Stochastic resonance of drawdown risk in energy market prices

Physica A 540 (2020) 123098

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Physica A journal homepage: www.elsevier.com/locate/physa

Stochastic resonance of drawdown risk in energy market prices✩ ∗

∗

Yang Dong a , Shu-hui Wen a , , Xiao-bing Hu c,d , Jiang-Cheng Li b , a

Faculty of Management and Economics, Kunming University of Science and Technology, Kunming 650032, China School of Finance, Yunnan University of Finance and Economics, Kunming 650221, China c China-France Research Center of Applied Mathematics for ATM, China d College of Electronic Information and Automation, Civil Aviation University of China, Tianjin 300300, China b

article

info

Article history: Received 5 June 2019 Received in revised form 25 September 2019 Available online 15 October 2019 Keywords: Econophysics Financial market Maximum drawdown Signal power amplification Stochastic resonance

a b s t r a c t We investigate the stochastic resonance dynamic behavior characteristics of price drawdown time series driven by internal and external periodic information and discuss the influence of information resonance on price drawdown risk, based on price drawdown, periodic Heston model and signal power amplification. A theoretical model of the periodic price drawdown time series is proposed to describe the energy price drawdown time series with the periodic Heston model. Then the signal power amplification (SPA) is employed to measure the stochastic resonance phenomenon in the internal and external periodic information environment of complex energy dynamics system. Combined with the real data of WTI spot price of NYMEX, the least square method of drawdown time series distribution is used to estimate the parameters of the model. The probability density functions of drawdown between the theoretical model and the real data are compared, and a good agreement can be found between both the simulated data from the proposed model and the real data. After the stochastic simulation of signal power amplification under internal and external periodic information respectively, the results show that: (i) In the functions of SPA versus price volatility parameters (noise correlation strength or periodic information strength), inverse resonance phenomenon can be observed, that is to say, there is the best system price volatility parameter, noise correlation strength and information strength corresponding to the least risk of the price drawdown; (ii) In the SPA versus amplitude of volatility fluctuations, the correlation induce multiple inverse stochastic resonance phenomenon can be observed; (iii) The increase of growth rate strengthens SPA and weakens drawdown risk of energy market prices. © 2019 Published by Elsevier B.V.

1. Introduction Energy is an important pillar of economic development, and crude oil market as an important part of the energy market has a high proportion of [1]. The price fluctuations of crude oil are increasingly affecting the economic development and social stability [2,3]. Energy price fluctuations and risks are also directly or indirectly transmitted to financial markets [4– 6]. Based on the complex relationship between energy market and financial market and the behavior of dynamic system, ✩ This work was supported by the Humanities and Social Science Fund of Ministry of Education of China (Grant No.: 19YJAZH045). ∗ Corresponding authors. E-mail addresses: [email protected] (S.-h. Wen), [email protected] (J.-C. Li). https://doi.org/10.1016/j.physa.2019.123098 0378-4371/© 2019 Published by Elsevier B.V.

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energy finance direction is formed [7]. The research on the complex dynamic behavior of crude oil market has attracted more and more attention from domestic and foreign scholars and the industry, such as international crude oil price [8], pipeline risk management [9], risk aversion and risk premiums [10], forecasting value-at-risk in oil prices [11], the shortterm forecasting of oil prices [12] etc. At the same time, drawdown of risk assets is an important indicator of investment risk and performance. Drawdown has also been the focus of scholars and industry studies in the pricing and management of risk, such as the maximum drawdown measuring investment risk [13,14], portfolio selection with drawdown [15], the maximal drawdown and distribution of Brownian motion [16], Capital Asset Pricing Model with drawdown [17], maximum drawdown and distribution of Brownian motion [16], conditional crash rate with drawdown [18], etc. In addition, the same as the natural system, the periodic information in financial market is natural [19,20]. The actual financial market contains intrinsic [21,22] and extrinsic [23,24] periodic information. Therefore, it is important to explore the effect of periodic information on drawdown risk in energy prices. Stochastic resonance (SR) is a common phenomenon for complex dynamical systems [25,26]. The resonance dynamic behavior and characteristics of complex system have been widely concerned and studied, such as a gene transcriptional regulatory system [27], the foraging colony system [28], effects of noise and delay on regime shifts in a ecosystem [29–31], noises enhanced stability and induced regime shifts [32], a stochastic bistable system [33,34], noise- and delay-induced regime shifts of vegetation [35], impact of time delays on stochastic resonance of a describing vegetation [36], roles of bounded noises [37,38], SR phenomenon with additive noise [39–41], SR phenomenon with the presence of multiplicative noise [42–44], generalized Lotka–Volterra (glv) models [45] etc. Not only in natural systems, but also in social and economic systems, researchers can also widely find the phenomenon of SR, such as an interacting-agent model of stock market [46], resonance of corporation finance [47], the suprathreshold SR [48], the SR of the stock prices [19,20], SR for financial market crashes and bubbles [49], etc. Meanwhile, the signal power amplification (SPA) can characterize the strength of stochastic resonance of weak periodic information by noise [25,26]. Then SPA is usually used to describe the resonance, such as in financial system [19,20], intracellular calcium oscillation system [50] and mammalian cochlea [51]. Therefore, it is important to explore the resonance behavior in the energy market. However, the periodicity of information is difficult to be observed from real market data in financial system. Meanwhile, the Heston model [52] can describe the stock price dynamics characteristics very well in econophysics [53,54]. For instance, the time-dependent probability distribution of stock price returns [55], probability distribution of returns for Nasdaq, &P500 and Dow–Jones [56], the probability distribution of returns with empirical high-frequency data [57], the stochastic volatility and exponential tails [58], the mean escape time [59,60] and the exact expressions for survival probability [61,62] all be discussed based on the Heston model. In addition, a generalized nonlinear Heston model has been investigated by focusing on behavior of time series of returns and stability of prices in financial markets [63–65], and in these papers the well known phenomenon of noise enhanced stability [66–68] also can be observed. To this end, the periodic Heston model [19] is employed to describe the energy price dynamics in this paper. Based on previous studies, the research in this paper is different and innovative as follows. First, we use the drawdown time series in this article to characterize the drawdown risks in the oil price in energy markets. Combining the thoughts and methods of Econophysics [18], we obtain drawdown time series model driven by periodic information based on Magdon-Ismail and Abu-Mostafa [16] and periodic Heston model [19]. Secondly, we use signal power amplification to measure the price drawdown resonance behavior characteristics of energy price driven by internal and external periodic information. Finally, we combine the actual energy price data, calculate the estimated parameters of the proposed model and give an empirical comparison. The rest of this paper is organized as follows. Section 2 presents the methods of maximum drawdown, signal power amplification and estimation. Section 3 discusses the SR with intrinsic and extrinsic periodic information. A brief discussion is given in Section 4. 2. Methods The energy market crash is one manifestation of the financial crisis. Its appearance is affected and impacted by the complex inside and outside the economic environment, and it is an appearance of the outbreak of systemic risk. One big reason is that drawdowns in energy prices and stock markets have triggered a chain of leveraged bets in energy markets. This further causes the stock market investor’s panic mentality to intensify. The result is a chain reaction of selling energy assets. In Ref. [18], it is presented that the conditional crash rate to discuss drawdown risk. In this paper, we will discuss more about the internal dynamic mechanism of crash risk. Market crash may be derived from the stochastic resonance behavior between price drawdown and the intrinsic and extrinsic periodic information. The specific method and discussion are described in detail below. 2.1. Maximum drawdown Based on maximum drawdown by Magdon-Ismail and Abu-Mostafa [16], the maximum drawdown rate of a stock within at a given time T : MDD = MDD(T , θ ) = sup [ sup

t ∈[0,T ] s∈[0,t ]

S(s) − S(t) S(s)

]

(1)

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and the logarithmic maximum drawdown ratio LNMDD = LNMDD(T , θ ) = sup [ sup x(s) − x(t)],

(2)

t ∈[0,T ] s∈[0,t ]

x(t) = log(S(t)/S(0)), is logarithmic price. Then we can obtain LNMDD = − log(1 − MDD), MDD = 1 − exp(−LNMDD), MDD ∈ [0, 1], LNMDD ∈ [0, +∞)

(3)

If N stocks are considered, the maximum drawdown ratio of (MDD1 , . . . , MDDN ) or (LNMDD1 , . . . , LNMDDN ) within a given time of T can be obtained. According to the sample, based on the frequency method or kernel density estimation, we can get the probability density distribution function PDF (MDD, T )or PDF (LNMDD, T ). When N −→ ∞, the density distribution function of the maximum drawdown ratio tends to be true, and the following conditional crash rate can be estimated via PDF [18]. We can define the drawdown D(t), t ∈ [0, T ] [16] in Eq. (2) as D(t) = sup [x(s) − x(t)].

(4)

s∈[0,t ]

D(t) is reflected Brownian motion on [0, T ]

{ dD(t) =

−dx(t), max[0, −dx(t)],

D(t) > 0 D(t) = 0

(5)

.

2.2. Period-driven drawdown model The actual financial market is driven by the intrinsic and extrinsic periodic information, and the dynamics of stock price can be described by the following coupled Ito stochastic differential equations [19]: dx(t) = [Ai cos(Ωi t + φi ) − dν (t) = a(b − ν (t))dt + c

ν (t)

]dt +

√ 2 ν (t)dη(t),

√ √ ν (t)Ae sin(Ωe t)dt + ν (t)dξ (t), (6)

where x(t) describes the log of the stock price, Ai are the amplitude of intrinsic (i.e., additive) periodic information, Ωi is the frequency of intrinsic periodic information, φi is the initial phase difference between intrinsic and extrinsic periodic information, Ae is the amplitude of extrinsic (i.e., multiplicative) periodic information and Ωe is frequency of extrinsic periodic information, ν (t) denotes the volatility of the stock price, a denotes the mean reversion of the ν (t), b denotes the long-run variance of the ν (t), c is often called the v olatility of v olatility i.e., it is the amplitude of volatility fluctuations. The deterministic solution of the ν (t) process has an exponential transient with characteristic time equal to a−1 , after which the process tends to its asymptotic value b [69], ξ (t) and η(t) are correlated Wiener processes and have the following statistical properties:

⟨dξ (t)⟩ = ⟨dη(t)⟩ = 0, ⟨dξ (t)dξ (t ′ )⟩ = ⟨dη(t)dη(t ′ )⟩ = δ (t − t ′ )dt , ⟨dξ (t)dη(t ′ )⟩ = ⟨dη(t)dξ (t ′ )⟩ = ρδ (t − t ′ )dt ,

(7)

ρ denotes the cross correlation coefficient between ξ (t) and η(t). Using Heston model to describe the logarithmic stock price, the corresponding D(t) is

⎧ ⎨

dD(t)

=

⎩

dν (t)

=

√ √ −[Ai cos(Ωi t + φi ) − ν 2(t) ]dt − ν (t)Ae sin(Ωe t)dt − ν (t)dξ (t), √ √ ν (t) max {0, −[Ai cos(Ωi t + φi ) − 2 ]dt − ν (t)Ae sin(Ωe t)dt − ν (t)dξ (t), √ a(b − ν (t))dt + c ν (t)dη(t).

{

D(t) > 0 D(t) = 0

,

(8)

2.3. The signal power amplification In order to investigate the roles of time delay on stochastic resonance in the market system[Eq. (8)], we employ the SPA to characterize the stochastic resonance of the system [19,25]:

η = 4A−2 |⟨eiΩ t X (t)⟩|,

(9)

where X (t) is obtained from the ensembles average over the stochastic path D(t) realizations. Through integrating Eq. (8) with a forward Euler algorithm, D(t) can be obtained. After fast Fourier transformation of X (t), we obtain the amplitude of the first harmonic of the output information and η from Eq. (9).

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Fig. 1. Theoretical and empirical comparison of probability density functions (PDF) of drawdown with the kernel density estimation and the bandwidth 0.1.

2.4. Data and estimation Data Parameters in Eq. (6) can be estimated by minimizing the following mean-square deviation [55,70]: (Di ) − Di | P Model Data P (Di )|, where the sum is taken over all available discrete drawdown Di at one day, P (Di ) is calculated from the real market data, and P Model (Di ) is calculated from the Eqs. (6)–(8). In this paper, in order to empirically study, we have chosen WTI spot price of NYMEX as s original sequence. Since it is easy to obtain the daily data, we select the data of WTI crude oil with one trading day as the cycle. The daily closing values from January 2, 1990 to September 28, 2018 yields 7298 samples for one day. The reasons for choosing this data are as follows. On the one hand, WTI crude oil has a high market share. Second, daily data is relatively easy to obtain. Finally, in order to analyze the dynamic behavior of energy markets in recent decades, we selected samples with a long time span. The statistical characteristics of return of this real data are mean = 0.000164, standard dev iation = 0.0226, skew ness = −0.4065 and Kurtosis = 10.703. We can find that WTI futures sequence shows a significant leptokurtic and fat tailed character. This indicates that the volatility range of WTI futures is relatively severe. We study two monotonous cases respectively:

∑

Case 1: Only considering the roles of intrinsic periodic information (i.e., let Ae = 0.0 and φi = 0) and our considered market data set, estimates of µ, a, b, c, ρ , Ai and Ωi are given by µ ˆ = −0.007763923, aˆ = 0.099971879, bˆ = 0.018230891, ˆ i = 0.287776422 respectively. cˆ = 0.049987502, ρˆ = 0.274386777, Aˆi = 0.030721901 and Ω Case 2: Only considering the roles of extrinsic periodic information (i.e., let Ai = 0.0) and our considered market data set, estimates of µ, a, b, c, ρ , Ae and Ωe are given by µ ˆ = −0.007253551, aˆ = 0.100733192, bˆ = 0.015028300, cˆ = 0.050325863, ρˆ = 0.214312462, Aˆe = 0.101603762 and Ωˆ e = 0.283696181, respectively.

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Fig. 2. Logarithmic SPA as a function surface diagram of logarithmic a and ρ for intrinsic periodic information.

To compare probability density function (PDF) of drawdown between the theoretical and true values based on the previous analysis, we take the estimated parameters of the above two cases. We use the Box-Muller method to generate random processes from a Gaussian distribution in simulating the noise sources via Eq. (7) with time integration step △t = 0.01 (as a trading day). The PDF of drawdown is evaluated by numerically simulated over 105 paths based on Eqs. (7) and (8), which is presented in Fig. 1 via the line. As an illustration, the previous real data set is used to calculate the PDF of drawdown with the kernel density estimation and the bandwidth 0.1, which is given in Fig. 1 via the quadrate symbol. Examination of Fig. 1 indicates that our proposed two methods can well fit the real cases. 3. Results and discussion Assets in energy markets are subject to systematic and unsystematic risks and exhibit complex dynamic behaviors. When energy asset prices drawdown sharply, if they exceed the threshold of market leverage levels, it is very easy to trigger a chain crash that leads to an energy financial crisis [18]. There are many factors that cause price drawdown, among which the systematic and non-systematic risks in and out of the market and their resonance behavior are an important factor causing the crisis [19]. Therefore, in this part, we adopt the method of statistical physics, which is the signal power amplification in the previous part, to discuss the resonance behavior of various random risks and internal and external periodic information in the energy market. Based on the above part of the proposed model and estimated parameters, we obtain signal power amplification by means of stochastic simulation. Detailed discussions are described below. 3.1. SR with intrinsic periodic information Risk assets in the energy market are subject to various intrinsic inquire information [19]. In this part, we study the resonance kinetic behavior characteristics of intrinsic periodic information and energy prices, and the results are presented in Figs. 2–6. In order to analyze the resonance effect of noise correlation and mean reversion of volatility, SPA is calculated as a function of noise correlation intensity ρ and mean reversion a of volatility and presented in Fig. 2. In SPA versus a, it can be observed that there is a minimum value corresponding to the minimum SPA value, namely the phenomenon of inverse resonance. This means that there is an optimal a corresponding value for the energy price drawdown with a minimum resonance, implying a lower crash risk. At the same time, it can be found that the minimum value in SPA versus a becomes weaker as the correlation strength ρ increases. That is to say there is a correlation to reduce the drawdown resonance of energy prices. In order to analyze the impact of noise correlation and the long-run variance, we calculate SPA as a function of noise correlation intensity ρ and the long-run variance b of volatility, which is presented in Fig. 3. In the negative correlation region, SPA versus b shows non-monotonic behavior characteristics, and there is an optimal b which makes the drawdown of energy price have the minimum resonance strength. Conversely, in the positive correlation region, SPA versus b shows monotonically increasing behavior, i.e., an increase of b enhances the resonance strength of energy price drawdown. That is to say, there is a positive correlation that induces reverse resonance in SPA versus b. At the same time, it can be found that as the decrease of b, the SPA versus correlation strength ρ shows the characteristic from monotonic behavior to

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Fig. 3. Logarithmic SPA as a function surface diagram of logarithmic b and ρ for intrinsic periodic information.

non-monotonic behavior. In the smaller b area, SPA versus ρ shows a maximum value, that is, there is a worst ρ which makes the energy price drawdown resonance strength maximum. To analyze correlation and the amplitude of the influence of volatility, we calculated the SPA as the noise correlation intensity ρ and amplitude c of volatility function, which is presented in Fig. 4. SPA versus c exhibits non-monotonic behavior, that is, there is an optimal c that minimizes the drawdown resonance strength of the energy price. At the same time, it can be found that with the increase of correlation strength of ρ , the maximum value of SPA versus c shows non-monotonic behavior characteristics. That is, there is an optimal ρ that minimizes the drawdown resonance strength of the energy price. In addition, under the appropriate ρ , in SPA as a function of the noise correlation ρ intensity and amplitude c of volatility, we can find that there are multiple inverse resonance phenomenons. In order to analyze the impact of the correlation and growth rate at macroeconomic scales, SPA is calculated as a function of noise correlation intensity ρ and growth rate µ, which is presented in Fig. 5. The SPA versus growth rate µ shows a monotonically decreasing behavior characteristic, that is, the increase of µ reduces the drawdown resonance strength of the energy price. At the same time, it can be found that with the increase of µ, the SPA versus correlation strength ρ changes from monotonic behavior to non-monotonic behavior. In the appropriate µ area, SPA versus ρ presents a minimum value, that is, there is an optimal ρ which minimizes the drawdown resonance strength of the energy price. To analyze the influence of noise correlation and intrinsic inquire information intensity on resonance behavior, we compute SPA as a function of noise correlation intensity ρ and Ai , which is presented in Fig. 6. In SPA versus Ai , it can be observed that there is a Ai matching the SPA minimum, namely the reverse resonance phenomenon. This means that there is an optimal Ai matching the minimum drawdown resonance of energy price, implying a lower risk of market crash. At the same time, it can be found that with the increase of correlation strength of ρ , the minimum value in SPA versus Ai increases, and SPA versus Ai changes from non-monotonic behavior to monotonic behavior. That said, there is a certain correlation to enhance the drawdown resonance of energy prices. In addition, in the resonance behavior of Figs. 2 and 6, we can find that the correlation strength ρ shows the opposite effects. 3.2. SR with extrinsic periodic information In this part, we study the resonance kinetic behavior characteristics of intrinsic periodic information and energy price. The results are shown in Figs. 7–11. In order to analyze the resonance effect of noise correlation and mean reversion of volatility, SPA is calculated as a function of noise correlation intensity ρ and mean reversion a of volatility, which is presented in Fig. 7. In SPA versus a, it can be observed that there is a minimum value corresponding to the minimum of SPA, namely the phenomenon of inverse resonance. This means that there is an optimal a corresponding to the drawdown resonance of energy prices, implying a lower risk of market crash. At the same time, it can be found that with the increase of correlation strength ρ , the minimum value in SPA versus a first decreases and then increases. In other words, there is an optimal ρ which makes the minimum value in SPA versus a minimum. That is to say there is a correlation to reduce the drawdown resonance of energy prices. The existence of the best ρ and mean reversion a of volatility minimizes the drawdown resonance strength of energy prices, which means the risk of a potential crash is minimal. In order to analyze the impact of noise correlation and the long-run variance, we calculate SPA as a function of noise correlation intensity ρ and the long-run variance b of volatility as shown in Fig. 8. In the negative correlation region,

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Fig. 4. Logarithmic SPA as a function surface diagram of logarithmic c and ρ for intrinsic periodic information.

Fig. 5. Logarithmic SPA as a function surface diagram of logarithmic µ and ρ for intrinsic periodic information.

Fig. 6. Logarithmic SPA as a function surface diagram of logarithmic Ai and ρ for intrinsic periodic information.

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Fig. 7. Logarithmic SPA as a function surface diagram of logarithmic a and ρ for extrinsic periodic information.

Fig. 8. Logarithmic SPA as a function surface diagram of logarithmic b and ρ for extrinsic periodic information.

SPA versus b shows non-monotonic behavior characteristics, and there is an optimal b which makes the drawdown of energy price have the minimum resonance strength. Conversely, in the positive correlation region, the monotonically increasing behavior characteristics of SPA versus b are enhanced b to enhance the drawdown resonance strength of the energy price. That is to say, SPA versus b is correlated to induce the reverse resonance phenomenon. At the same time, it can be found that with the weakening of b, the SPA versus correlation strength of ρ changes from monotonic behavior to non-monotonic behavior. In the smaller b area, SPA versus ρ shows a minimum value, that is, there is an optimal ρ which minimizes the drawdown resonance strength of energy prices, that is, the low risk of a potential crash. In the smaller b area, in the resonance behavior of Figs. 3 and 8, the correlation strength ρ shows the opposite effect. To analyze the influence of correlation and the amplitude of volatility, we calculate the SPA as the noise correlation intensity ρ and amplitude c of volatility function, which is presented in Fig. 9. SPA versus c exhibits non-monotonic behavior, that is, there is an optimal c that minimizes the drawdown resonance strength of the energy price. At the same time, it can be found that with the increase of correlation strength ρ , the maximum value of SPA versus c shows non-monotonic behavior characteristics That is, there is an optimal ρ that minimizes the drawdown resonance strength of the energy price. In addition, under the appropriate ρ , in SPA as a function of the noise correlation intensity ρ and amplitude c of volatility, we can find that there are multiple inverse resonance phenomenon. In order to analyze the impact of the correlation and growth rate at macroeconomic scales, SPA is calculated as a function of noise correlation intensity ρ and growth rate µ, which is presented in Fig. 10. The SPA versus growth rate µ shows a monotonically decreasing behavior characteristic, that is, the increase of µ reduces the drawdown resonance strength of the energy price. At the same time, it can be found that with the increase of µ, the SPA versus correlation

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Fig. 9. Logarithmic SPA as a function surface diagram of logarithmic c and ρ for extrinsic periodic information.

Fig. 10. Logarithmic SPA as a function surface diagram of logarithmic µ and ρ for extrinsic periodic information.

strength of ρ changes from monotonic behavior to non-monotonic behavior. In the appropriate µ area, SPA versus ρ presents a minimum value, that is, there is an optimal ρ which minimizes the drawdown resonance strength of the energy price. To analyze the influence of noise correlation and intrinsic inquire information intensity on resonance behavior, we compute SPA as a function of noise correlation intensity ρ and Ae , which is presented in Fig. 11. In SPA versus Ae , it can be observed that there is a minimum value corresponding to the minimum value of SPA, namely the phenomenon of inverse resonance. This means that there is an optimal Ae corresponding to the drawdown resonance of energy prices, implying a lower risk of market crash. At the same time, it can be found that with the increase of correlation strength of ρ , the minimum value in SPA versus Ae increases, and SPA versus Ae changes from non-monotonic behavior to monotonic behavior. That said, there is a certain correlation to enhance the drawdown resonance of energy prices. In addition, in the resonance behavior of Figs. 7 and 11, we can find that the correlation strength ρ shows the opposite effects. At the end of this section, we further explore the internal mechanism of complex dynamic behavior and its economic significance. From Fig. 2–11, we can find some interesting non-monotonous behavior features. The most important behaviors are inverse resonance and multiple inverse resonance. Like the literature [19], the essential characteristic of inverse resonance comes from antisynchronization between a stochastic time scale (determined by market system) and a deterministic time scale (determined by inquire information), when the internal frequency and periodic information frequency of the system affected by fluctuations of internal and external systems converge. Here, the inverse resonance behavior of drawdown risk is beneficial to energy financial markets. When inverse resonance occurs, it implies that the drawdown risk of the energy market has been greatly suppressed. That said, cyclical information and noise can limit the

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Fig. 11. Logarithmic SPA as a function surface diagram of logarithmic Ae and ρ for extrinsic periodic information.

Fig. 12. Comparison of the return correlation for intrinsic periodic information in (a) and extrinsic periodic information in (b) between real data and proposed model.

risk of a sharp pullback in energy markets. This also provides feasible support for policy control. The results suggest that regulators can use the corresponding periodic information and random information to regulate the market and make the energy market tend to be stable.

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Fig. 13. Comparison of the absolute return correlation for intrinsic periodic information in (a) and extrinsic periodic information in (b) between real data and proposed model.

3.3. Empirical comparison In this section, we further show (i) the probability density function (PDF) of the first passage times (FPT) of the returns and compare with the empirical real market data; (ii) the probability distribution function of volatility and compare with real market data; (iii) the return correlation, and the absolute return correlation. The results are presented in Figs. 12–15. First, we compare the return correlation and the absolute return correlation between the proposed model in Section 2.2 and the real data in Section 2.4 as shown in Figs. 12 and 13. We consider unknown parameter set θi = c(µ, a, b, c, ρ , Ai , Ωi ) only for the case of intrinsic periodic information and θe = c(µ, a, b, c, ρ , Ae , Ωe ) only for the case of extrinsic periodic information. For the return correlation between the real data set and proposed model, we use the same estimates in Section 2.4. For the absolute return correlation between the real data set and proposed model, we calculate the parameter estimates of θi = c( − 0.1791, 0.6749, 0.0101, 0.0503, −0.9514, 0.4941, 0.4049) and θe = c( − 0.2488, 0.5702, 0.00996, 0.0498, 0.0224, 0.0373, 0.3158). Then the return correlation and the absolute return correlation are simulated. The good agreements of the return correlation and the absolute return correlation between the real set and proposed model are observed. These results are consistent with those in Figure 6 in Ref. [63], and are similar to those in Refs. [71,72]. Next, we present an empirical comparison of the PDF of volatility between the proposed model and the real data as shown in Fig. 14. For the calculation of PDF of actual data, we use absolute return volatility to depict the real data’s volatility [73,74]. For the theoretical result, we also calculate the parameter estimates of θi = c( − 0.2102, 0.1561, 0.0339, 0.0932, −0.1284, 0.1221, 0.1921) and θe = c(0.1692, 0.1638, 0.0348, 0.0953, −0.1139, 0.0155, 0.7416). The good agreements of PDF of volatility between the real data set and proposed model are also found, which is consistent with the results in Ref. [63–65]. Finally, To compare the PDFs of FPT of the returns, we first calculate the FPT time series from the real data with a threshold range (Θi = −0.1σr and Θf = −0.5σr ) similar to that in Figure 2 in Ref. [63] and then show the PDF of FPT of real data as the square shown in Fig. 15. For the theoretical result, we calculate the parameter estimates of θi = c(2.112,

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Fig. 14. Comparison of PDF of volatility between real data and proposed model for intrinsic periodic information in (a) and extrinsic periodic information in (b) between real data and proposed model.

13.495, 1.604, 4.088, 0.0068, 0.0975, 0.0682) and θe = c(2.415, 20.572, 1.846, 4.483, −0.1315, 0.0806, 0.3017), and then simulate the PDF of FPT of model as line shown in Fig. 15. The good agreements of PDF of FPT between the real data set and proposed model are also observed. 4. Conclusions Energy is an important pillar of economic development and one of the important influencing factors. Crude oil market has always been an important part of the energy market and occupies a high proportion. Volatility in energy prices, particularly drawdown risks, has seriously affected market stability. Therefore, the study on the price fluctuation of crude oil and its dynamic behavior is of great significance. As a complex dynamic system, the energy market is driven by the periodic information fluctuations inside and outside the market, and is prone to resonance with the systematic and nonsystematic risks in the market. Therefore, in this paper, we discuss the resonance dynamic behavior characteristics of the drawdown time series driven by internal and external periodic information. We get the theoretical model of the drawdown time series of prices based on the periodic Heston model and use it to describe the drawdown risk of energy prices. We adopt signal power amplification to analyze the resonance phenomenon that the internal and external periodic information in the drawdown time series is driven by the noise of the complex energy dynamics system. The influence of information resonance on drawdown risk is further discussed by analyzing the behavior of resonance. Combined with the real data of WTI spot price of NYMEX, we use the least square method of drawdown time series distribution to estimate the parameters of the model. We compare the probability density functions of drawdown for the simulated data from the proposed model and the WTI data, which indicates a good agreement between both. Combined with the

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Fig. 15. Comparison of PDF of FPT between real data and proposed model for intrinsic periodic information in (a) and extrinsic periodic information in (b) between real data and proposed model.

estimation parameters based on the random simulation of signal power amplification under the internal and external periodic information respectively, we can find that: (i) In SPA versus a, SPA versus b, SPA versus Ai and SPA versus Ae , there are optimal values of a, b, Ai , Ae matching the minimum SPA, which means the inverse resonance can be observed; (ii) In SPA versus c, under the appropriate ρ , the presence of multiple inverse resonances can be observed, that is, in SPA versus c, the correlation induces multiple inverse resonances can be found. (iii) In the case of appropriate a, b, c, µ, the reverse resonance phenomenon can also be observed in SPA versus ρ , that is, the presence of the optimal correlation strength can greatly reduce the drawdown risk of energy market price; (iv) The increased growth rate µ strengthens SPA and weakens drawdown risk of energy market price; (v) In the smaller b area, the opposite performance is observed in the resonance behavior of the SPA versus ρ in the performance of intrinsic and extrinsic information. References [1] Y. Mu, C. Wang, W. Cai, The economic impact of China’s INDC: Distinguishing the roles of the renewable energy quota and the carbon market, Renew. Sustain. Energy Rev. 81 (2018) 2955–2966. [2] M. Wang, C. Ying, L. Tian, S. Jiang, Z. Tian, R. Du, Fluctuation behavior analysis of international crude oil and gasoline price based on complex network perspective, Appl. Energy 175 (175) (2016) 109–127. [3] Z. Cheng, The effects of fluctuation of oil prices on related industries, in: International Conference on Artificial Intelligence, 2011. [4] A. Charles, O. Darn, The efficiency of the crude oil markets: Evidence from variance ratio tests, Energy Policy 37 (11) (2009) 4267–4272. [5] J.I. Miller, R.A. Ratti, Crude oil and stock markets: Stability, instability, and bubbles, Energy Econ. 31 (4) (2009) 559–568. [6] I.A. Onour, Crude oil price and stock markets in major oil-exporting countries: evidence of decoupling feature, Int. J. Monet. Econ. Finance 5 (1) (2012) 1–10. [7] H. Lingyun, L. Chuanzhe, Energy finance: Research advances and analytical framework, J. Guangdong Univ. Finance 5 (2009).

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