributions that can adequately reflect the empirical distributions of returns.

- Multivariate modelling of volatility and correlation can be extremely complex, and practical models are often only applicable to very small dimensions.

- These volatility methods do not take into account changes in the news background of the digital information environment.

In the high-frequency volatility approach (HAR-RV model), the realised volatility plotted from the highest frequency data should give the best possible estimate of the cumulative volatility, that is, Iv as m ^ro. However, in practice, the sampling rate is limited by the actual supply or transaction frequency. Moreover, very high-frequency prices are affected by market microstructure, such as supply and demand rebound effects, price discreteness, etc., leading to bias and inconsistency in realised volatility. These methods for assessing volatility also do not consider changes in the news background of the digital information environment.

These findings prompt the conclusion that the systematic analysis of news information is useful for predicting the volatility of returns on financial assets, but its effectiveness depends on the individual company. Future studies are encouraged to explore the usefulness of the systematic analysis of news information in predicting the volatility of returns on financial assets in other markets and for other asset classes.

Acknowledgments

The research was financed as part of the project "Development of a methodology for instrumental base formation for analysis and modeling of the spatial socio-economic development of systems based on internal reserves in the context of digitalization" (FSEG-2023-0008)

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The article was submitted 02.05.2023, approved after reviewing 28.05.2023, accepted for publication 03.06.2023.

Статья поступила в редакцию 02.05.2023, одобрена после рецензирования 28.05.2023, принята к публикации 03.06.2023.

About authors:

1. John Gayomey, Lecturer, Department of Accountancy, Tamale Technical University, Tamale, Ghana. [email protected]

2. Andrey Zaytsev, Doctor of Economics, Professor, Graduate School of Industrial Economics, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russian Federation. https://orcid.org/0000-0002-4372-4207, [email protected]

Информация об авторах:

1. Джон Гайомей, преподаватель кафедры бухгалтерского учета Технического университета Тамале, Тамале, Гана. [email protected]

2. Андрей Зайцев, доктор экономических наук, профессор, Высшая инженерно-экономическая школа, Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург, Российская Федерация. https://orcid.org/0000-0002-4372-4207, [email protected]