CC BY
3© Irina. Z. Yarygina
Doctor of Economic Sciences, Professor, Professor of the Chair of World Economy and World Finance, Financial University under the Government of the Russian Federation,
Leningradsky Ave., 49, Moscow, 125167 E-mail: i.yarigina@fa.ru
© Vladimir B. Gisin
Candidate of Physical and Mathematical Sciences, Professor of the Department of Data Analysis and Financial Technologies, Financial University under the Government of the Russian Federation
Leningradsky Ave., 49, Moscow, 125167 E-mail: v.gisin@fa.ru
<E Bruno de Conti
Doctor of Economic Sciences, Professor, State University of Campinas
Brazil, Sao Paulo, Avenida Albert Einstein, 901, CEP 13083-852 E-mail: brunodeconti@unicamp.br
Fractal asset pricing models for financial risk evaluation
This article presents the results of the analysis of problems and prospects of using the theory of fractal market to mathematically predict the price dynamics of assets as part of the implementation of a financial risk management strategy. As part of the study, it was noted that the use of financial mathematics in banking practice contributes to the formation of conditions for the stable development of the economy. The methods of mathematical modeling of the price dynamics of financial assets have proved their viability, since they are built on a substantive hypothesis and supported using an adequate apparatus of fractal pair pricing models to reveal the characteristics of market relations of business entities. The purpose of the article is to disclose the features of the value of banking assets and develop recommendations aimed at assessing financial risks based on mathematical methods for predicting economic processes. Theoretical and empirical research methods contribute to the achievement of the goal of the work. The article reveals the features of mathematical modeling of economic processes related to asset pricing in a volatile market. The prospects of using forecast models to minimize the financial risks of derivative financial instruments are noted. It is concluded that the use of the methods in question contributes to the management of financial risks and the improvement of forecasts, including operations with derivatives. In addition, the fractal volatility parameters studied in the work showed predictive power regarding extreme events in financial markets, such as the collapse of the American investment bank Lehman Brothers in 2008. The relevance of the article is since a favorable investment climate and the use of modern financing methods largely depend on the effective management of financial risks.
Keywords: banking, asset valuation, economic and mathematical methods, financial risk, hedging.
This work is based on the research supported by the National Institute for the Humanities and Social Sciences, South Africa. JEL Classification: D81, F30, G32
For citation: Yarygina I.Z., Gisin V.B., De Conti B. Fractal asset pricing models for financial risk evaluation. World Economy and World Finance, 2024, vol. 3, no. 1, pp. 69-76. DOI: 10.24412/2949-6454-2024-0090
Ярыгина Ирина Зотовна
доктор экономических наук, профессор,
профессор кафедры мировой экономики и мировых финансов, Финансовый университет при Правительстве Российской Федерации
Ленинградский пр-т., 49, Москва, 125167 E-mail: i.yarigina@fa.ru
Гисин Владимир Борисович
кандидат физико-математических наук,
профессор кафедры анализа данных и финансовых технологий, Финансовый университет при Правительстве Российской Федерации
Ленинградский пр-т., 49, Москва, 125167 E-mail: v.gisin@fa.ru
Де Конти Бруно
доктор экономических наук, профессор, Государственный университет Кампинаса
Бразилия, Сан-Пауло, Avenida Albert Einstein, 901, CEP 13083-852 E-mail: brunodeconti@unicamp.br
Методы управления финансовыми рисками: фрактальные модели ценообразования активов
В настоящей статье представлены результаты анализа проблем и перспектив использования теории фрактального рынка в целях математического прогнозирования ценовой динамики активов в рамках реализации стратегии управления финансовыми рисками. В рамках исследования отмечено, что использование финансовой математики в банковской практике способствует формированию условий стабильного развития экономики. Методы математического моделирования ценовой динамики финансовых активов доказали свою жизнеспособность, так как строятся на содержательной гипотезе и подкрепляются использованием адекватного аппарата фрактальных парных моделей ценообразования в целях раскрытия особенностей рыночных отношений субъектов хозяйствования. Целью статьи является раскрытие особенностей стоимости банковских активов и разработка рекомендаций, направленных на оценку финансовых рисков на базе использования математических методов прогнозирования экономических процессов. Теоретические и эмпирические методы исследования способствуют достижению цели работы. В статье раскрыты особенности математического моделирования экономических процессов связанных с ценообразованием активов в условиях волатильного рынка. Отмечены перспективы использования прогнозных моделей в целях минимизации финансовых рисков производных финансовых инструментов. Сделан вывод, что использование рассматриваемых методик способствует управлению финансовыми рисками и улучшению прогнозов, в том числе операций с деривативами. Кроме того, параметры фрактальной волатильности, исследуемые в работе, показали предсказательную силу относительно экстремальных явлений на финансовых рынках, таких как крах американского инвестиционного банка Lehman Brothers в 2008 г. Актуальность статьи обусловлена тем, что благоприятный инвестиционный климат и использование современных методов финансирования во многом зависят от эффективного управления финансовыми рисками.
Ключевые слова: банковская деятельность, оценка стоимости активов, экономико-математические методы, управление финансовыми рисками, хеджирование.
Статья основана на исследовании, проведенном при поддержке Национального института гуманитарных и социальных наук Южной Африки.
Для цитирования: Ярыгина И.З., Гисин В.Б., Де Конти Б. Методы управления финансовыми рисками: фрактальные модели ценообразования активов // Мировая экономика и мировые финансы. 2024. Т. 3. № 1. С. 69-76. DOI: 10.24412/2949-6454-2024-0090
Introduction
World experience has shown that the traditional approach to the study of asset price dynamics is based on the identification of patterns of an economic nature and mathematical modeling of the manifestation of such patterns to manage financial risks. For example, the classical Black-Scholes-Merton model is associated with the efficient market hypothesis (EMH), which assumes that the price of an asset is determined by multiple random factors. In turn, the mathematical model of asset price dynamics helps to reveal the features of price dynamics. The use of such a model in practice helps to minimize financial risks and ensures the safety of banking activities in a volatile market.
It is important to note that financial mathematics over the last century has proved that a mathematical model, to be viable, must be based on a meaningful hypothesis and must be supported by an adequate mathematical apparatus. Models that do not contain both components, "unpaired" models, turn out to be unviable. For example, the mathematical apparatus used by Bachelier in 1900 in his model of price dynamics was ahead of its time, and the Bachelier model remained unclaimed for more than 60 years. After developing the efficient market hypothesis, the Bachelier model served as the basis for building modern pricing models. In a sense, the opposite example is given by the fractal market theory, which arose simultaneously with the efficient market hypothesis [Mandelbrot, 1968], but the mathematical apparatus of this theory (a model based on fractal Brownian motion) "did not keep pace" with the meaningful concept [Mandelbrot, 1971]. The lack of an adequate mathematical model of fractal price dynamics at the time of the formation of the fractal market hypothesis prevented the formation of a full-fledged theory.
Attempts to revise the classical theory are due to the peculiarities of the development of market relations and the observed volatility of asset price dynamics under the influence of stylized factors of market participants [Cont, 2007], namely:
• excessive volatility of asset returns, which cannot be assessed by traditional methods of economic processes.
• the appearance of "heavy tails" of distributions indicating the asymmetry of the market, contributing to the growth of risks and the likelihood of extreme events;
• autocorrelation of asset returns, in which homogeneous assets can demonstrate the absence of dependence of return increments and the presence of a significant long-term memory of economic processes that can find expression in homogeneous processes of market relations;
• clustering of volatility, in which jumps in profitability are followed by jumps in the opposite direction, significant for the market and price dynamics of assets, contributing to the likelihood of significant losses;
• the relationship between the trading volume of assets and market volatility, in which there is not only a positive correlation between trading volume and volatility, but also a similar type of long-term memory.
The study of these phenomena began in the 80s of the twentieth century [Pagan, 1996]. However, mathematical modeling of individual stylized facts was first carried out by researchers at the beginning of the 21st century [Ding, Granger, Engle, 1993; Guillaume, Dacorogna, Dave, Muller, Olsen, Pictet, 1997; Cont, 2001]. Currently, representatives of various scientific schools have shown that the peculiarities of market development are directly related to risk assessment and the need to use predictive mathematical models for adequate asset management solutions aimed at stable manifestation of economic processes. It is important to note that a universal mathematical model of the price dynamics of market assets has not yet been found. For example, research conducted within the framework of the European Central Bank in 2014 based on the analysis of data from developed economies of the European Union countries is aimed at finding a theoretical model explaining the observed phenomena of market relations [Hiebert, Jaccard, Schüler, 2018]. In turn, it is not possible to use the considered approach to predict the processes of emerging markets. In addition, a study of the price dynamics of cryptocurrencies conducted by representatives of the European Mathematical School in 2017 showed the peculiarities of forecasting the use of assets in cyberspace [Bariviera, Basgall, Hasperue, Naiouf, 2017]. In this regard, an interesting observation was made in 2019 in the field of stochastic financial mathematics [Restocchi, McGroarty, Gerding, 2019]. Analyzing the stylized facts of economic development based on a large statistical material, the authors concluded that emerging markets behave like markets where a variety of political forecasts are implemented, which confirms the role of general and specialized information in banking. An attempt to link stylized facts of market phenomena with the behavioral characteristics of economic agents is made using multi-agent models, including those involving artificial intelligence, in which market participants implement a relatively rational asset management strategy aimed at maintaining profits and managing risks [Gisin, Shapoval, 2008; Pruna, Polukarov, Jennings, 2016; Dhesi, Ausloos, 2016]. However, criticisms 71 of multi-agent forecasting models, especially in emerging markets, remain valid [LeBaron, 2000].
It is important to note that in complex forecasting models of a highly volatile non-traditional market, the use of "non-standard" models is promising. Thus, the basic theorem of asset dynamics pricing has been proved for markets for which the use of mathematical modeling was not possible [Acciaio, Beiglbock, Penkner, Schachermayer, 2016]. In 2018, representatives of the scientific school of the University of Jerusalem introduced the concept of a full-part market and attempted to mathematically predict an asset hedging strategy [Dolinsky, Neufeld, 2018]. In this regard, it is important to note that to calculate the price dynamics of assets and manage financial risks, it is necessary to use comprehensive information on the prices of real and virtual derivative financial instruments.
The variety of methods and models used in modern financial mathematics shows that the unifying concept that generalizes the classical and explains the stylized facts of market relations is not represented in modern science. The most systematic and consistent explanation of the stylized facts of economic development is obtained within the framework of the fractal market concept, which assumes the dependence of the forecasted value of asset price dynamics on the history of market development. This article is devoted to the analysis of this concept.
Models based on self-similar processes
The main assumption in fractal market theory is the assumption of self-similarity of dynamic price ranges of assets. The price dynamics of financial market assets is modeled, as a rule, using self-similar processes. This is supported by statistical observations and economic arguments [Shiryaev, 2004]. Self-similarity is a consequence of the presence in the market of many participants with different investment horizons and operating under the same conditions. Moreover, market participants act in a similar way on their investment horizons, which ensures a kind of invariance of market characteristics relative to the time scale of asset use. The statistical characteristic of scale invariance is the Hurst exponent H [Shiryaev, 2004], the value of which is in the range from zero to one. For the Brownian motion underlying classical models of a volatile market, the value of the Hurst index is 0.5. If the value of H is in the range from 0.5 to 1, the time series of asset price characteristics is persistent (trend-resistant); if H is in the range from 0 to 0.5, the time series is anti-persistent and demonstrates the property of returning to the average value.
The mathematical apparatus for describing self-similar random processes was proposed by A.N. Kolmogorov. Based on this, methods for obtaining accurate numerical market forecasts related to asset pricing processes have been developed for about half a century, but decisive results such as the Black-Scholes model have not yet been obtained. The reason is that using fractal Brownian motion to model asset pricing in the stock market faces the need to solve a difficult task. Unlike classical mathematical modeling, models based on fractal Brownian motion have arbitrage possibilities that cannot be described by rational pricing theory.
For a long time, the belief has dominated among researchers that the availability of arbitrage opportunities is inextricably linked to autocorrelation and memory of financial time series. A deeper insight into the mathematics of the fractal market shows that arbitrage-free and autocorrelation and self-similarity are due to different factors [Cheridito, 2004] provides examples of Gaussian random processes that have the same long-term memory as processes based on fractal Brownian motion with a Hearst exponent greater than 0.5, and at the same time lead to non-arbitrage market models. Note that the idea of a moving average was used to build a price model in [Cheridito, 2004], which successfully links the mathematical apparatus with the realities of the market understandable to the financier.
Nevertheless, most researchers find it more promising to use fractal Brownian motion to build a market model. Replacing Ito integral with Wiki integral helps to circumvent the problem of arbitrage opportunities [Biagini, Hu, Oksendal, Zhang, 2008; Rostek, Schobel, 2013]. Practice has shown that so far it has not been possible to find a convincing economic interpretation for modified integral, therefore, it is advisable to treat the use of mathematical modeling using Wiki integral with caution.
The solution to the problem of minimizing financial risks using mathematical modeling of price indicators of derivative financial instruments can be found based on a more complete account of the specifics of trading financial instruments in a particular financial market. A fractal market with proportional transaction costs is non-arbitrage. It is fundamentally impossible to accurately determine the price of derivative financial instruments in such a market; it is only possible to accurately set the price boundaries that do not allow arbitrage. However, the fractal market theory is attractive to market participants due to the possibility of using it to minimize the financial risks of asset management.
Classical predictive models assume that the price dynamics of a risky asset is described by a random process with underlying Brownian motion. Namely, let S(t) be the price of a risky asset at time t. Then the yield over a period At is represented in the following form
where n + y is the expected return, cr is the volatility of the return, AW(t)=W(t+At)-W(t), and W(t) is the so-called Wiener process (Brownian motion). The value AW(t) is normally distributed with an average value of zero and a variance of At. It is assumed that for different values of t, the increments of AW(t) are independent (unless the time intervals overlap).
Wiener processes belong to the class of self-similar random processes. In general, a stochastic market process is self-similar if a change in the time scale leads to a change in the spatial scale, and the probabilistic characteristics of the process remain unchanged. More precisely, a stochastic process, X(t), t>0, is called self-similar if for every a>0 it is possible to find b>0 so that stochastic processes X(at) and bX(t) have the same probabilistic characteristics. If, moreover, parameter b is related to parameter a in such a way that b=aH for some constant H for all a>0, the constant H is called the Hurst exponent and it is said that the process is self-similar to the Hurst exponent H. For the Wiener process, the Hearst index is 0.5.
If we consider changes in profitability over non-overlapping time intervals to be independent, Levy processes are used in the models. Models based on Levy processes provide a good approximation of real price series, in some cases much better than classical models [Shoutens, 2003]. Their application makes it possible to consider such features of financial time series as asymmetry and heavy tails of probability distributions, and thus more adequately assess risks (for example, ignoring heavy tails leads to underestimation of risks associated with extreme events). This is achieved since Levy processes are determined by a larger number of parameters than Wiener processes. As a rule, four parameters are used. Two parameters are in a certain sense like the parameters of the Wiener process: ^ is the position parameter (analogous to the average value, which the Levy process may have and is indefinite), ct is the scale parameter (analogous to the average deviation, which the Levy process may also have and is indefinite). Two more parameters allow us to consider the features of time series that are not captured by Wiener processes: the ^ — skewness parameter (allows us to consider the asymmetry manifested in the differences between probability distributions in the loss zone and in the zone of exceeding expectations). It is shown in [Shoutens, 2003] that the use of Levy processes to describe the returns of world stock indices gives quite satisfactory results. At the same time, it is possible to consider the dynamic features of financial series that escape in classical models. Similar results are obtained in relation to the Russian market [Gisin, Konnov, Sharov, 2012]. An important property of the model is its predictive ability. For the model to be considered qualitative and predictively valuable, it is necessary that it be sufficiently stable with respect to small fluctuations in the initial data and relatively small shifts along the time axis. In this regard, increasing the number of parameters allows for more accurate calibration based on historical data, but the stability of estimates is problematic. Data analysis shows that for periods of 1-2 months, models with a normal distribution show good result. With a forecast period of more than 200 days, both classical models and models based on Levy processes turn out to be not completely reliable. Finally, for periods of 100-150 days, models based on Levy processes give the best result [Borusyak, 2008]. Note that the use of non-classical models for the Russian market is more significant. For example, for the DJA index, the distributions in the corresponding Levy processes are close to normal, and both are consistent with empirical data. This is no longer the case for the RTS index due to high transaction costs (we also include costs due to insufficient liquidity).
The main example of a self-similar stochastic process with dependent increments is given by fractal Brownian motion. The dependence of the increments makes it possible to model processes with long-term memory using fractal Brownian motion. Thus, within the framework of such models, phenomena related to the formation of trends are explained.
The application of financial time series models based on self-similar processes faces fundamental difficulties, regardless of which processes we are talking about — processes with dependent or independent increments. The fact is that pricing in the classical Black-Scholes-Merton model is since this model has an equivalent martingale probability measure for price stochastic processes. Meaningfully, the existence of such a measure can be interpreted as the existence of a rational forecast in a certain sense, and the price of a derivative instrument is determined as if considering this forecast relative to its future prices. In general, for self-similar stochastic processes with independent increments, there is an infinite family of "rational predictions". Accordingly, an interval of prices appears which can be interpreted as "fair". In some cases, but not always, it is possible to estimate the boundaries of these intervals. But often these boundaries turn out to be meaningless. In models using fractal Brownian motion, with a Hearst index other than 0.5, there is no "rational forecast" (an equivalent martingale measure) at all and there are arbitrage opportunities. It is possible to build pricing models within the framework of such models 73
only considering the specifics of the actual functioning of the financial market. These features include transaction costs.
In classical models, the price of a derivative is determined using replication strategies. In the presence of transaction costs, accurate replication may be too expensive, and it is replaced by an approximate one obtained because of solving the problem of stochastic control using dynamic programming methods. Solving this problem in many cases turns out to be too difficult (even with today's computing power). Simplifications are achieved by narrowing the class of acceptable investment strategies, for example, portfolio restructuring can only be carried out at fixed intervals. In this case, using upper and lower hedges, it is possible to obtain acceptable estimates of the boundaries of price corridors [Kabanov, 2009]. Fundamentally important results were obtained in [Gerhold, Guasoni, Muhle-Karbe, Schachermayer, 2014], where it was possible to obtain estimates of the boundaries of price corridors under general assumptions. The authors managed to link trading volumes, liquidity and dynamic parameters of price movement and obtain estimates that allow them to build optimal trading strategies [Nika, Rasonyi, 2018]. These works make the issue of more consistent use of the so-called market time in models relevant. Technically, this concept has been used in many works. The results obtained in these studies open new possibilities for the application of the Tobin tax. Research, in our opinion, quite clearly indicates that it is advisable to link the passage of time in financial market models with financial events, and not only with the rotation of the Earth around the Sun [Gerhold, Guasoni, Muhle-Karbe, Schachermayer, 2014; Nika, Rasonyi, 2018].
It should be noted that when forecasting the price dynamics of assets to manage financial risks in difficult market conditions, the use of fractal modeling methods is promising.
Directions for further research
Let's focus on the results related to pricing in markets with transaction costs. In [Gerhold, Guasoni, Muhle-Karbe, Schachermayer, 2014; Nika, Rasonyi, 2018], it was possible to find an approach to describing optimal strategies in markets with transaction costs.
Under general assumptions, the share of capital invested in the risk component should be within the boundaries.
p-A , p+A
= and n, = t—r.
(2)
with
X = ya2(l-n,2(l-7T,)2) £1/3 + 0( e)
(3)
where is excess profitability, y is relative risk aversion, s is the spread between supply and demand prices, and
For example, calculations using formulas (13) and (14) for Sberbank ordinary shares in early 2014 gave the values n_ =45,6%, and =48,2%. The liquidity premium calculated using the method from [Gerhold, Guasoni, Muhle-Karbe, Schachermayer, 2014] turned out to be 0.04%. For less attractive and liquid assets, the purchase and sale boundaries turned out to be significantly lower, and the liquidity premium increased sharply. For example, for the Primorye Bank, it was 0.15%.
Recently, a significant number of studies have been devoted to modeling volatility using fractal Brownian motion. Within the framework of the constructed models, it is possible to explain the effects of short-term and long-term memory, the paradox of the "smile of volatility" and some other features [Guennoun, Jacquier, Roome, Shi, 2018].
The concept of rough fractal volatility (RFSV, Rough Fractional Stochastic Volatility) has become widespread [Bayer, Fri, Gatheral, 2016; Gatheral, Jaisson, Rosenbaum, 2018]. The RFSV concept generalizes models with stochastic volatility that have been used for more than 20 years (see [Comte, Renault, 1998]). In the standard model of stochastic volatility described by the equations
^ = fi(t,S(t))dt + a<it)dW(1\ty. d(lntr(t)) = k(9 - In a(t))dt + ydW{2\t)
(4)
(5)
it is proposed to use fractal Brownian motion instead of the Wiener process W(2) (t). Research in this direction was stimulated by the fact that a stable pattern was empirically revealed: the dynamics of volatility is fractal in nature, the Hearst index of the process is 0.1 for fixed income instruments. Such a Hurst indicator corresponds to very high volatility variability with a tendency to return to its average values. This observation makes it possible to significantly improve volatility forecasts, and, most importantly, to describe the possible risks and implied volatility of asset price dynamics much more accurately than using other models. The proposed approach is also promising in the formation of
predictive models of the price dynamics of assets of derivative financial instruments [Gatheral, Jaisson, Rosenbaum, 2018]. In addition, fractal volatility parameters demonstrate predictive power relative to extreme events in the financial sector. An example is the collapse of Lehman Brothers and other US investment banks in 2008, which was the cause of the global financial and economic crisis [Bayer, Fri, Gatheral, 2016].
The presented justification of the expediency of using fractal models of asset price dynamics and their practical application in the financial sector can help minimize risks and strengthen the stable development of market relations.
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