Methods for forecasting the effect of exogenous risk on stock markets
Karina Arias-Calluari, Fernando Alonso-Marroquin, Morteza Nattagh-Najafi, Michael Harré
MMethods for forecasting the effect of exogenous risks on stock markets
Karina Arias-Calluari a ,1 , Fernando Alonso-Marroquin a , Morteza. N. Najafi b and Michael Harré c a School of Civil Engineering, The University of Sydney, Australia b Department of Physics, University of Mohaghegh Ardabili, Ardabil, Iran c Complex Systems Research Group, Faculty of Engineering, The University of Sydney, Australia
A R T I C L E I N F O
Keywords :Probability theoryAnomalous diffusionData analysisDiffusionStochastic processes
Abstract
Markets are subjected to both endogenous and exogenous risks that have caused disruptions tofinancial and economic markets around the globe, leading eventually to fast stock market declines.In the past, markets have recovered after any economic disruption. On this basis, we focus on theoutbreak of COVID-19 as a case study of an exogenous risk and analyze its impact on the Standardand Poor’s 500 (S&P500) index. We assumed that the S&P500 index reaches a minimum beforerising again in the not-too-distant future. Here we present two cases to forecast the S&P500 index.The first case uses an estimation of expected deaths released on 02/04/2020 by the University ofWashington. For the second case, it is assumed that the peak number of deaths will occur 2-monthssince the first confirmed case occurred in the USA. The decline and recovery in the index wereestimated for the following three months after the initial point of the predicted trend. The forecast is aprojection of a prediction with stochastic fluctuations described by 𝑞 -gaussian diffusion process withthree spatio-temporal regimes. Our forecast was made on the premise that any market response canbe decomposed into an overall deterministic trend and a stochastic term. The prediction was basedon the deterministic part and for this case study is approximated by the extrapolation of the S&P500data trend in the initial stages of the outbreak. The stochastic fluctuations have the same structure asthe one derived from the past 24 years. A reasonable forecast was achieved with 85% of accuracy.
1. Introduction
In the investigation of the stock market dynamics, thereare two well-defined approaches: the “descriptive" and “struc-tural" models [1]. Nowadays, these models attempt to cap-ture the endogenous and exogenous systemic risks of marketmovements[1, 2, 3, 4].
Endogenous risks have been modeledin a microeconomic fashion by considering the interactionsbetween agents and how these change over time, causingnon-linear disruptions even if system parameters change in asmooth fashion [5, 6, 7]. New “descriptive models" present endogenous systemic risks as a dynamic factor affected bythe flow of market orders and time scale [2], excessive prof-its, and excessive losses [8] and large positive or negativevariation in stock markets index [9, 10] due to market ac-tivity. The recent “structural models" which are based onphysical model systems such as a combination of oscillationwithin a basin of free energy or external force [3, 11], spinglasses [12], and kinetic Ising model to model stock marketnetwork [13] proposed a natural non-equilibrium system. Inmost cases, they introduce the exogenous systemic risks asa global risk component which evolves all the time affectingprices of traded assets. The effects of these systemic risks arelikely causes of abrupt changes at the macro-level of marketdynamics [14, 15, 16, 17].In both systemic risks ( endogenous and exogenous ) themarket response can be decomposed into two parts, a re-sponse function that models changes in the overall trend ofthe system, called the skeleton [18], and the other is thestochastic term [19]. In this work, we analyse the nonlin- [email protected]
ORCID (s): https://orcid.org/0000-0001-6013-5490 ear properties of both of these terms.The endogenous systemic risks are an inherent part ofthe nonlinear dynamics of a market and may have detectableprecursor signals that act as warnings similar to those usedin other nonlinear systems such as climate and ecology [20].As an example, the 1987 market crash is likely an endoge-nous event [21], as it had a measurably different effect onmarket dynamics than the September 11 attacks [22] and the1995 Kobe earthquake [21] which pose systemic exogenous risks to markets. The last two events cannot be endogenizedinto market prices by ‘rational’ agents ahead of time becausethey are not foreseeable [23].This paper presents the solution of a critical outstandingproblem in finding how the market is coupled to the globe,and how exogenous and unpredictable global events producedeterministic trends in the market. In Section 2, we decom-pose the price return into a deterministic component and a 𝑞 -stationarity fluctuating term. In Section 3, the solution wasapplied to the forecasting of S&P500 index’s response dueto the outbreak of COVID-19. In Section 4, we present adiscussion on how this model provides an overview of theimpacts of an exogenous market shock that can be used as areference in developing predictive models with more accu-racy.
2. The Model
For this analysis, it is assumed that COVID-19 poses sys-tematic exogenous risks that affect stock markets behaviour.Therefore, our forecast models considered epidemiologicalresearch results to project the impact of COVID-19. Epi-demiological researchers from around the world have pro-
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Page 1 of 9 a r X i v : . [ q -f i n . S T ] J u l ethods for forecasting the effect of exogenous risk on stock markets duced an extensive array of analyses in order to model thespread, growth, peak, and ultimately the decline of the dis-ease [24, 25, 26, 27, 28]. For countries like China, Japan,Italy, and Iran, their epidemiological curve of COVID-19progression displays a peak before the second month sincethe first cases were detected [29, 30, 31]. In countries likethe UK, Australia and Germany, the governments have takenmitigation measures to slow the impact [25, 24].The simulta-neous reaction of governments, companies, consumers andmedia, have created a demand and supply shock, makingCOVID-19 a qualitatively different economic crisis than pre-vious crises [32]. This economic ‘wedging’ of falling supplyand demand is caused by rapidly escalating unemploymentdecreasing consumer demand where, in the US for exam-ple, 17 million Americans have applied for unemploymentinsurance in the first three weeks of the crisis [33], and busi-nesses are closing their doors reducing the supply of man-ufactured goods across the globe [34]. As a consequencefinancial markets around the world have fallen precipitouslyand market volatility is at near-all-time highs [35]. For ex-ample the S&P500 has registered the worst one-day fall inthe last years and the third biggest percentage loss in itshistory.In the uncertain environment of COVID-19 it is difficult toforecast the fluctuations of the S&P500 index. We then needassumptions such as the mortality rates due to COVID-19or the duration of the current shutdown of economic activ-ity. Our central assumption is based on a predicted peak forCOVID-19 deaths. Current results in Figure 1 based on thedaily World Health Organization reports [36] show that theS&P500 responds to the inflection points of the cumulativeamount of deaths and confirmed cases with no lag. This factsupports our assumption that when a peak number of deathsoccurs, the market reaches a stationary point.We developed two cases to forecast the fluctuations ofthe S&P 500. In the first case the peak number was locatedon the 23rd of May by the University of Washington [37].The second case is where the peak number of deaths is con-sidered 2-months since the first death occurred [38]. Afterthis point we expect a recovery period as economic activitystarts to return to normal. To construct the forecast, we as-sume that the stock market index can be decomposed intoa deterministic trend and a stationary stochastic fluctuation.The statistics of the fluctuation have been obtained by an-alyzing the S&P500 index during the 24 year period fromJanuary 1996 to March 2020 [39]. 𝐼 ( 𝑡 ) is the initial pointof the stock market index for some time point 𝑡 , and the in-dex 𝐼 ( 𝑡 ) for 𝑡 > 𝑡 is its time evolution. In these predictionswe take 𝑡 = 24∕03∕2020 and 𝑡 = 28∕02∕2020 for the twodifferent forecast. The price return at time 𝑡 is defined by: 𝑋 ( 𝑡 ) = 𝐼 ( 𝑡 + 𝑡 ) − 𝐼 ( 𝑡 ) . (1)In earlier work we used a detrended fluctuation approach[39] to decompose the price return 𝑋 ( 𝑡 ) into a deterministiccomponent 𝑋 ( 𝑡 ) and a stationary fluctuating component 𝑥 ( 𝑡 ) / / / / / / / / / / / / / / / / / / / / / / time(dd/mm/yy) C u m u l a ti v e I( t ) I(t)s(t) cum d(t) cum
Inflection
Figure 1: The S&P500 decreased after the WHO (WorldHealth Organization) reported an increment in confirmedcases 𝑠 ( 𝑡 ) and number of deaths 𝑑 ( 𝑡 ) due to COVID-19. Theincrements in the number of cases are represented with ablack filled point, which are an abrupt change of slope ofthe first derivative of 𝑠 ( 𝑡 ) 𝑐𝑢𝑚 and 𝑑 ( 𝑡 ) 𝑐𝑢𝑚 . 𝑋 ( 𝑡 ) = 𝑋 ( 𝑡 ) + 𝑥 ( 𝑡 ) . (2)The trend 𝑋 ( 𝑡 ) was obtained by averaging the index over amoving time window. The size of the window was one year,that was optimized to guarantee that the fluctuations aroundthe trend exhibit stationary behavior. The governing equa-tion of this stationary behaviour is [40]: 𝑡 𝜉 𝜕𝑝𝜕𝑡 = 𝜉𝐷 𝜕 𝑝 𝑞 𝜕𝑥 , (3)By using a curve-fitting analysis on the S&P500 data overthe past 24 years (see Appendix), we show that the proba-bility density function (PDF) of the detrended price is welldescribed by the functional form: 𝑝 ( 𝑥, 𝑡 ) = 1( 𝐷𝑡 ) 𝛼 𝑔 𝑞 ( 𝑥 ( 𝐷𝑡 ) 𝛼 ) , (4)where 𝛼 = 𝑞𝜉 . Being 𝐷 , 𝑞 and 𝛼 time-dependent fittingexponents which analysis is displayed in Figure A.8. The 𝑔 𝑞 term is the 𝑞 -Gaussian function defined as: 𝑔 𝑞 ( 𝑥 ) = 1 𝐶 𝑞 𝑒 𝑞 (− 𝑥 ) (5)and the 𝑞 -exponential function is 𝑒 𝑞 ( 𝑥 ) = [1 + (1 − 𝑞 ) 𝑥 ] 𝑞 that reduces to the exponential function when 𝑞 → . Thenormalization constant 𝐶 𝑞 for < 𝑞 < is given by: 𝐶𝑞 = √ 𝜋𝑞 − 1 Γ((3 − 𝑞 )∕(2( 𝑞 − 1)))Γ(1∕( 𝑞 − 1)) . (6) K Arias Calluari et al.:
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18 01 / / time(dd/mm/yy) I( t ) S&P E-mini futures The dot-com bubble SP500 float adjustment Housing boom peaksObama elected president Trump elected presidentTrump's tax reformH-K.returned to China WorldCom's bankruptcy TARP passed collapsed Chinesemarket turbulenceBrexitreferendumChina retaliated US imposed tariffsTrumptweeted"I am aTariff Man"COVID-19 outbreakRussian Crisis S&P downgrade GreekWorld tradecentre attacks
Figure 2: The S&P500 index 𝐼 ( 𝑡 ) from 02/01/96 to 24/03/20 (24 years). In this data set the largest daily percentage loss of −11 . was registered on 16/03/20. Some other key events are also shown for reference.The cumulative distribution function (CDF) of the PDF Eq.(4)of the detrended price return is; 𝐹 ( 𝑥, 𝑡 ) = ∫ 𝑥 −∞ 𝑝 ( 𝑥, 𝑡 ) 𝑑𝑥. (7)The complementary of the CDF is used here to quantify riskunder extreme events [41]. This is defined as probability that 𝑋 − 𝑋 lies outside the interval [− 𝑥, 𝑥 ] is 𝑃 ( 𝑋 ( 𝑡 ) − 𝑋 ( 𝑡 ) > | 𝑥 | ) = 1 − ∫ 𝑥 − 𝑥 𝑝 ( 𝑥, 𝑡 ) 𝑑𝑥. (8)For this paper we introduce the standarization of the 𝑞 -errorfunction erf 𝑞 ( 𝑥 ) = 2 ∫ 𝑥 𝑔 𝑞 ( 𝑦 ) 𝑑𝑦 (9)The standard error function is a special case of the 𝑞 -errorfunction for 𝑞 = 1 . Considering this definition, Eq.(8) iswritten as: 𝑃 ( 𝑋 ( 𝑡 ) − 𝑋 ( 𝑡 ) > | 𝑥 | ) = 1 − erf 𝑞 ( 𝑥 ∕( 𝐷𝑡 ) 𝛼 ) . (10)We use Eq.(10) to develop a forecast of the market basedon the historical fluctuations in the S&P500. Figure 2 shows some key dates over the last 24 years of the S&P500. Themarket dynamics consists of periods of a bull market (sys-tematic increase of the index) and a bear market (system-atic decrease). The crashes in this plot are discontinuous.They occur in a very short period of time that we attributeas noise that it is added to the deterministic component. Forexample, during the Global Financial Crisis (GFC) there isa long-term decline in the value of the S&P500 that we caninterpret as the deterministic trend, but near the point whereLehman Brothers collapses there is a discontinuous marketcrash. In fact, there are several very large daily price move-ments that are not always attributable to a specific event.These ‘crashes’ (both up and down) are more similar to largestochastic movements, not a part of the deterministic com-ponent of the market dynamics.
3. Effect of COVID-19 to S&P500
To establish the effect of COVID-19 on the market weassume that it has a deterministic, exogenous impact. Thisimpact is assumed to be a response function that correspondsto a COVID-19 induced bear market followed by a smoothtransition to a bull market. As a further refinement we as-
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Page 3 of 9ethods for forecasting the effect of exogenous risk on stock markets sume that this transition is dictated by a key quantitative fac-tor: the estimated date on which the mortality rate peaks.Our first estimation of this trend (a private communicationbetween authors on April 2, 2020) was based on the time-frame from to in which a bear-market decline in 𝐼 ( 𝑡 ) had already been observed. For theconstruction of the deterministic trend ̃𝐼 ( 𝑡 ) the parabola andhyperbola functions were used. For the parabola three con-ditions were applied and are illustrated in Figure 3: (i) Theinitial point of the predicted trend at 𝑡 is ̃𝐼 ( 𝑡 ) , (ii) the slopeof ̃𝐼 ( 𝑡 ) at 𝑡 is obtained from a linear fitting during the in-terval to , (iii) the point where therecovery is predicted at days satisfies ̃𝐼 ′ ( 𝑡 +60 days ) = 0 ,i.e. ̃𝐼 ( 𝑡 ) 𝑑𝑡 ||||| 𝑡 +60 days = 0 . (11) / /
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Time (days) I( t ) ii. i*.iii*.i. iii. Figure 3: A downward trend followed by a market recoveryis estimated by applying a parabolic fitting (magenta line)and hyperbola fitting (blue line). Three steps were applyingto define each deterministic trend. The step ii is the same forboth cases.For the hyperbola three similar conditions were applied: Theslope of the market recovery (bull market) (i ∗ ) is assume tobe 0.5 of the slope of the market’s collapse similar to the pre-vious crashes that have occurred over the past 24-years. Theslope (ii) is the same as the parabola and point (iii ∗ ) is theintersection of the slopes (i ∗ ) and (ii) . For both methods thepoint (iii) was based on the public information available bythe University of Washington where the death rate was esti-mated to peak on 24/05/2020 (60 days from 𝑡 ) ). To obtainthis prediction of the trend we assumed that there is no timelag between the peak of mortality rate and the time wherethe market starts to recover.Up to this point we have evaluated the systemic risk asthe deterministic aspect of the market evolution, which is of- -2 -1 t (days) -0.8-0.6-0.4-0.200.20.40.60.8 x Zone AZone BZone B Zone B Zone C C r o ss ov e r Figure 4: Three different zones were determined based onan abrupt slope change of 𝛼 and 𝑞 values. The circles rep-resent the end points of the strong super-diffusion regime(zone A) from 𝑡 to 𝑡 = 35 minutes. The remaining areaduring the first minutes to days corresponds to a weaksuper-diffusion regime (zone B). A normal diffusion processis reach after thirty days. The gray dashed lines represent thetransitions between each zones.ten neglected by analysts in the determination of the impactof exogenous events such as the COVID-19 pandemic. Inwhat follows we evaluate the risk associated with the stochas-tic fluctuations of the index by ‘dressing’ the deterministicrisk with a 𝑞 -Gaussian diffusion process. In a previous pub-lication [40] we found that the 𝑞 -Gaussian fluctuations ofthe S&P500 index around the trend have distinctive super-diffusion spatio-temporal regimes (‘space’ refers to the sizeof the fluctuations). To allow long-term forecasting on theorder of days, we have extended the analysis for longer times.Three well-defined regimes are observed in Fig. 4,• Zone A: Strong superdiffusion process with short-timecorrelations• Zone B: Weak superdiffusion process with weak timecorrelations• Zone C: Normal diffusion process with no time corre-lations.These zones are well-distinguished in time, where ZoneA goes from 0 to 38 minutes, Zone B from 10 days to 28 daysand after a crossover period Zone C starts at 30 days (Fig. 4).Note that the time series tends to a classical diffusion processfor large times, in agreement with the classical Central LimitTheorem. See [40] for a complete analysis of these zones andtheir derivation.Two techniques were used to calculated the optimal 𝑞 foreach 𝑝 ( 𝑥, 𝑡 ) , one focusing on the PDF and the other focusing K Arias Calluari et al.:
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Page 4 of 9ethods for forecasting the effect of exogenous risk on stock markets on each CDF of detrended price return. For the first tech-nique two methods were used. The least square method and 𝑞 -moments method. The least square method applies Eq.(4)as the fitting function. The 𝑞 -moments method uses a sys-tem of two equations and two variables, 𝑞 and 𝛼 . The firstequation is given by the “second moment" or variance ⟨ 𝑥 ⟩ .The second equation refers to the “escort second moment" or 𝑞 -variance ⟨ 𝑥 ⟩ 𝑞 . In general, the 𝑞 -moments are appliedto PDFs with asymptotic decays because they provide finitevalues [42]. The second technique is based on each CDF,where a least squares method is applied using Eq.(18). Thequality of the fitting models was evaluated after reproducingthe time evolution of 𝐶𝐷𝐹 ( 𝑥, 𝑡 ) with the time-dependent fit-ting parameters previously obtained, then we compare themwith the 𝐶𝐷𝐹 of detrended price return. The best resultswere obtained with the second technique, which displays asmoothness and more accurate time evolution of the
𝐶𝐷𝐹 of detrended price (Figure 5).Figure 5: Comparison of time evolutions of
𝐶𝐷𝐹 𝑠 betweenthe price return of detrended stock market data and the ana-lytical model which considers the time-dependent fitting pa-rameter obtained from the second technique.We extend the analysis of this derivation in the supple-mentary materials section.The corresponding 𝑞 , 𝛼 and 𝐷 values that fit each of theseequations are displayed in Fig.A.8(a-c-d). The 𝑞 and 𝛽 val-ues were calculated directly after fitting Eq.(18). The 𝛼 val-ues are calculated as the power law of the 𝛽 value, by averag-ing the power law over a moving time window smaller thanthe transition zone. The 𝐷 value is calculated by replacing 𝛼 in Eq.(16). The results for 𝑞 , 𝛼 and 𝐷 obtained by applyingthe least squares method of the CDFs are consistent with theones calculated focused on the PDFs. The convergence of 𝑞 → and 𝛼 → shows that the PDF of 𝑥 is Gaussian when 𝑡 → ∞ .Then, we construct a forecast of S&P500 using two pre-dicted trends, the parabola and the hyperbola shown in Fig.6.We note that these trends are approximations based upon theprices for the following trading days. For that reason bothtrends were re-calculated using 𝑡 = 28∕02∕2020 ; the dateon which the first death occurred in USA; and ̃𝐼 ′ (60) = 0 ;two months after the first case was confirmed. These newtrends display a better performance, notably the hyperbolawhich fits better than the others.The uncertainty was modeled with the analytical formof the detrended price by replacing the 𝛼 ( 𝑡 ) , 𝑞 ( 𝑡 ) , and 𝐷 ( 𝑡 ) calculated in the Appendix into Eq.(4) with their numericalestimates. Once the stochastic term is simulated it is addedinto the deterministic trend to see the full stochastic path of 𝐼 ( 𝑡 ) . / /
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20 11 / / Time (days) I( t ) Figure 6: The decline and recovery on four different trendsdue to COVID-19 are observed. The first forecast methodconsiders 𝑡 = 24∕03∕2020 and the second one 𝑡 =28∕02∕2020 . For both cases a parabola and hyperbola fit-ting was used, considering the method previously illustratedin Fig. 3. The hyperbola method with 𝑡 = 28∕02∕2020 displays the most accurate result at the moment.Figure 7 shows the forecast result identifying the rangeof possibilities during the market’s decline and subsequentrecovery. The predicted trend was plotted as a hyperbolawith 𝑡 = 28∕02∕2020 (Figure 6). The uncertainty is pre-sented by a shaded contour plot with a scale from 0 to 1.These values represent the probability of a possible varia-tion of 𝑃 ( 𝑋 − 𝑋 > | 𝑥 | ) which can be visualized as a “coneof uncertainty”. This probabilistic path of the S&P500 overthe following 60 days (2 months) shows the evolution of thePDF of 𝐼 ( 𝑡 ) , which diffuses as 𝑡 increases. Also we see thatthe real data for index after 𝑡 (the dashed line) lies within a K Arias Calluari et al.:
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Figure 7: The decline and recovery on 𝐼 ( 𝑡 ) due to COVID-19 is observed from to the following months. Adownward and upward market predicted trend is calculatedby applying a hyperbolic fitting with no time lag. The uncer-tainty is represented by a scale from to , where representthe 𝑃 ( 𝑋 − 𝑋 > | 𝑥 | ) of each contour line. The economy willstart its recovery after this epidemic peak of deathstiny interval in the vicinity of the predicted trend, mainly inthe most probable (dark) regions, verifying that our model isworking and is reliable for forecasting with of accuracy.We construct a forecasting based on essential public in-formation provided by the University of Washington on theirweb page[36]. A good forecasting is always an iterative pro-cess that can be daily updated considering new factors andchanges.
4. Conclusions
In summary, we have presented a model of the stochasticand systematic risk in the stock market and applied it to fore-cast the stock market response to COVID-19. We have as-sumed that the stochastic risk is a 𝑞 -Gaussian diffusion pro-cess. The systemic risk is the deterministic aspect of the mar-ket evolution that is often neglected by market analysts buthere we have assumed that COVID-19 has a deterministic,exogenous impact on the market. We have dressed this non-linear skeleton with a 𝑞 -Gaussian diffusion process that wedeveloped in previous work. The response function we haveused for forecasting is a simple behavioural model based onthe assumption that markets recover once prices drop to suf-ficiently low enough levels, in future work it can be improvedby developing better tools to evaluate systematic risk usingbehavioural [43] or macroeconomic [44] models. In addi-tion, our stochastic method is still heuristic and it needs to be developed by formulating the governing equations thatallows transition from strong- to weak- superdiffusion andconverging to a normal diffusion process for large times, asdictated by the classical Central Limit Theorem. However,this method opens up a potential opportunity for risk controlin other areas such as climatology, seismology (earthquakes)and communication networks, where large multivariate datasets provide a window on the dynamics of the underlyingsystem.
5. ACKNOWLEDGMENTS
We acknowledge the Australian Research Council grantDP170102927. K.A.C. thanks The Sydney Informatics Hubat The University of Sydney for providing access to HPC-Artemis for financial data processing. We thanks Sornette,Constantino Tsallis and Christian Beck for inspiring discus-sions.
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A. Supplementary materials
The probability density function (PDF) of the detrendedprice is well described by the functional form: 𝑝 ( 𝑥, 𝑡 ) = √ 𝛽 𝑔 𝑞 (√ 𝛽𝑥 ) , (12)where, 𝛽 = ( 𝐷𝑡 ) −2∕ 𝛼 , allows to recovery Eq.(4).Then, a system of two equations based on variance ⟨ 𝑥 ⟩ and 𝑞 -variance ⟨ 𝑥 ⟩ 𝑞 were used to obtain the 𝑞 and 𝛽 values.The variance ⟨ 𝑥 ⟩ has a finite value for 𝑞 < ; ⟨ 𝑥 ⟩ = 1 𝛽 (5 − 3 𝑞 ) . (13)The normalized 𝑞 -variance ⟨ 𝑥 ⟩ 𝑞 is defined as [42] ⟨ 𝑥 ⟩ 𝑞 = ∫ ∞−∞ 𝑥 𝑝 ( 𝑥, 𝑡 ) 𝑞 𝑑𝑥 ∫ ∞−∞ 𝑝 ( 𝑥, 𝑡 ) 𝑞 𝑑𝑥 , (14) In general, the q-moments are calculated on PDFs with heavytails because they provide finite values. The q-moments andthe standard moments are equal for 𝑞 = 1 .To obtain the analytical solution of the integral in thenumerator and denominator of Eq.( 14), we notice that 𝐾 ( 𝛽, 𝑞 ) ≡ ∫ ∞−∞ [1 − (1 − 𝑞 ) 𝛽𝑥 ] 11 − 𝑞 𝑑𝑥 = 𝐶 𝑞 √ 𝛽 , and also the second moment of 𝑥 is proportional to 𝜕𝛽𝐾 ( 𝛽, 𝑞 ) ,so the analytical solution of the numerator is: 𝜕𝜕𝛽 𝐾 ( 𝛽, 𝑞 ) = − ∫ ∞−∞ 𝑥 [1 − (1 − 𝑞 ) 𝛽𝑥 ] 𝑞 𝑞 𝑑𝑥 = − 12 𝛽 −3∕2 𝐶 𝑞 Then, the analytical solution of the denominator is: ∫ ∞−∞ 𝑝 ( 𝑥, 𝑡 ) 𝑞 𝑑𝑥 = 3 − 𝑞 Inserting this identity into the Eq.(14), we obtain: ⟨ 𝑥 ⟩ 𝑞 = 1(3 − 𝑞 ) 𝛽 . (15)The equation system is defined by Eqs(13) and (15) and isused to calculate 𝛽 and 𝑞 as a function of time. The 𝛼 and 𝐷 values are calculated using: 𝛽 = ( 𝐷𝑡 ) −2∕ 𝛼 (16)We divide the data of S&P500 in overlapping time windowsnot longer than the transition zones. In each window theexponent of the power law relation is calculated and thenthe Eq.(16) is used to calculate 𝐷 and 𝛼 . The results for 𝑞 , 𝛼 and 𝛽 are shown in Subfigures a-1,b-1,c-1 and d-1.In Subfigure a-1 an abrupt transition in 𝑞 is noticed when 𝑡 = 30 days. This abrupt change leads to a discontinuity inthe fitted function 𝑞 ( 𝑡 ) .We have used an alternative method to extract the parame-ters, for which this discontinuity is absent. This method isbased on the fitting of the 𝐹 ( 𝑥, 𝑡 ) of real price return 𝑥 . Theanalytical expression of 𝐹 ( 𝑥, 𝑡 ) is obtained in terms of the erf 𝑞 ( 𝑥, 𝑡 ) . First, Eq.(7) is replaced in Eq.(8), 𝑃 ( 𝑋 ( 𝑡 ) − 𝑋 ( 𝑡 ) > | 𝑥 | ) = 2 − 2 𝐹 ( 𝑥, 𝑡 ) . (17)Then, Eq.(17) and Eq.(10) are compared, resulting to( 𝑠 = 𝑥 √ 𝛽 ); 𝐹 ( 𝑥, 𝑡 ) = 0 . 𝑞 ( 𝑠 )2 (18)Where, erf 𝑞 ( 𝑠 ) = 2 √ 𝛽𝐶 𝑞 𝑥 𝐹 [ , 𝑞 − 1 , , (1 − 𝑞 ) 𝑠 ] . (19) K Arias Calluari et al.:
Preprint submitted to Elsevier
Page 7 of 9ethods for forecasting the effect of exogenous risk on stock markets a-1 -2 a-2 -3 -2 -1 t(days) -6 -4 -2 ( t ) Zone B Zone C b-1 -3 -2 -1 -6 -4 -2 Zone B Zone C b-2 t(days) ( t ) c-1 t(days) ( t ) c-2
20 40 60 80 100 120 140 t(days) D ( t ) d-1
20 40 60 80 100 120 140 t(days) D ( t ) d-2 Figure A.8: Calculation of 𝑞 and 𝛽 by applying a Equation system [Eqs(13) and (15)] and fittings of 𝐹 ( 𝑥, 𝑡 ) . (a-1, a-2)Estimation of 𝑞 ( 𝑡 ) value, for (a-1) a normalization of units was made. (b-1,b-2) The value of 𝛽 ∼ 𝑡 −2∕ 𝛼 . (c-1,c-2) Theexponent of the power law relation of 𝛽 vs 𝑡 per time window is calculated as 𝛼 ( 𝑡 ) . (d-1,d-2) The 𝐷 value is calculated byreplacing 𝛼 in Eq.(16). The convergence to a Normal distribution function is observed when 𝑡 → ∞ with 𝑞 = 1 and 𝛼 = 2 .(coefficients are in Table 1) K Arias Calluari et al.:
Preprint submitted to Elsevier
Page 8 of 9ethods for forecasting the effect of exogenous risk on stock markets
Fitting 𝑝 ( 𝑥, 𝑡 ) Fitting 𝐹 ( 𝑥, 𝑡 ) 𝑎 = 0 .
556 ± 0 . 𝑎 = 0 .
005 ± 0 . 𝑞 𝑏 = 1 .
485 ± 0 . 𝑏 = 2 .
473 ± 0 . 𝑐 = 1 .
811 ± 0 . 𝑏 = 0 .
009 ± 3 . −4 𝑚 = −1 .
23 ± 0 . 𝑏 = 0 .
010 ± 3 . −5 𝑚 = −1 .
32 ± 0 . 𝛽 𝑏 = 0 .
008 ± 3 . −5 𝑚 = −1 .
10 ± 0 . 𝑏 = 0 .
003 ± 7 . −4 𝑚 = −1 .
23 ± 0 . 𝑏 = 0 .
010 ± 2 . −4 𝑚 = −1 .
06 ± 0 . 𝑏 = 0 .
007 ± 8 . −4 𝑚 = −0 .
97 ± 0 . 𝑎 = 0 .
027 ± 0 . 𝑏 = 1 .
620 ± 0 . 𝑎 = 0 .
035 ± 0 . 𝑏 = 1 .
553 ± 0 . 𝛼 𝑎 = 0 .
004 ± 0 . 𝑏 = 1 .
839 ± 0 . 𝑎 = 0 .
008 ± 0 . 𝑏 = 1 .
831 ± 0 . 𝑏 = 2 .
00 ± 0 . 𝑏 = 2 .
00 ± 0 . 𝑎 = 0 .
008 ± 0 . 𝑏 = 0 .
130 ± 0 . 𝑎 = 0 .
022 ± 0 . 𝑏 = 0 .
098 ± 0 . 𝐷 𝑎 = −0 .
001 ± 7 . −5 𝑏 = 0 .
445 ± 0 . 𝑐 = 2 . −6 ± 0 . −7 𝑐 = −3 . −4 ± 1 . −5 𝑐 = 0 .
012 ± 0 . 𝑐 = 0 .
363 ± 0 . 𝑎 = −0 .
002 ± 6 . −5 𝑏 = 0 .
525 ± 0 . Table 1: Coefficients values of fittingsThe Eq.(18) is used to calculate 𝑞 and 𝛽 value as a functionof time. The 𝛼 values are calculated again with Eq.(16) byoverlapping time windows, and then, 𝐷 is obtained by re-placing 𝛼 in 16.These results are shown in the second column of Figure A.8.The Subfigure (a-2) does not display an abrupt transition of 𝑞 value. For 𝛽 , 𝛼 and 𝐷 the transitions are smooth in eachzone. K Arias Calluari et al.: