Characterizing Catastrophic events in Financial Stock Markets

Arun SharmaQueen Mary Uni. of London

April 10, 2015

Abstract

This paper used a harmonic damped oscillator to model the 1987 Black Monday, 2008 Sub-prime and2014 Commodity price collapse. Analysis of the results found that the Nikkei 225 was the most likelycandidate behind the 1987 crash, and failed to recover quickly afterwards either due to trading softwareerrors, aggressive trading, or both. While the 2008 Sub-prime crisis changed the global economic climate,dealing majority of its damaged through secondary shockwaves. Financial based indices performed thebest during this shock, most likely to due to their financial knowledge, but the non-financial NASDAQdemonstrated how indices are able to mimic the performance of a financially dominated index, if powerfulenough. The FTSE100 revealed that it became more risk adverse to large market shocks, helping thegrowth of hedge funds. On the other hand emerging economies became more volatility in comparison todeveloped economies, largely fueled by a change in investor interest. Indices on average responded well tothe commodity collapse and most likely hedged their exposure with the use of derivatives. Nevertheless asthe crisis escalated, the failure of hedging strategies and growing investor uncertainty damaged developedeconomies more than emerging, but these repercussions increased the volatility for emerging markets astime went on.

1 Introduction

Understanding the volatility found in financial stock markets has become a key area of concern for institu-tions, governments and investors. Speculative asset price movement in an index is not enough to explainthe violent oscillations that are seen, and thus has prompt the need for further clarity.[1] has demonstratedhow the Effective Market Hypothesis can be used to explain certain volatile phenomena found in smoothmarkets, however there still lacks a clear understanding for what is known as Catastrophic market events.

Catastrophic events in financial markets play turmoil on the value of an index and individual companies,so much so they have helped give birth to what is known as Risk Management. It is still unknown as towhy these events occur, but the most damaging have arisen from investors expectation, human error and themarket environment/resources [2, 3, 4, 5, 6]. Moreover, financial media are forecasting that the likelihoodof another crash soon is imminent, and could occur as early as summer 2015. Therefore safeguarding themoney, resources and jobs that hold society together is vital to ensure our worlds long term stability. [7]

There have been many unsuccessful attempts to model such phenomena until [2] proposed a Log PeriodicPower Law model that could have successfully predicted the 2007 Hang Seng index crash. Since, [8] hasdeveloped a variation of the LPPL model, implementing a harmonic damped oscillator to characterize theviolent drops and chaotic oscillations.

Therefore, by fitting a harmonic damped oscillator to various different indices, one could develop andunderstanding for the characteristics of a catastrophic event, and potentially mitigate a future one. Fur-thermore by comparing several indices for different time periods, one could begin to match any similaritiesbetween markets and uncover reasons as to why some economies perform better or worse than others.

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1.1 The Event

The following 3 events have been chosen due to their global impact and nature, which allows them to po-tentially happen again.

1.1.1 Black Monday October 1987

The first shall be the Black Monday crash of 1987, which is believed to be caused by a fault in Asianfinancial trading software. This caused the world largest markets to plummet, for example the Dow Joneswho dropped 22.6% in a single day. However the actual cause for the crash is unknown and is still disputedtoday.[5, 9, 10] As financial stock markets have become ever more automated there is always the likelihoodof a similar event occurring, therefore modelling the event may shed light on what actually happened.

1.1.2 Sub-prime Mortgage September 2008

The next shall be the 2008 Sub-prime mortgage crisis, which was the largest crash the world has ever seenand has been estimated to of cost the world $10 trillion. A systematic error with the way loans wereissued by lenders and traders exchanged financial derivatives, created whats known as a bubble. This bubblequickly accumulated debt that could not be paid off and when people begun to default on their payments,the financial lenders took the hit with the taxpayers money. This undoubtedly sent shockwaves aroundfinancial markets and caused the collapse of the Lehman Brothers [8, 11, 12, 14]. As human error andinvestor confidence are unpredictable controls, being able to analytically study such behaviours will aid inunderstanding the events characteristics.

1.1.3 Crude Oil Collapse September 2014

Last of all shall be the recent collapse of global commodity prices that was caused by crude oil dropping by55% from $115 in 2014 to $47 per barrel in January 2015. This collapse was caused by a change in demand,political uncertainty and the appreciation of the US dollar, instigating the sell-off of commodities globally[15]. Despite prices stabilizing in early 2015, indices suffered greatly throughout 2014 [16]. Thus betterunderstanding the relationship between index performance and global commodity volatility, could have hugeimplications.

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1.2 The Indices

Due to their global reach, spectrum of industries and economic status (Developed or Emerging), the following12 indices have been selected.

FTSE100 - United KingdomDow Jones - USANYSE Composite - USANYSE US 100 - USAS&P 500 - USANASDAQ - USADAX - GermanyMXX - MexicoBVSP - BrazilNikkei 225 - JapanHSI - ChinaSensex - India

2 The Model - Damped Harmonic oscillator

The model that shall be used stems from [17]s work on the Log-periodicity model and incorporates a har-monic damped oscillator,

mP = k(P P ) P + et (1)

where m is the mass, P is the price of a stock, P is what the market considers to be a reasonable pricefor the stock and k is the spring coefficient. P is the rate at which the stocks price drops with a constantcoefficient , P is the stock prices acceleration leaving the quickly decaying shock to the market, et.

To further understand this equation it shall be re-written it in terms of two first-order differential equa-tions,

U = V

V = kP + et V kU

and their trajectories for the initial conditions U = 1, V = 1,m = 1, k = 5, P = 1, = 1, = 1, t = 1, = 1shall be displayed. These conditions have been chosen, as they resemble a trajectory with an initial shock,and a strong, quickly decaying oscillating factor.

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It is clearly visible that V s trajectory violently crashes at first but quickly oscillates to the markets fairprice, which is a characteristic of an indexs price soon after a significant shock [8].

Following on to now briefly study the equations dynamics to large shocks, the minimumss for V s tra-jectory, for k value between 0.3-10 shall be generated. Recall, k is the coefficient for the difference betweenthe initial and collapsed price, and helps to describe the amount of value lost. (The other conditions frombefore have been kept the same)

As the Crash Testing diagram above depicts, the initial minima/shock quickly grows but begins to deceleratefor increasing values of K. This behaviour resembles the evolution of the natural logarithm over time, and

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demonstrates that the model begins to absorb more of the shock if more is applied. This can be comparedin real life to the 2008 Sub-prime crisis and how the American government was required to bail out bankssuch as JP Morgan. Without the funds the bank would have defaulted and the American economy would ofcollapsed, therefore by helping to absorb some of the shock the economy was somewhat stabilized [11]. Asthe model already incorporates a shock absorber, which can be related to real techniques used in the market,we can already being to see its feasibility for modelling such events.

Now solving the original differential equation (1) leads to the solution

P (t) = A+Bet + Cet cos(wt ), (2)

which shall be the equation that is worked on for the remainder of the paper.

In this model A denotes the median of the index price over the time period, B determines the strengthof the initial shock while regulates how fast it decreases; C determines the oscillations amplitude while controls how fast it decays, leaving w and ; w is the frequency of the oscillations and determines wherethe peaks of the oscillations occur.

3 Method for fitting - GaussNewton Least Squares

As the main analysis is based on the calculated parameters, it important to demonstrate the effectivenessof the chosen optimization method, and due to its simplicity and potentially quick convergence, the Gauss-Newton method for non-linear least square problems has been used to fit the model to the data. This methodminimizes the square of the error between a function value and the actual data point, in turn calculatingthe best parameters to mimic the data.

The method is a modification of the famous Newton method that assumes the Hessian to be naturallysmall for good initial conditions, and thus ignores it to save computation time. The model only requires aninitial guess for each desired parameter and chosen value of accuracy/error [18, 19].

Therefore the goal is to minimize the following function found in [19],

R =

mi=1

(P (i) yi)2 (3)

where R denotes the residuals, P (i) is the ith function and yi are the real data.

To allow the indices to be compared, all the data points have been normalized by dividing each data point bythe average value over the time period. As only the first 20 days after each crash are modelled, no problemsshould arise from this simplification.

To illustrate how the equation was modelled, a fitting for the main European index the EURONEXT forthe 2014 commodity collapse will be illustrated.

The method begins by selecting the initial conditions, and to provide quick convergence the following wereused as a starting point for all indexes:A = 1, B = 1, = 0.1, C = 0.1, = 0.01, w = 1 and = 1;These parameters were chosen as they describe a solution that drops due to a large shock and oscillatesstrongly, consequently providing each model with a realistic guess, saving considerable amounts of compu-tational time.

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The above diagram demonstrates the models value against the actual data points and converged in 1 iteration(0.24 seconds) with a final residual of 0.0002. However this plot highlights a potential weakness for the Gauss-Newton algorithm, which is that the method only converges towards the local minima when minimizing[18, 19]. This reluctantly can be avoided by slightly tweaking the initial conditions until the model convergestowards a improved minimum, or the global minimum. For instance,

This time the model converged in 5 iterations (0.19 seconds) with a final Residual of 0.000008. This diagramclearly performed more effectively than the previous, though there is still an improvement to be made.

As the above solution does not oscillate like required, the oscillating parameters w and can be tweakedto force the solution into an oscillating, more realistic trajectory. Like so,

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Despite converging in 11 iterations (0.18 seconds), the residual is smaller (0.00002), and the trajectory clearlymimics the data points and provides the more realistic answers. The initial guesses for the final plot wereA = 1B = 0.8, = 0.1, C = 0.1, = 0.01, w = 1.5 and = 0.3, and lead to A = 1.191, B = 0.061, =0.096, C = 0.004, = 0.17, w = 0.777 and = 4.376.

4 Residual Analysis

Given a reliable technique for computing the parameter values, the nature of the residuals for the EU-RONEXT for 2014 will briefly be explored.

4.1 Individual Residuals

Firstly, the accuracy of the individual residuals shall be studied for the initial guess, the parameters used inEURONEXT 3 2014, as it will shed light on how the model behaves for a good initial guess.

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The top plot portrays how the model struggled to match some of the oscillations, but generally improvedtowards the end. This is likely due to the solution being a line or curve with no oscillations that failed toaccurately model to data.

The second diagram highlights how for larger data points the model once again struggled, and emphasizeshow the initial stock caused problems for the model.

Lastly, the bottom plot demonstrates how the individual residuals improve on the previous, but thesquare/circular motion demonstrates that the errors improved slightly in relation to the previous.

From this it can be concluded that the initial guess performed poorly despite being deemed good. Nowmoving onto the individual residuals for the final solution.

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The top diagram reveals how effect the Gauss-Newton method has been in minimizing the error for nearlyall of the data points despite the small spike, which is visible in the EURONEXT 3 price plot above. Whencomparing the residual plot for the guess with the final solutions, it is clear that the final solutions residualsare on average smaller by a factor of 102.

The second, like before, describes the relationship between the data point and the corresponding residual,and in this case the data points have no relation to the performance of the residual. This indicates thatthe model has successfully aligned itself with the data and as time goes on, will continue to follow a similartrajectory.

The last diagram depicts how the errors in relation to each other considerably change. The triangularmotion highlights how a small error goes on to lead to a larger one in relation, which is also visible in theEURONEXT 3 price plot as the model neglects various key points between t = 4 14.

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From this we can conclude how well the Gauss-Newton algorithm has done in fitting the model to thedata, not just for the average residual, but in relation to each data point.

4.2 Iterated Residuals

The final residual analysis will be conducted on the sum of the individual residuals for each iteration; this isto explore the methods convergence rate for several repetitions.

The figures above express the catastrophic error that can arise from the optimization method if the wrongsearch direction is taken. Nevertheless, the algorithm quickly corrects itself and finds a local minimum. Thesecond plot shows in relation to the previous how big the error grew.

Now that the model has proved how reliable is on average at fitting the damped oscillator to the data,the main part of studying the individual indices will now follow

5 Black Monday 1987

The following indices have been modelled from the 15th of October to the 11th of November and werecomputed to a final residual of atleast order 105

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5.1 Implementation & Analysis

(I) FTSE100. (II) Dow Jones.

(III) NYSE Composite. (IV) S&P 500.

(V) NASDAQ. (VI) Nikkei 225.

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(VII) Hang Seng.

Table I: Black Monday Parameters

Index A B C w FTSE100 0.909 0.518 0.251 0.117 0.194 4.535 2.727DowJones 0.983 0.242 0.565 -0.114 0.106 1.148 4.113NYSEComp 0.978 0.004 0.209 0.114 0.163 5.056 -1.853S&P500 0.978 -0.057 0.125 0.134 0.188 1.06 0.729NASDAQ 0.953 0.445 0.383 -0.067 0.095 4.971 1.279Nikkei 0.968 0.416 0.469 0.103 0.135 4.07 0.23HSI 0.893 2.456 0.67 1.057 0.524 2.12 5.719

Table 1 holds the solutions relating to each index for the Black Monday crash. As all of the values for Aare < 1, it can be established that each index begun to stabilize to a value below the average of its oscillations.

For the HIS index a large B,C and correspond to a strong shock with a quickly peaking but decayingoscillation, signifying that the market quickly stabilized after the initial drop. This demonstrates howChineses investors saw no potential market risk, and thus confidence was unchanged and prevented anybullish activity. The Nikkei however has a large w and small , corresponding to an oscillator with highfrequency. This suggests that Japanese trading software could have started the 1987 collapse and took severaldays to fix the fault. Likewise, it could be the result of bullish traders capitalizing on the crash, purchasinglarge amounts of stock after the initial drop, driving the index back up and selling high, plunging it backdown.

5.2 Black Monday Global Average

Now an average trajectory for the Black Monday crash for each optimized parameters will be generatedusing the data from Table 1. The average values for the crash are A = 0.952, B = 0.575, = 0.367, C =0.192, = 0.201, w = 3.280 and = 1.849. These lead to,

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This is a quickly decaying oscillator with large shock and validates all of the models above, as they all haveeither stabilized or begun to do so. This establishes that globally, investors were not worried about the crashescalating and continued to invest/trade as normal soon after.

6 Sub-prime Mortgage crash 2008

The following indices have been modelled from the 3rd of September to the 24th of October and werecomputed to a final residual of atleast order 105. The specific start dates vary as some markets droppedlater than others.

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6.1 Implementation & Analysis

(VIII) FTSE100. (IX) Dow Jones.

(X) NYSE Composite. (XI) NYSE US 100.

(XII) S&P 500. (XIII) Dax.

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(XIV) NASDAQ. (XV) Mexico IPC.

(XVI) Bovespa. (XVII) Nikkei 225.

(XVIII) Hang Seng. (XIX) Sensex.

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Table II: Sub-prime Mortgage Parameters

Index A B C w FTSE100 0.925 0.32 0.186 0.032 0.004 5.205 2.186DowJones 1.43 -0.286 -0.037 0.009 -0.064 2.118 11.648NYSEComp 0.928 0.422 0.268 0.058 0.022 5.004 0.985NYSE100 0.95 0.385 0.317 0.038 0.009 5.166 1.929S&P500 0.893 0.373 0.151 0.035 0.002 4.853 0.551Dax 0.942 0.262 0.217 0.023 -0.05 1.11 2.237NASDAQ 0.905 0.362 0.165 -0.025 -0.025 1.441 8.656MXX 0.875 0.354 0.118 0.012 -0.085 4.869 0.603BVSP 1.01 -0.0004 -0.327 0.462 0.289 0.447 1.186Nikkei 0.81 0.51 0.113 0.023 -0.047 5.165 1.826HSI 1.253 -0.105 -0.075 0.026 -0.054 1.044 2.641Sensex 0.189 1.029 0.023 -0.018 -0.069 1.066 2.026

Looking at the values for A,B and for the DowJones and HIS index, it is clear that they both describesimilar trajectories. A small B but large correspond to an index whos price suffered due to an after shockfrom the initial crash, while A highlights how the average of the oscillations were still positioned abovethe actual average. This reveals that the aftershock caused the price to collapse. This should be expectedas the DowJones index only concerns manufacturing companies, so the collapse of the America financialinstitutions would have caused shockwaves eventually impacting on all businesses. Furthermore, the ChineseHIS has strong ties with American banks who heavily invest in china, nevertheless American banks wouldhave begun to liquidate their assets to prevent defaulting. This would have caused the HIS index to fall ata slower rate until the true extent of the crisis was known. This together verifies that this crisis altered theglobal economic climate, affecting all types of stakeholders linked to American markets.

Comparing the FTSE100 with the German Dax can highlight an important relationship between this crisisand financially dependent economies. The FTSE100 2008 exhibits a trajectory that initially drops withviolent oscillations, however reverts to stable mean of A = 0.925. On the other hand the neighbouringGerman Dax has a negative value for , and this of course corresponds to a diverging, exponentially growingoscillating term, which can be seen in Dax 2008. The FTSE100 has always had a strong financial positionas many of the many companies consist of banks, hedge funds, pension funds, etc., unlike with the Dax.Characterizing how regardless of geographically location or index price, a financial index will overall performbetter in a crisis brought by financial institutions. This is most likely due to the index having superiorfinancial knowledge over a non-financial index.

This relationship also extends to the following indices in Table 2, the NYSEComp, NYSE US100 and theS&P 500, whom all also happen to be financially dependent indices [11, 16]. Leaving the remaining indicesin Table 1 to correspond to economies that are either emerging or rely on automotive, manufacturing,electronics, etc., who generally performed worse in this case.

6.2 Sub-prime Global Average

Now like before, an average trajectory for the Sub-prime mortgage crash for each optimized parameter willbe generated using the data from Table 2. The average values for the crash are A = 0.926, B = 0.302, =0.093C = 0.056, = 0.006, w = 3.124 and = 3.039. Therefore the trajectory resembles:

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The diagram above has been extended to show the trajectory for 120 days or from the end of September2008 to the end of January 2009. The plot is forecasting that the global economy took some time to beginstabilizing after the crisis, and begun to settle after 90 days or towards the end of 2008. However towardsthe end of January 2009 (between 95-120 days from the initial shock), the economic climate became morevolatile once again.

The above chart shows the index price for the S&P 500 from September 2008 to March 2009 [16], and clearlyshows a second major dip approximately 120 days or 4 months after the initial crash. This coincides withthe 2008 average diagram above, indicating the potential use of damped harmonic oscillators characteringoncoming secondary shocks in financial markets, given the right conditions.However as the model only used 20 data points, it graph should not be taken too seriously for forecastingthe following 100 days.

7 Crude Oil Collapse 2008

7.1 Implementation & Analysis

The following indices have been modelled from the 23rd of September to the 22nd of October and werecomputed to a final residual of atleast order 105. The specific start dates vary as some markets droppedlater than others.

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(XX) FTSE100. (XXI) Dow Jones.

(XXII) NYSE Composite. (XXIII) NYSE US 100.

(XXIV) S&P 500. (XXV) Dax.

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(XXVI) NASDAQ. (XXVII) Mexico IPC.

(XXVIII) Bovespa. (XXIX) Nikkei 225.

(XXX) Hang Seng. (XXXI) Sensex.

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Table III: Oil Collapse Parameters

Index A B C w FTSE100 1.098 -0.061 -0.043 0.007 0.008 1.141 1.637DowJones 0.96 0.082 0.082 -0.008 -0.051 0.553 -4,992NYSEComp 1.132 -0.096 -0.028 -0.001 -0.145 5.641 -5.139NYSEUS100 1.025 -0.004 -0.148 0.005 0.008 8.048 2.546S&P 500 1.025 -0.003 -0.155 0.005 -0.007 1.743 2.283Dax 1.407 -0.336 -0.018 0.00003 -0.357 2.116 -2.085NASDAQ 1.031 -0.005 -0.149 -0.005 -0.043 4.569 1.233MXX 1.03 -0.009 -0.099 0.01 0.038 0.968 1.276BVSP 1.026 -0.002 -0.206 -0.037 0.106 0.774 10.832Nikkei 0.869 0.203 0.045 0.007 -0.081 1.152 4.434HSI 1.072 -0.036 -0.061 -0.019 0.111 0.515 4.474SENSEX 1.026 -0.012 -0.064 -0.013 0.046 -7.419 -21.521

The As for each index are on average above 1, indicating that majority of the indices forecasted to goonto stabilize or absorb the impact from the commodity shock. Nevertheless the DowJones and Nikkei arethe only two with an A < 1 and are largely based off leading industrial companies, whom of course operateof commodities such as oil, coal, etc. This demonstrates how the collapse in global commodity prices candamage developed manufacturing economies more than emerging, like the HSI.

The Sensex has a considerably large negative C,w and , signifying that the index oscillated violently,frequently and peaked early on. This reinforces how investors initially perceived the drop in oil to be anadvantage for Indias manufacturing, emerging economy, but as commodities continued to collapse, defla-tionary fears rose and investors sought safer investments [20]. Comparing the Sensex to the Dax, Nikkeiand DowJones it is clear that emerging manufacturing economies are more volatile to price movements overdeveloped manufacturing economies.

Overall, Table 3 has the lowest set of C values for any of the 3 crashes, meaning that markets on averagesaw modest oscillations and did not peak aggressively. This new risk adverse environment can be explainedby a growing trend, and what today is known as a derivative. Derivatives are financial contracts which hedgethe risk associated to volatility, and have become common practice in todays market environment [21]. Forinstance the budget airline Ryanair actively hedges oil to reduce their exposure [22]. Therefore markets arebecoming more aware of financial shocks and have begun to devise techniques to insure themselves, and isthe leading indicator as to why markets were met with mild climates 20 days after the initial shock.

7.2 Crude Oil Global Average

Now the Crude oil crash average for each optimized parameter will be generated using the data from Table3. The average values for the crash are A = 1.073, B = 0.023, = 0.070, C = 0.004, = 0.031, w =1.65, = 1.3750, and lead to

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(The value for for Sensex has not been included as it skews the data and is too extreme to incorporatewith the other indices.)

The negative B,,C and represent a trajectory that drops and oscillates negatively more over time,whereas the remaining parameters denote a mildly oscillating path. These collectively correspond to a mar-ket environment that slowly collapses and goes onto never recover, which did initially happen as commodityprices continued to drop up until January 2015, were they then stabilized [15]. For that reason it is clear fromthis plot and Table 3 that this crash reveals how markets initially react to shocks that can be hedged, butonce the full extent of the crisis is known, even hedging techniques cannot repress the forthcoming collapse.

8 Historical and Geographical relationships

8.1 America

The following section moves onto splitting the American financial markets to help understand their individualroles and relations to past performance.

Table IV: The American Parameters

Year Index A B a C b w y1987 0.983 0.242 0.565 -0.114 0.106 1.148 4.1132008 DowJones 1.42 -0.286 -0.037 0.009 -0.064 2.118 11.6482014 0.96 0.082 0.082 -0.008 -0.051 0.553 -4.9921987 0.978 0.004 0.209 0.114 0.163 5.056 -1.8532008 NYSEComp 0.928 0.442 0.268 0.058 0.022 5.004 0.9852014 1.132 -0.096 -0.028 -0.001 -0.145 5.641 -5.1391987 - - - - - - -2008 NYSEUS100 0.95 0.385 0.317 0.038 0.009 5.166 1.9292014 1.025 -0.004 -0.148 0.005 0.008 8.048 2.5461987 0.978 -0.057 0.125 0.134 0.188 1.06 0.7292008 S&P 500 0.893 0.373 0.151 0.035 0.002 4.853 0.5512014 1.025 -0.003 -0.155 0.005 -0.007 1.743 2.2831987 0.953 0.445 0.383 -0.067 0.095 4.971 1.2792008 NASDAQ 0.905 0.362 0.165 -0.025 -0.025 1.441 8.6562014 1.031 -0.005 -0.149 -0.005 -0.043 4.569 1.233

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(XXXII) Dow Jones. (XXXIII) NYSE Composite.

(XXXIV) NYSE US 100. (XXXV) S&P 500.

(XXXVI) NASDAQ.

From the above plots and Table 4, it is evident that the DowJones, NYSE Composite and S&P 500all behaved similarly to Black Monday market shock, with the NASDAQ sharing some characteristics.Nevertheless for both the Sub-prime mortgage and crude oil crash, the NASDAQ begins to closely mimic thebehaviour of the NYSE Composite, NYSEUS 100 and the S&P 500, while the DowJones no longer follows

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their behaviour. This can be explained by the swell in value for technology companies and contraction for theauto motive industry in the earth 21th century [23]. Thus allowing the technology industries performance toheavily influence the neutral indices (NYSE composite, S&P 500, etc.) and technology index (NASDAQ),while the weighting from auto motives began to decline, altering the DowJones respectively.

This suggests that markets react differently in relation to specific industries, but then can react moreuniformly for larger companies with more financial value in a shock.

Furthermore using Table 3 and the 4 plots above, it is evident that American indices were equally aseffected by the 1987 and 2008 crash with respect to their share price, while the 2014 shock was minimal incomparison. This reinforces the concept of derivatives earlier mentioned and their growing role in hedgingrisk/damping shocks.

Moreover the 2008 crash saw smaller s, which translate to longer periods of oscillation compared to thelarge C s found for the 1987 crash, corresponding to a violent drop. This demonstrates that American stocksmarkets quickly dealt with the Black Monday crash, but failed to cope with the fundamental error whichcaused the Sub-prime mortgage crisis, which was able to grow in volatility over time.

8.2 Europe/Asia (Developed)

The following section focuses on the developed European and Asian financial markets to help understandtheir individual roles and relations to past performance.

Table V: European & Asian Parameters

Year Index A B C w 1987 0.909 0.518 0.251 0.117 0.194 4.535 2.7272008 FTSE100 0.925 0.32 0.186 0.032 0.004 2.118 11.6482014 1.098 -0.061 -0.043 0.007 0.008 1.141 1.6371987 - - - - - - -2008 Dax 0.942 0.262 0.217 0.023 -0.05 1.11 2.2372014 1.407 -0.336 -0.018 0.00003 -0.357 2.116 -2.0851987 0.968 0.416 0.469 0.103 0.135 5.165 1.8262008 Nikkei 0.81 0.51 0.113 0.023 -0.047 5.165 1.8262014 0.869 0.203 0.045 0.007 -0.081 1.152 4.434

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(I) FTSE. (II) Dax.

(III) Nikkei.

The FTSE100 and Nikkei both saw large values for B,,C and w as well as smaller parameters forthe 1987 crash against their 2008 performance, highlighting that British and Japanese markets in 1987violently crashed, frequently oscillated but quickly stabilized. However this conveys that initially the 2008shock was not as destructive, but the forthcoming shock waves were more harming and chaotic, regardlessof geographical location.

Now studying the FTSE100 Triple Crash plot reveals that the British market behaved similarly toAmerica and suffered far more in respect to the 1987 than in 2008 crash initially. This is due to the 1987shock having a larger B, and , corresponding to a quickly stabilizing crash. The 2008 FTSE100 sawmarket values on average to be half of what they were in 1987, resulting in prolonged, volatile climate. Thisreinforces that British investors have become a lot more risk adverse with the market, as with less confidencecomes more volatility. Moreover this will fuel the needs for risk managing traders/hedge funds, who mayspeculate more in times of hardship, causing the initial recovery to take longer [21].

8.3 Emerging Markets

The following section focuses on the emerging financial markets to help understand their individual rolesand relations to past performance.

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Table VI: Emerging Market Parameters

Year Index A B C w 1987 - - - - - - -2008 MXX 0.875 0.354 0.118 0.012 -0.085 4.869 0.6032014 1.03 -0.009 -0.099 0.01 0.038 0.968 1.2761987 - - - - - - -2008 BVSP 1.01 -0.0004 -0.329 0.462 0.289 0.447 1.1862014 1.206 -0.002 -0.206 -0.037 0.106 0.774 10.8321987 0.893 2.456 0.67 1.057 0.524 2.12 5.7192008 HSI 1.253 -0.105 -0.075 0.026 -0.054 1.044 2.6412014 0.869 0.203 0.045 0.007 -0.081 1.152 4.4341987 - - - - - - -2008 SENSEX 0.189 1.029 0.023 -0.018 -0.069 1.066 2.0262014 1.026 -0.012 -0.064 -0.013 0.046 -7.419 -21.521

(I) Mexico IPC. (II) Bovespa.

(III) Hang Seng. (IV) Sensex.

The BVSP was the only index to crash this violently for the 2008 event when compared to its own 2014trajectory, explained by the negative , and C, forcing the market to rapidly drop with no oscillations.The 2008 crash saw more modest values and a positive C, allowing the oscillator to drive the share price upat times. This indicates that both investor confidence and the Brazilian economy began to deteriorate due to

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the 2008 financial crisis, but remained relatively stable for the commodity collapse. Emphasizing that Brazil,a country known for its commodity exports, had a stronger relationship to global financial institutions in2008 than it did with commodity importers in 2014.

9 Conclusion

The Nikkei saw aggressive price movements for a number of days after the 1987 crash while global marketsquickly stabilized. This suggests the Nikkei may have caused the shock, and a combination of this and traderactivity, repressed the market from calming as effectively as others. Furthermore, the Black Monday crashwas the most destructive over a short period of time, relative to the amount of value lost when the crashoccurred. However the Sup-prime mortgage crisis created a volatile market environment for an extendedperiod, which was caused by secondary shockwaves.

Financial based indices perform more effectively in financial crashes compared to manufacturing/indus-trial indices, at least for the initial time period. This phenomenon is likely due to the additional expertise andinsider knowledge retained by financials over non-financial companies. Nevertheless, this has been contestedby the NASDAQ who has grown to perform like the leading American indices during a crisis, largely dueto the exponential growth of technology companies in recent years. This underlines how larger companiesperform more uniformly compared to smaller businesses during times of financial hardship, regardless ofindustry.

The FTSE100 has become more risk adverse to aggressive market fluctuations, which is most likely dueto investors decreasing their tolerance to risk, fueling the need for hedge funds. Emerging markets on theother hand have seen increased volatility for the more recent shocks when compared to developed indices.This can be explained by investors looking elsewhere for safer heavens to store investment, while othersspeculate, causing the indices to oscillate.

Despite being a global crisis, the effect of the Commodity collapse may have been damped by the useof financial derivatives. However aggressive commodity movements can have a bigger impact on developedeconomies compared to emerging. Though repercussions from investor interest can have direct volatilityconsequences on emerging economies.

26

References

[1] Shiller,R., The use of Volatility measures in assessing Market Efficiency, National Bureau of EconomicResearch, No. 565, October 1980.

[2] Bree, D., Joseph, N. Testing for financial crashes using the Log Periodic Power Law model, Institutefor Scientific Interchange, April 2013. arXiv:1002.1010v2

[3] Subrahmanyam, A., Titman, S., Financial Market Shocks and the Macroeconomy, University of Cal-ifornia, Los Angeles, February 2013.

[4] White, W., Financial markets: shock absorbers or shock creators?, Bank for International Settlements,4th Geneva Conference on the World Economy, May 2002.

[5] Shiller, R., Investor behaviour in the October 1987 stock market crash: Survey Evidence, NationalBureau of Econimic Research, No. 2446, November 1987.

[6] Jaffee, D., Russell, T., Catastrophe Insurance, Capital Markets, and Uninsurable Risks, The Journalof Risk and Insurance, Vol.64, No2, pp.205-230, 1998.

[7] Farrell, P., Stock-market crash of 2016: The countdown begins, Market Watch, March 2015.

[8] L, P., Shocks in financial markets, price expectation, and damped harmonic oscillators,Insper, Insti-tuto de Ensino e Pesquisa, September 2011. arXiv:1103.1992v2

[9] Johansen, A., Stock market crashes are outliers, Department of Earth and Space Science and Instituteof Geophysics and Planetary Physics University of California, December 1997. arXiv:cond-mat/9712005

[10] Doukas, J., Arshanapalli, B,. International stock market linkages: Evidence from the pre- and post-October 1987 period, Journal of Banking and Finance, Vol.17, pp.193-208, May 2010.

[11] Ferguson, C., Inside Job, Sony Pictures Classics, May 2010.

[12] Clowers, N., Financial Crisis Losses and Potential Impacts of the Dodd-Frank Act, Financial Regu-latory Reform, United States Government Accountability Office, January 2013.

[13] Rogoff, K., The Shifts and the Shocks, Prospect Magazine, Harvard University, August 2014.

[14] Schwartz, A., Origins of the Financial Market Crisis of 2008, National Bureau of Economic Research,Vol.29, No.1, Winter 2009.

[15] Cosic, D., Bui, T,. Gong, X,. Vorisek, D. Commodity Markets Outlook, World Bank Group, Quarter1, January 2015.

[16] Yahoo Finance Yahoo Finance: Historical Prices, Accessed: March 2015.

[17] Sornette, D., Johansen, A,. Significance of log-periodic precursors to financial crashes, Institute ofGeophysics and Planetary Physics and Department of Earth and Space Science, February 2008.

[18] Nonlinear Least Squares Data Fitting, George Mason University, Department of Mathematical Sci-ences, Lecture Notes.

[19] Gratton, S., Lawless, A,. Nichols, N,. Approximate Gauss-Newton methods for nonlinear least squaresproblems, Numerical Analysis Report, Department of Mathematics, University of Reading, September2004.

[20] Sinha, V., Why Sensex Crashed 900 Points, its Biggest Fall in Over 5 Years, NDTV Profit, Market,January 2015.

[21] Fung, W., Hsieh, D,. The Risk in Hedge Fund Strategies: Theory and Evidence from Trend Followers,The Review of Financial Studies, Vol.14, No.2, 2011.

27

[22] Lundgren, K. Ryanair Lifts Profit Goal, Sees Fuel Hedges Slowing Gains, Bloomberg Business,Bloomberg Vol.14, February 2015.

[23] Mowery, D., Rosenberg, N., Paths of Innovation: Technological Change in 20th-Century America,Cambridge University Press, USA 1998.

Appendix A Instructions for using the Gauss-Newton Matlab code

First ensure the GaussNewton, Gradient, Resplot functions and at least one script including the data canall be found in your current directory. The codes for each of these have been included below.

The GaussNewton function works for 2 different methods. The first is the Mathematician method, whichrequires the user to calculate the subject functions derivatives for each of its parameters, while the second isthe Economist method and calculates them for the user. The advantage of Type 1 is that the computationaltime is considerably faster, while non-mathematicians who cannot calculate their functions derivatives canuse Type 2.

A.1 Mathematician Method (Type = 1)

If the user wants to use the current model please skip part 1.

1. Type out the subject function in the way done in GaussNewton, including all of the parameters. Ensurewherever the function Fun has been called matches with the subject function at the start. Ignore dFun

2. Select the amount of data points to fit, T, the amount of maximum iterations, It, Type = 1, and a 1or 0 depending on if they would like to study the models residuals.

3. Select the residual stopping criteria, , and the initial conditions for each of the subject functionsparameters. (The initial conditions used in this paper have been included at the end)

4. Call the script containing the relevant data points, and run.

A.2 Economist Method (Type = 2)

If the user wants to use the current model please skip parts 1 and 2.

1. Type out the subject function in the way done in GaussNewton, including all of the parameters. Ensurewherever the function Fun has been called matches with the subject function at the start. Also re-writethe subject function with capitals for the parameters, and as done for the subject function, ensure therest of the script matches up with the parameters found in dFun.

2. Modify Gradient to now include the new parameters.

3. Mathematical Method part 2

4. Mathematical Method part 3

5. Mathematical Method part 4

Appendix B Matlab Code

B.1 Gauss-Newton

function x = GaussNewton

% This is the main subject function.Fun = @(x1,x2,x3,x4,x5,x6,x7,t) x1 + x2.*exp(-x3.*t) + x4.*exp(-x5.*t).*cos(x6.*t-x7);

28

% This is the subject function that is used for differentiating (same function as above)dFun = @(X1,X2,X3,X4,X5,X6,X7,t) X1 + X2.*exp(-X3.*t) + X4.*exp(-X5.*t).*cos(X6.*t-X7);

runtime =cputime; % This used to calculate the computation time

Res=[0];Residual=[];C=[0];

T=20; % Time or amount of data points the model will fit toIt=50; % Desired amount of maximum iterationsType=1; % Type 1 for Math, 2 for EconomistResOn=0; % Select 1 for initial individual Residual analysisResOn2=0; % Select 1 for final individual Residual analysisResFin=0; % Select 1 for final iterated Residual analysis

% Below is the desired final residual condition and the 7 initial conditions

e=0.00001;x1(1)=0.9;x2(1)=0.8;x3(1)=0.01;x4(1)=0.1;x5(1)=0.1;x6(1)=5;x7(1)=0.2;

% Below is where the data scripts are stored and run

DataFTSE2008%DataDJ2008%DataNYSECOMP2008

if ResOn==1Resplot(Fun,x1,x2,x3,x4,x5,x6,x7,y,t) % Allows us to examine the residuals

end

j=1;

for i=1:length(t)

f(i) = Fun(x1(j),x2(j),x3(j),x4(j),x5(j),x6(j),x7(j),t(i));

F(i) = f(i) - y(i);

Res= [Res + F(:,i)2];

end

Residual = [Residual Res];

if Type == 2

% Calculates the functions derivatives for Econ method and stores them

DFstore = grad1(dFun,T);end

while 0.5*Res>e && C

f(i) = Fun(x1(j),x2(j),x3(j),x4(j),x5(j),x6(j),x7(j),t(i));

F(i) = f(i) - y(i);

end

if Type == 1

% Pre-calclated derivatives

df1 = ones(length(t),1);df2 = exp(-t*x3(j));df3 = -t*x2(j).*exp(-t*x3(j));df4 = exp(-t*x5(j)).*cos(x7(j) - t*x6(j));df5 = -t*x4(j).*exp(-t*x5(j)).*cos(x7(j) - t*x6(j));df6 = t*x4(j).*exp(-t*x5(j)).*sin(x7(j) - t*x6(j));df7 = -x4(j).*exp(-t*x5(j)).*sin(x7(j) - t*x6(j));

dF=[ df1 df2 df3 df4 df5 df6 df7];

elseif Type == 2

X1=x1(j);X2=x2(j);X3=x3(j);X4=x4(j);X5=x5(j);X6=x6(j);X7=x7(j);

dF = eval(subs(DFstore));

end

Deltaf = dF'*F';

p = -(dF'*dF)\Deltaf;

x1(j+1)=x1(j)+p(1);x2(j+1)=x2(j)+p(2);x3(j+1)=x3(j)+p(3);x4(j+1)=x4(j)+p(4);x5(j+1)=x5(j)+p(5);x6(j+1)=x6(j)+p(6);x7(j+1)=x7(j)+p(7);

X=[x1(j+1)' x2(j+1)' x3(j+1)' x4(j+1)' x5(j+1)' x6(j+1)' x7(j+1)'];

C=C+1;

for s=1:length(t)

Res=[0];

f(s) = Fun(x1(j+1),x2(j+1),x3(j+1),x4(j+1),x5(j+1),x6(j+1),x7(j+1),t(s));

F(s) = f(s) - y(s);

Res= [Res + F(:,i)2];

end

Residual = [Residual Res];

j=j+1;

end

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f = Fun(x1(j),x2(j),x3(j),x4(j),x5(j),x6(j),x7(j),t);

if ResOn2==1Resplot(Fun,x1(j),x2(j),x3(j),x4(j),x5(j),x6(j),x7(j),y,t) % Allows us to examine the residuals

end

if ResFin ==1

% plots the final residuals over time

if length(Residual)>1figuresubplot(2,1,1)plot(1:length(Residual),Residual(1:end))title('Residuals over time')xlabel('Time')ylabel('Residual')

% plots the current residual against the next

subplot(2,1,2)plot(Residual(2:end),Residual(1:end-1))title('Residual comparison')xlabel('Residual(i+1)')ylabel('Residual(i)')

endend

Parameters = {'A';'B';'a';'C';'b';'w';'y'};Guess = [x1(1);x2(1);x3(1);x4(1);x5(1);x6(1);x7(1)];FinalVals = round([X'],3);

% Table of results

Results = table(Guess, FinalVals,...'RowNames',Parameters)

fprintf('Final residual is %4.8f\n',Res*0.5)fprintf('%hd iteration(s)\n', C) ;fprintf('Total runtime is %4.2f seconds\n', cputime-runtime);

if sum(isnan(X)) == 0 % Only plots the solutions if they workfigureplot(t,f,'-ob')title('Sensex 2014')hold on

plot(t,y(t),'-or')xlabel('Time')ylabel('Price')legend('Model','Real')hold off

endend

B.2 Gradient code

function df = Gradient(dFun,T)

syms X1 X2 X3 X4 X5 X6 X7 t

Vars = [X1 X2 X3 X4 X5 X6 X7]; % User enters variables here

f = dFun(X1,X2,X3,X4,X5,X6,X7,t);

t=1:T; % amount of data points

31

for m=1:length(t)for i = 1:length(Vars)

t=m;

df(m,i) = subs(diff(f,Vars(i)));

endend

dF = eval(subs(df));

end

B.3 ResPlot code

function z = Resplot(Fun,x1,x2,x3,x4,x5,x6,x7,y,t)

Res=[0];Resi=[];

x1(1)=x1;x2(1)=x2;x3(1)=x3;x4(1)=x4;x5(1)=x5;x6(1)=x6;x7(1)=x7;

j=1;

for i=1:length(t)

f(i) = Fun(x1(j),x2(j),x3(j),x4(j),x5(j),x6(j),x7(j),t(i));

F(i) = f(i) - y(i);

Resi =[Resi F(:,i)2];

Res= [Res + F(:,i)2];

End

% These graph show how the initial residuals behave for the initial conditions, for a single iteration

figuresubplot(3,1,1)plot(1:length(Resi),Resi)title('Individual residuals over time')xlabel('Time')ylabel('Indv Res')

subplot(3,1,2)plot(f,Resi)title('Individual residuals vs models value')xlabel('Model vals')ylabel('Indv Res')

subplot(3,1,3)plot(Resi(2:end),Resi(1:end-1))

title('Individual residuals comparison')xlabel('Indv Res(i+1)')ylabel('Indv Res(i)')

end

B.4 Example of a Data script for T=20: FTSE2008

32

t=[1:T]';

FTSE2008 = [4900.00; 4870.34; 4980.34; 4589.19; 4605.22;4366.69; 4313.80; 3932.06; 4256.90; 4394.21;4079.59; 3861.39; 4063.01; 4282.67; 4229.73;4040.89; 4087.83; 3883.36; 3852.59; 3926.38];

FTSE2008av = sum(FTSE2008)/length(FTSE2008);

for i=1:length(FTSE2008)FTSE2008Norm(i) = FTSE2008(i)/FTSE2008av;

end

y=FTSE2008Norm';

Appendix C Initial Conditions used in this paper

% FTSE1987 FTSE2008 FTSE2014

x1(1)=1; x1(1)=0.9; x1(1)=1;x2(1)=1; x2(1)=0.8; x2(1)=1;x3(1)=0.1; x3(1)=0.01; x3(1)=0.1;x4(1)=0.1; x4(1)=0.1; x4(1)=0.1;x5(1)=0.01; x5(1)=0.1; x5(1)=0.01;x6(1)=4.5; x6(1)=5; x6(1)=1;x7(1)=3.5; x7(1)=0.2; x7(1)=1;

% DJ1987 DJ2008 DJ2014

x1(1)=1; x1(1)=1; x1(1)=1;x2(1)=1; x2(1)=1; x2(1)=1;x3(1)=0.2; x3(1)=0.1; x3(1)=0.1;x4(1)=0.2; x4(1)=0.1; x4(1)=0.1;x5(1)=0.01; x5(1)=0.01; x5(1)=0.01;x6(1)=1.1; x6(1)=1; x6(1)=1;x7(1)=3.6; x7(1)=1; x7(1)=0;

% NYSECOMP1987 NYSECOMP2008 NYSECOMP2014

x1(1)=1; x1(1)=0.8; x1(1)=1;x2(1)=1; x2(1)=0.8; x2(1)=1;x3(1)=0.1; x3(1)=0.3; x3(1)=0.1;x4(1)=0.1; x4(1)=0.3; x4(1)=0.1;x5(1)=0.01; x5(1)=0.01; x5(1)=0.04;x6(1)=5.9; x6(1)=5; x6(1)=7;x7(1)=5.8; x7(1)=1; x7(1)=0.84;

% NYSEUS1002008 NYSEUS1002014

x1(1)=1; x1(1)=1;x2(1)=1; x2(1)=1;x3(1)=0.1; x3(1)=0.05;x4(1)=0.1; x4(1)=0.01;x5(1)=0.01; x5(1)=0.01;x6(1)=5; x6(1)=8;x7(1)=1; x7(1)=0.5;

% SP5001987 SP5002008 SP5002014

x1(1)=1; x1(1)=0.9; x1(1)=1;x2(1)=0.8; x2(1)=0.4; x2(1)=1;x3(1)=0.01; x3(1)=0.15; x3(1)=0.1;x4(1)=0.1; x4(1)=0.05; x4(1)=0.1;x5(1)=0.01; x5(1)=0.01; x5(1)=0.01;

33

x6(1)=1; x6(1)=4.8; x6(1)=1.6;x7(1)=1.4; x7(1)=0.6; x7(1)=0.5;

% DAX2008 DAX2014

x1(1)=1; x1(1)=1;x2(1)=0.5; x2(1)=1;x3(1)=0.1; x3(1)=0.1;x4(1)=0.1; x4(1)=0.1;x5(1)=0.01; x5(1)=0.01;x6(1)=1; x6(1)=2.5;x7(1)=1; x7(1)=4;

% NASDAQ1987 NASDAQ2008 NASDAQ2014

x1(1)=1; x1(1)=1; x1(1)=1;x2(1)=1; x2(1)=0.5; x2(1)=1;x3(1)=0.01; x3(1)=0.1; x3(1)=0.07;x4(1)=0.2; x4(1)=0.1; x4(1)=0.1;x5(1)=0.1; x5(1)=0.1; x5(1)=0.01;x6(1)=5; x6(1)=1; x6(1)=4.5;x7(1)=1.5; x7(1)=0.3; x7(1)=1;

% MXX2008 MXX2014

x1(1)=1; x1(1)=1;x2(1)=0.4; x2(1)=0.4;x3(1)=0.1; x3(1)=0.1;x4(1)=0.1; x4(1)=0.1;x5(1)=0.1; x5(1)=0.01;x6(1)=5; x6(1)=1;x7(1)=1; x7(1)=2;

% BVSP2008 BVSP2014

x1(1)=1; x1(1)=1;x2(1)=1; x2(1)=0.1;x3(1)=-0.5; x3(1)=0.1;x4(1)=0.5; x4(1)=0.1;x5(1)=0.2; x5(1)=0.01;x6(1)=0.5; x6(1)=1;x7(1)=1.1; x7(1)=12;

% N2251987 N2252008 N2252014

x1(1)=1; x1(1)=0.9; x1(1)=1;x2(1)=1; x2(1)=0.5; x2(1)=1;x3(1)=0.01; x3(1)=0.1; x3(1)=0.1;x4(1)=0.2; x4(1)=0.1; x4(1)=0.1;x5(1)=0.1; x5(1)=0.01; x5(1)=0.01;x6(1)=5; x6(1)=5; x6(1)=1;x7(1)=3.5; x7(1)=0.5; x7(1)=2.5;

% HSI1987 HSI2008 HSI2014

x1(1)=1; x1(1)=1; x1(1)=1;x2(1)=0.8; x2(1)=0.5; x2(1)=1;x3(1)=0.1; x3(1)=0.1; x3(1)=0.01;x4(1)=0.1; x4(1)=0.1; x4(1)=0.01;x5(1)=0.01; x5(1)=0.01; x5(1)=0.01;x6(1)=1.6; x6(1)=1; x6(1)=0.5;x7(1)=2.5; x7(1)=3; x7(1)=4;

% SENSEX2008 SENSEX2014

x1(1)=1; x1(1)=1;x2(1)=1; x2(1)=1;

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x3(1)=0.1; x3(1)=0.1;x4(1)=0.1; x4(1)=0.1;x5(1)=0.01; x5(1)=0.01;x6(1)=1.; x6(1)=5.;x7(1)=1; x7(1)=0;

35

IntroductionThe EventBlack Monday October 1987Sub-prime Mortgage September 2008Crude Oil Collapse September 2014

The Indices

The Model - Damped Harmonic oscillatorMethod for fitting - GaussNewton Least SquaresResidual AnalysisIndividual ResidualsIterated Residuals

Black Monday 1987Implementation & AnalysisBlack Monday Global Average

Sub-prime Mortgage crash 2008Implementation & AnalysisSub-prime Global Average

Crude Oil Collapse 2008Implementation & AnalysisCrude Oil Global Average

Historical and Geographical relationshipsAmericaEurope/Asia (Developed)Emerging Markets

ConclusionAppendix Instructions for using the Gauss-Newton Matlab codeMathematician Method (Type = 1)Economist Method (Type = 2)

Appendix Matlab CodeGauss-NewtonGradient codeResPlot codeExample of a Data script for T=20: FTSE2008

Appendix Initial Conditions used in this paper