Hedging and machine learning driven crude oil data analysis using a refined Barndorff-Nielsen and Shephard model
HHedging and machine learning driven crude oil data analysisusing a refined Barndorff-Nielsen and Shephard model
Humayra Shoshi ∗ , Indranil SenGupta † Department of MathematicsNorth Dakota State UniversityFargo, North Dakota, USA.May 1, 2020
Abstract
In this paper, a refined Barndorff-Nielsen and Shephard (BN-S) model is imple-mented to find an optimal hedging strategy for commodity markets. The refinement ofthe BN-S model is obtained with various machine and deep learning algorithms. Therefinement leads to the extraction of a deterministic parameter from the empirical dataset. The problem is transformed to an appropriate classification problem with a coupleof different approaches- the volatility approach and the duration approach. The analy-sis is implemented to the Bakken crude oil data and the aforementioned deterministicparameter is obtained for a wide range of data sets. With the implementation of thisparameter in the refined model, the resulting model performs much better than theclassical BN-S model.
Key Words:
Variance swaps, Quadratic hedging, Drawdown, Classification problems,Stochastic models.
Price risk in commodity trading refers to fluctuation in the price of asset. To reduce pricerisk, traders hedge the commodity price with commodity derivatives such as futures, options,or swaps. Hedging is an act of taking opposite position in the similar market to reduce theprice risk. With appropriate hedging of the underlying position, the loss from one marketis offset by another market.A commodity of fundamental importance is crude oil. Consequently a study of thefluctuation of crude oil price time series is of utmost importance (see [6, 7]). This allowsto evaluate the potential impacts of its shocks in several economies and on other financial ∗ Email: [email protected] † Email: [email protected] a r X i v : . [ q -f i n . M F ] A p r ssets. In [22], the authors analyze the efficiency of crude oil markets by means of esti-mating the fractal structure of these time series. In [20], it is shown that the efficiencyof energy futures markets is time-varying and changes drastically over the sample period.In particular, for futures contracts with one to four months to maturities, crude oil andgasoline are found to be more efficient compared to others. In [5], the authors discuss howthe traditional oil producers may react in counter-intuitive ways in face of competition fromalternative energy sources. The paper considers the big decline in oil prices, from around$110 per barrel in June 2014 to less than $40 in March 2016, and shows the significanceof competition between different energy sources. With the ongoing COVID-19 pandemicsituation, this analysis is very relevant. In [15], the authors present a sequential hypothesistesting on two streams of observations that are driven by L´evy processes. After that, ma-chine learning algorithms are implemented to analyze the oil price dynamics for the Bakkenregion in the United States.A frequently used stochastic volatility model for the commodity market analysis is theBarndorff-Nielsen and Shephard (BN-S) model (see [1, 2, 3, 4, 8, 10]). This model findsvarious applications in the derivative and commodity market. In recent literature, theBN-S model is implemented to find an optimal hedging strategy for the oil commodity(see [19, 23]). In [14], the BN-S model is implemented to the analysis of the S&P 500market using a K -component mixture of regressions model. In spite of having a lot ofadvantages, the classical BN-S model has some major disadvantages including a “short-rangedependence”. In the recent paper [18], a refinement of the BN-S model is proposed. It isshown that a machine learning driven refined BN-S model can be used as an improvement ofthe classical BN-S model. The analysis is further improved in [16], where a machine learningdriven sequential hypothesis testing is implemented to refine the BN-S model. In both thepapers ([16] and [18]), machine learning based techniques are implemented for extractinga deterministic component from the commodity price processes. Also, the refined BN-Smodel is shown to incorporate long range dependence without actually changing the model.In this paper, we investigate the refinement of the BN-S model by analyzing the underly-ing data set with a couple of different approaches- (1) volatility approach, and (2) durationapproach. In effect, these approaches provide a “jump-detection technique” for a financialtime series. The papers [11, 12, 13, 21], discuss various motivations for these approaches.In [11], it is observed that by fitting the log-periodic power law equation to a financial timeseries, it is possible to predict the event of a crash. The paper investigates the financial crisisof 2008, with the log-periodic power law. In [12], drawdowns, defined as the loss from thelast local maximum to the next local minimum, is introduced. It is shown that drawdownscan be used as a natural measure of real market risks than the variance, the value-at-riskor other measures based on fixed time scale distributions of returns. It is shown that verylarge drawdowns belong to a different class of their own and call for a specific amplifica-tion mechanism. In [13], drawdowns are implemented and crashes are classified as eitherevents of an endogenous origin preceded by speculative bubbles or as events of exogenousorigins associated to external shocks. However, the proposed classification does not rule outthe existence of other precursory signals in the absence of so-called log-periodic power lawsignatures. In [21], the price volatility before, during, and after financial asset bubbles are2nvestigated for possible commonalities. It is also empirically investigated whether volatilitymay be used as an indicator or an early warning signal of an unsustainable price increaseand the associated crash.The organization of the paper is as follows. In Section 2, a refined BN-S model ispresented. Some useful properties of variance swaps with respect to the refined BN-S modelis studied. In addition, a quadratic hedging procedure is discussed. In Section 3, the dataset is provided, and then two procedures, the volatility approach and the duration approach,in the classification problem are introduced. Various numerical results are also provided inthat section. Finally, a brief conclusion is provided in Section 4. Many models in recent literature try to capture the stochastic behavior of time series. Forexample, in the case of the Barndorff-Nielsen and Shephard (BN-S) model, the stock orcommodity price S = ( S t ) t ≥ on some risk-neutral filtered probability space is modeled by S t = S exp( X t ) , (2.1) dX t = b t dt + σ t dW t + ρ dZ λt , with b t = ( r − λκ ( ρ ) − σ t ) , (2.2) dσ t = − λσ t dt + dZ λt , σ > , (2.3)where the parameters ρ, λ ∈ R with λ >
0, and ρ ≤
0. Here r is the risk-free interest ratewhere a stock or commodity is traded up to a fixed horizon date T . In the expression for b t , the cumulant transform for Z under the new measure is denoted as κ ( · ). In this model W t is a Brownian motion and the process Z t is a subordinator. For a refined BN-S model(see [18]) the stock or commodity price S = ( S t ) t ≥ on some risk-neutral filtered probabilityspace (Ω , F , ( F t ) ≤ t ≤ T , Q ) is modeled by (2.1), with dX t = b t dt + σ t dW t + ρ (cid:16) (1 − θ ) dZ λt + θdZ ( b ) λt (cid:17) , (2.4)where θ ∈ [0 ,
1] is a deterministic parameter, and b t is given by (2.2). Machine learningalgorithms are implemented to determine the value of θ . The process Z ( b ) in (2.4) is asubordinator that is independent of Z . In addition, Z ( b ) has greater intensity than thesubordinator Z . W , Z and Z ( b ) are assumed to be independent, and ( F t ) is assumed to bethe usual augmentation of the filtration generated by ( W, Z, Z ( b ) ).In this case (2.3) is given by dσ t = − λσ t dt + (1 − θ (cid:48) ) dZ λt + θ (cid:48) dZ ( b ) λt , σ > , (2.5)where, as before, θ (cid:48) ∈ [0 ,
1] is deterministic . For simplicity, we assume θ = θ (cid:48) for the rest ofthis paper. 3s shown in [18], the dynamics given by (2.1), (2.4), and (2.5) incorporates a long-rangedependence. If the jump measures associated with the subordinators Z and Z ( b ) are J Z and J ( b ) Z respectively, and J ( s ) = (cid:82) s (cid:82) R + J Z ( λdτ, dy ), J ( b ) ( s ) = (cid:82) s (cid:82) R + J ( b ) Z ( λdτ, dy ); thenfor the log-return of the improved BN-S model given by (2.1), (2.4), and (2.5),Corr( X t , X s ) = (cid:82) s σ τ dτ + ρ (1 − θ ) J ( s ) + ρ θ J ( b ) ( s ) (cid:112) α ( t ) α ( s ) , (2.6)for t > s , where α ( ν ) = (cid:82) ν σ τ dτ + νρ λ ((1 − θ ) Var( Z ) + θ Var( Z ( b )1 )).We observe that the solution of (2.5) can be written as σ t = e − λt σ + (1 − θ ) (cid:90) t e − λ ( t − s ) dZ λs + θ (cid:90) t e − λ ( t − s ) dZ ( b ) λs . (2.7)This enforces positivity of σ t . Thus, the process σ t is strictly positive and it is boundedfrom below by the deterministic function e − λt σ . The instantaneous variance of log returnsis given by ( σ t + ρ (1 − θ ) λ Var[ Z ] + ρ θ λ Var[ Z ( b )1 ]) dt, and therefore simple calculation shows that the continuous realized variance in the interval[0 , T ] is σ R = 1 T (cid:90) T σ t dt + ρ (1 − θ ) λ Var[ Z ] + ρ θ λ Var[ Z ( b )1 ] . For the rest of this section we develop a procedure to show an effective hedging algorithmusing the refined BN-S model. In Subsection 2.1, we briefly introduce some results relatedto the variance swap. In Subsection 2.2 we develop results related to hedging algorithmwhere variance swaps and some specific options are used.
A variance swap is a forward contract on realized variance (see [8, 9, 10]). The payoff ofvariance swap at the maturity T is given by N ( σ R − K Var ), where K Var is the annualizeddelivery price or exercise price of the variance swap, and N is the notional amount of thedollars per annualized volatility point squared. Without loss of generality we take N = 1.The arbitrage free price of the variance swap is the expectation of the present value of thepayoff in the risk-neutral world and it is given by E Q (cid:2) e − r ( T − t ) ( σ R − K Var ) |F t (cid:3) , 0 ≤ t ≤ T ,where F t is the σ -field generated by the history of the process up to time t . When F t isgiven and s ≥ t a similar derivation as in (2.7) gives σ s = e − λ ( s − t ) σ t + (1 − θ ) (cid:90) st e − λ ( s − u ) dZ λu + θ (cid:90) st e − λ ( s − u ) dZ ( b ) λu . (2.8)We denote V t = (cid:82) t σ u du . For a fixed horizon date T , we consider P Var ( t, σ t , V t ) as afunction of t , σ t and V t with the final condition (independent of S ) given by P Var ( T, σ T , V T ) = σ R − K Var = V T T − K Var . σ R = 1 T (cid:90) T σ s ds + ρ (1 − θ ) λ Var[ Z ] + ρ θ λ Var[ Z ( b )1 ]= 1 T (cid:18)(cid:90) t σ s ds + (cid:90) Tt σ s ds (cid:19) + ρ (1 − θ ) λ Var[ Z ] + ρ θ λ Var[ Z ( b )1 ]= 1 T (cid:18) V t + 1 λ (1 − e − λ ( T − t ) ) σ t + 1 − θλ (cid:90) Tt (cid:16) − e − λ ( T − s ) (cid:17) dZ λs + θλ (cid:90) Tt (cid:16) − e − λ ( T − s ) (cid:17) dZ ( b ) λs (cid:19) + ρ (1 − θ ) λ Var[ Z ] + ρ θ λ Var[ Z ( b )1 ] . (2.9)Based on this result we can prove the following theorem. Theorem 2.1.
The arbitrage free price of the variance swap, with respect to the risk neutralmeasure Q , is given byP Var ( t, σ t , V t ) = e − r ( T − t ) (cid:104) V t + ( T − t ) (cid:16) κ (1 − θ ) + κ ( b )1 θ (cid:17) + 1 λ (cid:16) − e − λ ( T − t ) (cid:17) (cid:16) σ t − κ (1 − θ ) − κ ( b )1 θ (cid:17) + ρ (1 − θ ) λκ + ρ θ λκ ( b )2 − K Var (cid:21) , where κ and κ are the first cumulant (i.e., the expected value) and the second cumulant(i.e., the variance) of Z respectively; and κ ( b )1 and κ ( b )2 are the first cumulant (i.e., theexpected value) and the second cumulant (i.e., the variance) of Z ( b )1 respectively.Proof. The conditional expected value, given F t , of equation (2.9) gives the value E ( σ R |F t ) = 1 T ( V t + 1 λ (1 − e − λ ( T − t ) ) σ t + (1 − θ ) κ λ (cid:90) Tt (cid:16) − e − λ ( T − s ) (cid:17) λ ds + θκ ( b )1 λ (cid:90) Tt (cid:16) − e − λ ( T − s ) (cid:17) λ ds ) + ρ (1 − θ ) λ Var[ Z ] + ρ θ λ Var[ Z ( b )1 ]= 1 T ( V t + 1 λ (1 − e − λ ( T − t ) ) σ t + κ (1 − θ ) (cid:18) T − t − λ (cid:16) − e − λ ( T − t ) (cid:17)(cid:19) + κ ( b )1 θ (cid:18) T − t − λ (cid:16) − e − λ ( T − t ) (cid:17)(cid:19) ) + ρ (1 − θ ) λκ + ρ θ λκ ( b )2 . (2.10)Hence the theorem follows from simplification of (2.10). In this subsection, we show that there is an effective hedging procedure in relation to therefined BN-S model given by (2.1), (2.4), and (2.5). With respect to Q , the dynamics of S t is given by dS t S t = rdt + σ t dW t + (cid:90) R + ( e ρ (1 − θ ) x −
1) ˜ J Z ( λdt, dx ) + (cid:90) R + ( e ρθx −
1) ˜ J Z ( b ) ( λdt, dx ) , (2.11)5here we assume that random measures associated with the jumps of Z and Z ( b ) , andL´evy densities of Z and Z ( b ) are given by J Z , J Z ( b ) , and ν Z , ν Z ( b ) , respectively. The com-pensator for J Z ( λdt, dx ) is given by λν Z ( dx ) dt and we define ˜ J Z ( λdt, dx ) = J Z ( λdt, dx ) − λν Z ( dx ) dt . Similarly, the compensator for J Z ( b ) ( λdt, dx ) is given by λν ( b ) Z ( dx ) dt and wedefine ˜ J Z ( b ) ( λdt, dx ) = J Z ( b ) ( λdt, dx ) − λν Z ( b ) ( dx ) dt .As introduced in [19] and [23], we consider a “stable” commodity Y t given by (withrespect to Q ) a geometric Brownian motion dY t = Y t ( r dt + σ d ˜ W t ) , (2.12)with d ˜ W t · dW t = ρ (cid:48) dt , with W t defined in (2.4) (same as in (2.11)), and σ > Theorem 2.2.
Consider a European option with payoff H ( Y T ) where H : R + → R . Thenthe risk-minimizing quadratic hedge amounts to holding a position of the underlying S equalto φ t = ∆( t, S t , Y t ) , where ∆( t, S t , Y t ) = ρ (cid:48) σσ t Y t S t ∂C∂Y + A + Bσ t + λ (cid:82) R + ( e ρ (1 − θ ) x − ν Z ( dx ) + λ (cid:82) R + ( e ρθx − ν ( b ) Z ( dx ) , (2.13) where C is the Black-Scholes price of the option written on Y , and A = λ (1 − θ ) S t (cid:90) R + (cid:0) P ( t, σ t + x, V t ) − P ( t, σ t , V t ) (cid:1) ( e ρ (1 − θ ) x − ν Z ( dx ) , (2.14) B = λθS t (cid:90) R + (cid:0) P ( t, σ t + x, V t ) − P ( t, σ t , V t ) (cid:1) ( e ρθx − ν ( b ) Z ( dx ) . (2.15) Proof.
From (2.11), it is clear that the discounted commodity price ˆ S t = e − rt S t is a mar-tingale with respect to Q . We consider a self financing strategy ( φ t , φ t ) with φ ∈ L ( ˆ S ).The discounted value of the portfolio ( ˆΠ) is then a martingale with terminal value given byˆΠ T ( φ ) = (cid:90) T φ t d ˆ S t = (cid:90) T φ t ˆ S t (cid:18) σ t dW t + (cid:90) R + ( e ρ (1 − θ ) x −
1) ˜ J Z ( λdt, dx ) + (cid:90) R + ( e ρθx −
1) ˜ J Z ( b ) ( λdt, dx ) (cid:19) = (cid:90) T φ t ˆ S t σ t dW t + (cid:90) T φ t ˆ S t (cid:18)(cid:90) R + ( e ρ (1 − θ ) x −
1) ˜ J Z ( λdt, dx ) + (cid:90) R + ( e ρθx −
1) ˜ J Z ( b ) ( λdt, dx ) (cid:19) . (2.16)The arbitrage-free price of the option written on the commodity Y with payoff H ( Y T ) isgiven by C ( t, Y ) = e − r ( T − t ) E Q [ H ( Y T ) | Y t = Y ] .
6e denote ˆ C ( t, Y ) = e − rt C ( t, Y ) and Π = ˆ C (0 , Y ) = e − rT E Q [ H ( Y T )]. Then, by Itˆoformula we obtain ˆ C ( t, Y t ) − Π = (cid:90) t ∂C∂Y ( u, Y u ) ˆ Y u σ d ˜ W u . (2.17)On the other hand, if we consider a variance swap written on S t , and denote ˆ P ( t, σ t , V t ) = e − rt P ( t, σ t , V t ), ˆ˜ P ( t, σ t , V t ) = e − rt ˜ P ( t, σ t , V t ), and Π = e − rT ˜ P (0 , σ , V ) = P (0 , σ , V ),then, using Itˆo formula we obtain: e − rT ˜ P ( t, σ t , V t ) − Π = (1 − θ ) (cid:90) t (cid:90) R + (cid:16) ˆ P ( s, σ s − + x, V s ) − ˆ P ( s, σ s − , V s ) (cid:17) ˜ J Z ( λds, dx )+ θ (cid:90) t (cid:90) R + (cid:16) ˆ P ( s, σ s − + x, V s ) − ˆ P ( s, σ s − , V s ) (cid:17) ˜ J Z ( b ) ( λds, dx ) . (2.18)We denote Π = Π + Π , and (cid:15) ( φ, Π ) = ˆΠ T ( φ ) + Π − ˆ C ( T, Y T ) − ˆ˜ P ( T, σ T , V T ). Notethat ˜ P ( T, σ T , V T ) = P ( T, σ T , V T ), and thus we have (cid:15) ( φ, Π ) = ˆΠ T ( φ ) + Π − ˆ C ( T, Y T ) − ˆ P ( T, σ T , V T ) . (2.19)Considering expressions in (2.17) and (2.18) at t = T , adding those, and subtracting from(2.16) we obtain (cid:15) ( φ, Π ) = (cid:90) T φ t ˆ S t σ t dW t − (cid:90) T ∂C∂Y ˆ Y t σ d ˜ W t + (cid:90) T (cid:90) R + (cid:104) φ t ˆ S t ( e ρ (1 − θ ) x − − (1 − θ ) (cid:16) ˆ P ( t, σ t + x, V t ) − ˆ P ( t, σ t , V t ) (cid:17)(cid:105) ˜ J Z ( λdt, dx )+ (cid:90) T (cid:90) R + (cid:104) φ t ˆ S t ( e ρθx − − θ (cid:16) ˆ P ( t, σ t + x, V t ) − ˆ P ( t, σ t , V t ) (cid:17)(cid:105) ˜ J Z ( b ) ( λdt, dx ) . Using the isometry formula and observing E Q [ (cid:15) ( φ, Π )] = 0, we obtain the variance of (cid:15) ( φ, Π ) as E Q [ (cid:15) ( φ, Π )] = E Q (cid:20)(cid:90) T φ t ˆ S t σ t dt (cid:21) + E Q (cid:34)(cid:90) T (cid:18) ∂C∂Y (cid:19) ˆ Y t σ dt (cid:35) + E Q (cid:20)(cid:90) T (cid:90) R + (cid:104) φ t ˆ S t ( e ρ (1 − θ ) x − − (1 − θ ) (cid:16) ˆ P ( t, σ t + x, V t ) − ˆ P ( t, σ t , V t ) (cid:17)(cid:105) λν Z ( dx ) dt (cid:21) + E Q (cid:20)(cid:90) T (cid:90) R + (cid:104) φ t ˆ S t ( e ρθx − − θ (cid:16) ˆ P ( t, σ t + x, V t ) − ˆ P ( t, σ t , V t ) (cid:17)(cid:105) λν ( b ) Z ( dx ) dt (cid:21) − E Q (cid:20) ρ (cid:48) σ (cid:90) T φ t ˆ S t ˆ Y t σ t ∂C∂Y dt (cid:21) . φ t . Differentiating the quadratic expression we obtain the first order condition2 φ t ˆ S t σ t − ρ (cid:48) σ ˆ S t ˆ Y t σ t ∂C∂Y + 2 (cid:90) R + (cid:104) φ t ˆ S t ( e ρ (1 − θ ) x − − (1 − θ ) (cid:16) ˆ P ( t, σ t + x, V t ) − ˆ P ( t, σ t , V t ) (cid:17)(cid:105) ˆ S t ( e ρ (1 − θ ) x − λν Z ( dx )+ 2 (cid:90) R + (cid:104) φ t ˆ S t ( e ρθx − − θ (cid:16) ˆ P ( t, σ t + x, V t ) − ˆ P ( t, σ t , V t ) (cid:17)(cid:105) ˆ S t ( e ρθx − λν ( b ) Z ( dx ) = 0 . (2.20)Also, in this case the second order condition is positive, which confirms the minimization.Solution of (2.20) is given by (2.13).We conclude this section with the application of the above result to an explicit casewhen P ( t, σ t , V t ) is given by Theorem 2.1. Corollary 2.3.
Consider the refined BN-S model given by (2.1) , (2.4) and (2.5) (with θ (cid:48) = θ ). Consider a European option with payoff H ( Y T ) where H : R + → R . Then therisk-minimizing quadratic hedge amounts to holding a position of the underlying S equal to φ t = ∆( t, S t , Y t ) , where ∆( t, S t , Y t ) = ρ (cid:48) σσ t Y t S t ∂C∂Y + A + Bσ t + λ (cid:82) R + ( e ρ (1 − θ ) x − ν Z ( dx ) + λ (cid:82) R + ( e ρθx − ν ( b ) Z ( dx ) (2.21) where C is the Black-Scholes price of the option written on Y ,and A = (1 − θ ) S t e − r ( T − t ) (1 − e − λ ( T − t ) ) (cid:82) R + x ( e ρ (1 − θ ) x − ν Z ( dx ) B = θS t e − r ( T − t ) (1 − e − λ ( T − t ) ) (cid:82) R + x ( e ρθx − ν ( b ) Z ( dx ) .Proof. The proof follows directly with the application of Theorem 2.2 in the expressions forthe A and B in (2.13). In this section, at first in Subsection 3.1, we present an overview of the empirical dataset. After that, in Subsection 3.2, we develop a couple of procedures for the data analysis.Finally, the results of the data analysis and the implication of the results for the refinedBN-S model are presented in Subsection 3.3.
We consider crude oil price data over a period of 7 years. We use the daily Bakken crude oilprice data set for the period April 4, 2012 to July 11, 2017 (Figure 1) . There are a total of8 ,
329 available data in this set. For convenience, we index the dates (for available data) from0 (for April 4, 2012) to 1328 (for July 11, 2017). The following table (Table 1) summarizesvarious estimates for the data set. Figures 1, 2, and 3, show various characterization of thedata set. Table 1: Properties of the empirical data set.Daily Price Change Daily Price Change %Mean -0.03787 -0.02183 %Median -0.01000 0.019992 %Maximum 7.40 15.05 %Minimum -7.76 -15.36 %Figure 1: Line plot for the Bakken oil price from April 2012- July 2017.Figure 2: Distribution plot for the Bakken oil price.9igure 3: Histogram for the Bakken oil price.
For the data analysis we present here two different approaches, the aim of which is to finda θ with reasonable accuracy. First, we implement the following procedure, naming it, Volatility Approach , to create a classification problem for the data set.
Volatility approach
We work through the following steps (Step 1 through Step 7).1. We conduct exploratory data analysis.2. We consider the daily
Bakken Oil Price for the data, and we calculate the daily pricechange and the daily price log returns using it. Using the daily price log returns wecalculate the realized variance and the realized volatility respectively.3. We compute the realized volatility over 20 consecutive trading days for the oil prices.Since the computed realized volatility is very small, in order to properly utilize thevolatility movements we create a new feature (column) that contains the realizedvolatility return in percentage, and we call it “realized volatility return in percentage” .4. Using the realized volatility return in percentage feature we perform the followingsteps:4a. We consider twenty consecutive days starting from index 0 (day 1) to index 19 (day20). We compute the maximum realized volatility return in percentage for thosetwenty trading days. We then try to identify realized volatility return in percentage value(s), in those twenty trading days, which is strictly greater than or equal to themaximum. We assign V = 1 if we find such values, otherwise V = 0.10b. We continue step 4a for index 1 (day two) to index 20 (day twenty one) and so onrespectively until we have checked through all the data points in our realized volatilityreturn in percentage feature. We call V crash-like days .5. We create a new data-frame from the old one where the features will be twentyconsecutive daily change in prices. For example, if the daily change in prices are a , a , a , · · · , a , a , a , a , a , a , a , · · · ;then the first row of the data set will contain a , a , a , · · · , a , a , a ;second row of the data set will contain a , a , · · · , a , a , a , a ;6. We create a target column for the new data-frame (as created in the preceding step)as follows: θ = 1 for those set of twenty Bakken oil prices that immediately precede at least 1 (or more) crash-like days in the following twenty days. Otherwise we labelthe target column by θ = 0 .
7. We run various classification algorithms from machine learning where the input is the daily change in close price for twenty consecutive days and output is θ -value (0 or1). We evaluate the classification report and confusion matrix in each case.Figures 4-7 show various characterization of the data set related to the volatility ap-proach described above. The purpose of the heatmap in Figure 4 is to better understandthe realized volatility calculated over a period of twenty days for our entire data set. Thegoal is to use the numerical values and color pattern to observe any big changes for everymonth over the period of five years. As we can see that the realized volatility have verysmall values. This motivates us in computing realized volatility return in percentage. Thisis shown in Figure 5. For Figure 5, using the numerical values and color pattern from theheatmap we observe that over the five years the realized volatility return in percentage doesnot have any drastic change except for one outlier on July 2017. Figure 6 and Figure 7represent line plots which show us the jumps in the realized volatility return in percentageand the realized volatility over the five years, respectively. With the help of these figureswe can see the highest jumps over the years, which also provides help in writing Step-4 ofthe above procedure. 11igure 4: Heatmap for the realized volatility of the Bakken oil price over five years.Figure 5: Heatmap for the realized volatility return in percentage over the five years for theBakken crude oil price.Figure 6: Line plot for the realized volatility return in percentage for the Bakken oil price.12igure 7: Line Plot for the realized volatility of the Bakken oil price.Next, we present the second approach to our data analysis. We implement the followingprocedure, naming it, Duration Approach , to create a classification problem for the dataset.
Duration approach
We work through the following steps (Steps 1 through 7)1. We conduct exploratory data analysis.2. We consider the daily
Bakken oil price for the data. From the oil prices we calculatethe daily change and drawdowns for the prices. A drawdown is the total loss over con-secutive days from the last maximum to the next minimum of the price. A drawdownoccurring over n days is described as d = p min − p max p max with p max = p ( t ) > p ( t ) > · · · > p ( t n ) = p min , where t , · · · t n , and p ( t i ) are the time period over n days and Bakken oil prices re-spectively.It is to be noted that we will not include those prices in the drawdown calculationwhere the next minimum price occurs at the beginning of the data set before the last maximum price as well as the last maximum price that occurs at the very end of thedata set (for example- if first minimum of the oil price is on day 5 (index 4) and firstmaximum is on day 7 (index 6), we will drop the minimum price of day 5 from ourcomputation.)3. Our goal is to identify the dates when the drawdowns occurred in order to find the duration of each drawdown, i.e. how long the drawdowns lasted. We will use theduration of the drawdowns as a measure to identify crash like days in our data set.13. We fix a value for our duration, D , and we obtain the drawdowns that lasted for that D time period (for example, if our duration period is two days, i.e. D=2 , then wewill search for drawdowns that lasted for two days (or more), and take note of theircorresponding daily change prices.)5. We create a new data-frame from the old one where the features (columns) will beten consecutive daily change in oil prices. For example, if the daily change prices are a , a , a , · · · , a , a , a , a , · · · ;then the first row of the data set will contain a , a , a , a , a , a , a , , a , a , a ;second row of the data set will contain a , a , a , a , a , a , a , a , a , a ;etc.6. We create a target column for the new data-frame (as created in the preceding step)as follows: θ = 1 for those set of ten daily change prices that immediately precede at least two drawdowns with duration D , in the following ten days. Otherwise we labelthe target column by θ = 0 .
7. We run various classification algorithms from machine learning where the input is the daily change in close price for ten consecutive days and output is θ -value (0 or 1).We evaluate the classification report and confusion matrix in each case.Figures 8 and 9 show various characterization of the data set related to the durationapproach described above. With the help of the bar graph in Figure 8, we can see thatmost drawdowns last for short period duration. For example, a more likely duration is ofone or two days, compared to long duration of eight or nine days. In Figure 9, the spikes inthe line plot give us some idea about the changes associated with the drawdowns in termsof duration (in number of days), over the period of five years. For example, between 2014and 2015 most drawdowns lasted for one day or two days, and very few drawdowns wentpast four days. 14igure 8: A bar graph to show the duration in the number of days for the drawdownscomputed for the Bakken oil price.Figure 9: A line plot to show the duration over the span of five years.From these two approaches we will show that we can find θ with reasonable accuracyand use this for (2.4). In both the Volatility Approach and the
Duration Approach theresult can be improved by adjusting the number of days (in Step 5 for both) from twentyand ten respectively to a higher number. It is worth noting that the various deep learningmodels provide a value of θ between 0 and 1. In Step 6 (for both), we approximate that by0 or 1. However, the actual value of θ may be directly used in (2.4).We partition this data set in various ways. For each partition we use a train-test-split ,with respect to a given date. For the analysis using the Volatility Approach we use the maximum to detect crash-like days for each set of twenty data points, i.e. θ = 1 for theset of twenty daily change prices that immediately precede at least one crash-like days (ormore) in the following twenty days. Otherwise, we use θ = 0. For the analysis using the Duration Approach we use
D= 2 , i.e. θ = 1 for the set of ten daily change prices thatimmediately precede at least two drawdowns of duration D=2 (or more) in the followingten days. Otherwise, we use θ = 0.We run various supervised learning algorithms on the crude oil price data. We beginwith the logistic regression (LR) and the random forest (RF) classification of the data set.15fter that, we implement various deep learning techniques:(A) A neural network with three hidden layers (with activation functions consisting oftwo tanh and one ReLU respectively) and an output layer (with a softmax activationfunction). For simplicity we approximate θ in (2.4) with 0 (for example: “durationwith less than two days”) and 1 (for example: “duration with more than two days”).For these approximations, we take θ = 1 if the output probability for the softmaxactivation function corresponding to θ = 1 is more than 0 . Long short-term memory (LSTM) along with the neural network described in (A).(C)
LSTM along with a batch normalizer (BN) and the neural network described in (A).
For the following tables, we provide classification reports for various machine learning algo-rithms. The support is the number of samples of the true response that lie in that class. Forthe tables (Table 2 through Table 9), we provide classification reports for various machinelearning algorithms using the Volatility Approach. For tables (Table 10 through Table 18)we provide classification reports for various machine learning algorithms using the DurationApproach. Finally, for tables (Table 19 through Table 21), we provide classification reportsusing both Volatility Approach and Duration Approach over the same training and testingdates.As observed in [18], to incorporate long range dependence, a single L´evy subordinatoris not effective for the BN-S model. If θ = 1 is obtained with the help of machine learningalgorithms, we can modify the initial L´evy subordinator ( Z ) with the L´evy subordinator( Z ( b ) ) that corresponds to larger fluctuations. On the other hand if θ = 0 is obtained withthe help of machine learning algorithms, we can modify the L´evy subordinator Z ( b ) with Z .From the tables, it is obvious that the logistic regression is less efficient in detecting θ = 1based on the historical data. As observed in [18], for the majority of the cases the neuralnetwork technique (A), LSTM (B), or the LSTM with a batch normalizer (C), work betterthan the random forest classifier. To avoid complexity, only three hidden layers are used.The results improve if the number of hidden layers is increased and also if the learning rateof the gradient descent method used is decreased.After θ is obtained, its value can be implemented to (2.4). The machine learning algo-rithms can be performed on a real-time basis to continue or update with the backgrounddriving L´evy process in the BN-S model. The analysis shows that for the Bakken oil pricedynamics, the jump is not completely stochastic. Similar to the results obtained in WestTexas Intermediate (WTI or NYMEX) crude oil prices data set, as obtained in [15, 16, 18],there is a deterministic element that can be implemented to apply the existing models foran extended period of time. Consequently, the refined BN-S model incorporates long termdependence without changing the tractability of the model.16able 2: Various estimations for training date(index) : January 16, 2013(200) to June 11,2013 (300); and testing date(index) : June 12 (301) to July 10 (320).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.33 0.38 0.50 0.67 0.67Recall θ = 0 1.00 1.00 0.17 1.00 1.00f1-score θ = 0 0.50 0.55 0.25 0.80 0.80Support θ = 0 6 6 6 6 6Precision θ = 1 1.00 1.00 0.74 1.00 1.00Recall θ = 1 0.20 0.33 0.93 0.80 0.80f1-score θ = 1 0.33 0.50 0.82 0.89 0.89Support θ = 1 15 15 15 15 15Table 3: Various estimations for training date(index) : February 5, 2014 (465) to June 2,2014 (545); and testing date(index) : June 3, 2014 (546) to July 8, 2014 (570).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.32 0.35 0.17 0.40 0.36Recall θ = 0 1.00 1.00 0.12 1.00 1.00f1-score θ = 0 0.48 0.52 0.14 0.57 0.53Support θ = 0 8 8 8 8 8Precision θ = 1 1.00 1.00 0.65 1.00 1.00Recall θ = 1 0.06 0.17 0.72 0.33 0.22f1-score θ = 1 0.11 0.29 0.68 0.50 0.36Support θ = 1 18 18 18 18 18Table 4: Various estimations for training date(index) : June 9, 2015 (802) to December 9,2015 (930); and testing date(index) : December 10, 2015 (931) to February 8, 2016 (970).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.73 0.76 0.68 0.70 0.72Recall θ = 0 0.87 1.00 0.55 0.74 0.84f1-score θ = 0 0.79 0.86 0.61 0.72 0.78Support θ = 0 31 31 31 31 31Precision θ = 1 0 0 0.12 0 0Recall θ = 1 0 0 0.20 0 0f1-score θ = 1 0 0 0.15 0 0Support θ = 1 10 10 10 10 1017able 5: Various estimations for training date(index) : November 10, 2015 (910) to March8, 2016 (990); and testing date(index) : March 9, 2016 (991) to April 8, 2016 (1012).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.58 0.59 0.67 0.50 0.57Recall θ = 0 0.50 0.93 0.43 0.64 0.86f1-score θ = 0 0.54 0.72 0.52 0.56 0.69Support θ = 0 14 14 14 14 14Precision θ = 1 0.36 0 0.43 0 0Recall θ = 1 0.44 0 0.67 0 0f1-score θ = 1 0.40 0 0.52 0 0Support θ = 1 9 9 9 9 9Table 6: Various estimations for training date(index) : April 6, 2016 (1010) to August 5,2016 (1095); and testing date(index) : August 6, 2016 (1096) to September 6, 2016 (1116).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.60 0.59 1.00 0.68 0.68Recall θ = 0 0.92 1.00 0.08 1.00 1.00f1-score θ = 0 0.73 0.74 0.14 0.81 0.81Support θ = 0 13 13 13 13 13Precision θ = 1 0.50 0 0.43 1.00 1.00Recall θ = 1 0.11 0 1.00 0.33 0.33f1-score θ = 1 0.18 0 0.60 0.50 0.50Support θ = 1 9 9 9 9 9Table 7: Various estimations for training date(index) :September 12, 2016 (1120) to January12, 2017 (1205); and testing date(index) : January 13, 2017 (1206) to February 17, 2017(1230). LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.58 0.58 0.50 0.61 0.53Recall θ = 0 1.00 1.00 0.67 0.73 0.67f1-score θ = 0 0.73 0.73 0.57 0.67 0.59Support θ = 0 15 15 15 15 15Precision θ = 1 0 0 0.17 0.50 0.29Recall θ = 1 0 0 0.09 0.36 0.18f1-score θ = 1 0 0 0.12 0.42 0.22Support θ = 1 11 11 11 11 1118able 8: Various estimations for training date(index) :October 24, 2016 (1150) to March 27,2017 (1255); and testing date(index) : March 28, 2017 (1256) to May 26, 2017 (1298)LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.83 0.93 1.00 1.00 1.00Recall θ = 0 0.38 0.69 0.26 0.13 0.28f1-score θ = 0 0.53 0.79 0.41 0.23 0.44Support θ = 0 39 39 39 39 39Precision θ = 1 0.08 0.20 0.15 0.13 0.15Recall θ = 1 0.40 0.60 1.00 1.00 1.00f1-score θ = 1 0.13 0.30 0.26 0.23 0.26Support θ = 1 5 5 5 5 5Table 9: Various estimations for training date(index) : January 3, 2017 (1198) to March 20,2017 (1250); and testing date(index) : March 21, 2017 (1251) to April 18, 2017 (1270).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0 0.50 0.12 0 0Recall θ = 0 0 0.27 0.09 0 0f1-score θ = 0 0 0.35 0.11 0 0Support θ = 0 11 11 11 11 11Precision θ = 1 0.15 0.47 0.23 0.35 0.27Recall θ = 1 0.20 0.70 0.30 0.60 0.40f1-score θ = 1 0.17 0.56 0.26 0.44 0.32Support θ = 1 10 10 10 10 10Table 10: Various estimations for training date(index) : January 16, 2013(200) to June 11,2013 (300); and testing date(index) : June 12 (301) to July 10 (320).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.56 0.69 0.75 0.67 0.67Recall θ = 0 0.69 0.69 0.69 0.77 0.46f1-score θ = 0 0.62 0.69 0.72 0.71 0.55Support θ = 0 13 13 13 13 13Precision θ = 1 0.20 0.50 0.56 0.50 0.42Recall θ = 1 0.12 0.50 0.62 0.38 0.62f1-score θ = 1 0.15 0.50 0.59 0.43 0.50Support θ = 1 8 8 8 8 819able 11: Various estimations for training date(index) : January 14, 2014 (450) to June 16,2014 (555); and testing date(index) : June 17, 2014 (556) to August 12, 2014 (595).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.20 0.15 0.20 0.14 0.08Recall θ = 0 0.78 0.44 0.11 0.44 0.11f1-score θ = 0 0.32 0.23 0.14 0.21 0.10Support θ = 0 9 9 9 9 9Precision θ = 1 0.67 0.67 0.78 0.58 0.72Recall θ = 1 0.12 0.31 0.88 0.22 0.66f1-score θ = 1 0.21 0.43 0.82 0.32 0.69Support θ = 1 32 32 32 32 32Table 12: Various estimations for training date(index) : January 12, 2015 (700) to April 16,2015 (765); and testing date(index) : April 17, 2015 (766) to May 14, 2015 (785).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.27 0.19 0 0.17 0Recall θ = 0 0.75 0.75 0 0.50 0f1-score θ = 0 0.40 0.30 0 0.25 0Support θ = 0 4 4 4 4 4Precision θ = 1 0.90 0.80 0.79 0.78 0.78Recall θ = 1 0.53 0.24 0.88 0.41 0.82f1-score θ = 1 0.67 0.36 0.83 0.54 0.80Support θ = 1 17 17 17 17 17Table 13: Various estimations for training date(index) : May 14, 2015 (785) to November17, 2015 (915); and testing date(index) : November 18, 2015 (916) to December 16, 2015(935). LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.43 0.54 0.40 0.38 0.42Recall θ = 0 0.67 0.78 0.22 0.56 0.56f1-score θ = 0 0.52 0.64 0.29 0.45 0.48Support θ = 0 9 9 9 9 9Precision θ = 1 0.57 0.75 0.56 0.50 0.56Recall θ = 1 0.33 0.50 0.75 0.33 0.42f1-score θ = 1 0.42 0.60 0.64 0.40 0.48Support θ = 1 12 12 12 12 1220able 14: Various estimations for training date(index) : January 8, 2016 (950) to March 8,2016 (990); and testing date(index) : March 9, 2016 (991) to March 30, 2016 (1005).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.75 0.67 0.80 0.89 1.00Recall θ = 0 0.82 0.73 0.36 0.73 0.73f1-score θ = 0 0.78 0.70 0.50 0.80 0.84Support θ = 0 11 11 11 11 11Precision θ = 1 0.50 0.25 0.36 0.57 0.62Recall θ = 1 0.40 0.20 0.80 0.80 1.00f1-score θ = 1 0.44 0.22 0.50 0.67 0.77Support θ = 1 5 5 5 5 5Table 15: Various estimations for training date(index) : April 1, 2016 (1007) to August 5,2016 (1095); and testing date(index) : August 6, 2016 (1096) to October 3, 2016 (1135).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.85 0.85 0.91 0.83 0.84Recall θ = 0 0.97 0.97 0.60 0.83 0.91f1-score θ = 0 0.91 0.91 0.72 0.83 0.88Support θ = 0 35 35 35 35 35Precision θ = 1 0 0 0.22 0 0Recall θ = 1 0 0 0.67 0 0f1-score θ = 1 0 0 0.33 0 0Support θ = 1 6 6 6 6 6Table 16: Various estimations for training date(index) : September 12, 2016 (1120) to De-cember 13, 2016 (1185); and testing date(index) : December 14, 2016 (1185) to January 20,2017 (1210). LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.33 0.38 0.20 0.21 0.18Recall θ = 0 0.40 0.50 0.20 0.30 0.30f1-score θ = 0 0.36 0.43 0.20 0.25 0.22Support θ = 0 10 10 10 10 10Precision θ = 1 0.57 0.62 0.50 0.42 0.22Recall θ = 1 0.50 0.50 0.50 0.31 0.12f1-score θ = 1 0.53 0.55 0.50 0.36 0.16Support θ = 1 16 16 16 16 1621able 17: Various estimations for training date(index) : November 1, 2016 (1156) to Febru-ary 1, 2017 (1218); and testing date(index) : February 2, 2017 (1218) to March 1, 2017(1237). LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.80 0.76 0.86 0.93 1.00Recall θ = 0 0.86 0.93 0.43 0.93 0.14f1-score θ = 0 0.83 0.84 0.57 0.93 0.25Support θ = 0 14 14 14 14 14Precision θ = 1 0.60 0.67 0.38 0.83 0.33Recall θ = 1 0.50 0.33 0.83 0.83 1.00f1-score θ = 1 0.55 0.44 0.53 0.83 0.50Support θ = 1 6 6 6 6 6Table 18: Various estimations for training date(index) : January 5, 2017 (1200) to May 31,2017 (1300); and testing date(index) : June 1, 2017 (1301) to July 11, 2017 (1328).LR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.75 0.73 0.75 0.71 0.57Recall θ = 0 1.00 0.73 0.40 0.67 0.27f1-score θ = 0 0.86 0.73 0.52 0.69 0.36Support θ = 0 15 15 15 15 15Precision θ = 1 0 0.20 0.25 0.17 0.15Recall θ = 1 0 0.20 0.60 0.20 0.40f1-score θ = 1 0 0.20 0.35 0.18 0.22Support θ = 1 5 5 5 5 522able 19: Various estimations for training date(index) : April 18, 2012 (10) to August 16,2012 (95); and testing date(index) : August 17, 2012 (96) to September 28, 2012(125)Volatility ApproachLR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.39 0.40 0.42 0.44 0.39Recall θ = 0 1.00 1.00 0.83 1.00 1.00f1-score θ = 0 0.56 0.57 0.56 0.62 0.56Support θ = 0 12 12 12 12 12Precision θ = 1 0 1.00 0.71 1.00 0Recall θ = 1 0 0.05 0.26 0.21 0f1-score θ = 1 0 0.10 0.38 0.35 0Support θ = 1 19 19 19 19 19Duration ApproachLR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.39 0.41 0.40 0.38 0.39Recall θ = 0 0.85 0.92 0.46 0.77 0.85f1-score θ = 0 0.54 0.57 0.43 0.51 0.54Support θ = 0 13 13 13 13 13Precision θ = 1 0.33 0.50 0.56 0.40 0.33Recall θ = 1 0.06 0.06 0.50 0.11 0.06f1-score θ = 1 0.10 0.10 0.53 0.17 0.10Support θ = 1 18 18 18 18 1823able 20: Various estimations for training date(index) : August 23, 2012 (100) to February14, 2013 (220); and testing date(index) : February 15, 2013 (221) to April 15, 2013 (260).Volatility ApproachLR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.76 0.89 0.82 0.96 0.97Recall θ = 0 0.91 0.97 0.28 0.72 0.94f1-score θ = 0 0.83 0.93 0.42 0.82 0.95Support θ = 0 32 32 32 32 32Precision θ = 1 0.00 0.83 0.23 0.47 0.80Recall θ = 1 0.00 0.56 0.78 0.89 0.89f1-score θ = 1 0.00 0.67 0.36 0.62 0.84Support θ = 1 9 9 9 9 9Duration ApproachPrecision θ = 0 0.45 0.50 0.92 0.48 0.57Recall θ = 0 0.68 0.77 0.50 0.45 0.59f1-score θ = 0 0.55 0.61 0.65 0.47 0.58Support θ = 0 22 22 22 22 22Precision θ = 1 0.12 0.29 0.62 0.40 0.50Recall θ = 1 0.05 0.11 0.95 0.42 0.47f1-score θ = 1 0.07 0.15 0.75 0.41 0.49Support θ = 1 19 19 19 19 1924able 21: Various estimations for training date(index) : August 21,2013 (350) to March27,2014 (500); and testing date(index) : March 28,2014 (501) to May 23,2014 (540).Volatility ApproachLR RF Neural Network (A) LSTM (B) BN (C)Precision θ = 0 0.73 0.75 0.77 0.86 0.84Recall θ = 0 1.00 1.00 0.67 0.83 0.87f1-score θ = 0 0.85 0.86 0.71 0.85 0.85Support θ = 0 30 30 30 30 30Precision θ = 1 0 1.00 0.33 0.58 0.60Recall θ = 1 0 0.09 0.45 0.64 0.55f1-score θ = 1 0 0.17 0.38 0.61 0.57Support θ = 1 11 11 11 11 11Duration ApproachPrecision θ = 0 0.46 0.49 0.35 0.48 0.58Recall θ = 0 0.95 1.00 0.32 0.79 0.74f1-score θ = 0 0.62 0.66 0.33 0.60 0.65Support θ = 0 19 19 19 19 19Precision θ = 1 0.50 1.00 0.46 0.60 0.71Recall θ = 1 0.05 0.09 0.50 0.27 0.55f1-score θ = 1 0.08 0.17 0.48 0.37 0.62Support θ = 1 22 22 22 22 2225 Conclusion
Management of oil revenue is risky in recent years as the volatility of oil prices has increasedsignificantly in the last several years. Firms and organizations deal with these risks indifferent ways. A refined version of the major tractable stochastic model- the BN-S model-is implemented in the present paper for minimizing the quadratic hedging error. As shown inthis paper, there are certain advantages of this model relative to traditional and conventionalappropriates. The theoretical results are implemented for the data analysis of the Bakkenoil price. But, the procedure and analysis presented in this paper, in principle, can alsobe performed to other financial commodities. The procedure presented in this paper alsoshows a data science driven approach to deal with the stochastic models for the commoditymarket. It is shown that a data science driven approach can be used to effectively modifystochastic models. The resulting model can be enacted to better analyze the commoditymarkets.The two approaches discussed in the data analysis section of this paper are attempts toidentify crash-like days. At the same time, it portrays the potential of merging the datascience with stochastic models. In this paper, we apply various supervised and deep learningtechniques to identify θ by working in conjunction with realized volatility and duration ofdrawdown of oil prices, respectively. Nonetheless, there is still room for further refinement ofthese discussed approaches. For the Volatility Approach , rather than simply looking at the maximum realized volatility return in percentage over a period of twenty consecutive daysin Step-4 of this particular approach, one can look into the mean of the realized volatilityreturn in percentage over twenty consecutive days, or look at the highest positive jump in realized volatility over five years. These will result in two different approaches. As it canbe observed from this paper, data science driven approaches, and especially deep learningtechniques can be a valuable resource into effective modification and efficient analysis of thestochastic models for the commodity market.
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