I Know First Presentation (February 2013)

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Finance

Published on June 25, 2013

Author: iknowfirst

Source: slideshare.net

Description

I Know First co-founder, Dr. Lipa Roitman delivers a lecture at Tel Aviv University about his company's advanced stock market forecasting algorithm.

How Can We Predict the FinancialMarkets?Investing in the time of uncertaintyBy Dr. Lipa RoitmanIKnowFirst.comFebruary 2013

2SlideIKnowFirst.comI Know First - Israeli Forecasting Technologies is an Israeli start up company. Our mainproduct is a financial market forecasting algorithm that predicts daily more than 200 markets:Stocks, world indices, Currencies & Commodities.The company-I Know first-Introduction• The system is a predictivemodel based on artificialIntelligence (AI) andMachine Learning (ML),and incorporatingelements of ArtificialNeural Networks andGenetic Algorithms, builtwith insights of ChaosTheory and self-similarity,the Fractals.• I Know Firsttracks andpredicts the flowof money fromone market orinvestmentchannel toanotherI Know FirstPredicts 200investment channelsdailyTracks theflow of money ArtificialIntelligence(AI) andMachineLearning (ML),ArtificialNeuralNetworksGeneticAlgorithms

3SlideIKnowFirst.comLipa Roitman PhDEntrepreneur, algorithmic trading system developer20 years in the AI (artificial intelligence) and MachineLearning FieldsConsultant for startup companiesR&D Chemist with record in computer modeling ofprocess, new product and process development http://iknowfirst.com/Stock-forecast-articles l k rc ic he e

4SlideIKnowFirst.com"it is exceedingly difficult to make predictions,particularly about the future"Niels Bohr.

5SlideIKnowFirst.comOld Fallacies About MarketsEfficient: “Markets are efficient andunpredictable: today’s information isalready reflected in price. No one stockis a better buy then the other”.Random walk: “The patterns of stockmarket prices are purely random.Markets are unpredictable because theyare random. Playing markets is agamble“Markets are neither totally efficient, nor totallyrandom. They are complex and chaotic.

6SlideIKnowFirst.comTrading the Stock MarketFact: big trading houses likeGoldman Sachs, MorganStanley consistently make profittrading stocks.Some fail spectacularly, likeLehman Brothers, when theymake big bets that go wrong,and we read of them in thepaper.

7SlideIKnowFirst.comHigh-Frequency TradingHigh-frequency trading HFT: quick arbitrage (inmilliseconds)Computer algorithms place orders based oninformation that is received electronically, beforehuman traders can react.Example: arbitrage from the bid-ask spread→Supposed to provide liquidity, but sometimes errors in thealgorithms cause huge distortions in prices.→Example: May 6, 2010 Flash Crash→January 23, 2013 AAPL plunge

8SlideIKnowFirst.comRisk Management: Fat Tailed Distribution AAPL FlashDump High-frequencytradingalgorithms. How else could800,000 sharesworth nearly$300 million besold in 17-secondintervals?

9SlideIKnowFirst.comHigh-Frequency TradingProfits from high-frequency trading HFT inAmerican stocks havepeaked in 2009, andgoing down since:→ $1.25 billion in 2012, down35 percent from 2011 and74 percent lower than $4.9billion in 2009→The percentage of stocktrades handled by firmsthat specialize in H.F.T. fellto about 51 percent in 2012from 60 percent in 2009.

10SlideIKnowFirst.comHigh-Frequency TradingWhy HFT is slowing down?Lower trade volumeretail investor is not back yet(begins to come back recently)Mainly institutional investors are participatingTechnological costs: financial resources to compete in thisfield are now enormous (high latency, proximity toexchanges, investment in hardware and software)High competition-low profit

11SlideIKnowFirst.comTrading the Stock MarketStock market is not a walk in the park, but it’s nota random walk either.Everyone can get lucky with stock marketsometimes.To do it consistently requires knowledge ofthe market, the market direction and riskmanagement strategy.

12SlideIKnowFirst.comChaos is the Result of Complexity Why do stock prices move?→news stream constantly injects new information.→mixed messages→different market players psychology creates patterns.

13SlideIKnowFirst.comChaos is the Result of Complexity Objective factors→Different valuationmodelsFundamentalvaluationPrice momentum, etc.→Different time horizons short time horizon vs.the longer view.Reasons for chaotic behavior

14SlideIKnowFirst.comPsychology of TradingHuman factor: It is difficultif not impossible to make a100% rational decisionduring uncertainty.Prospect theory:(Kahneman and Tversky,1979): Losses have moreemotional impact than anequivalent amount of gains.Risk aversion.Reasons for chaotic behavior

15SlideIKnowFirst.comPsychology of TradingWhich stock do you check first when you analyzeyour portfolio performance daily ?→Last one?→First one?→The largest investment?

16SlideIKnowFirst.comPsychology of TradingLatest news biasOverreaction: can’t quantifyAnchoring: buying on dipsTrend lovers: herd mentalityWhy people lose money

17SlideIKnowFirst.comChaos is the Result of Complexity

18SlideIKnowFirst.comThere is an Order in the Chaos. Basic Money Law: Moneyis looking for: highest returns with lowest risk It constantly flows fromone market to another.→ However the way the flowoccurs is rarely smooth,but marked with periods ofturbulence.

19SlideIKnowFirst.comWhat is Chaos?Chaos is a complex non-linear evolving systemthat is sensitive to initial conditions. It has“memory”.There are times when the path is well definedand predictableThere are points in time (instability regime)where a minor perturbation can switch thefuture path between two opposite directions.Chaos can appear as randomness, but it isnot. This is the way we could tell them apart:

20SlideIKnowFirst.comRandom behavior cant be learned. It is random.→Past patterns dont repeat.Chaotic systems have memory. What happened inthe past affects the future.→ Can be learned and predicted to some extent (quasi-deterministic chaos).What is Chaos?

21SlideIKnowFirst.comChaos is a law ofnature. It appearsin complex dynamicsystems whereeach elementaffects the others.Astronomy: many-bodies system.WeatherWhat is Chaos?

22SlideIKnowFirst.comStock Market and Quantum Mechanics Double slit experimentParticle-wave duality

23SlideIKnowFirst.comStock Market and Quantum MechanicsFive years chart:Patterns can be seen.The prices jump from 1 level to another

24SlideIKnowFirst.comStock Market and Quantum MechanicsOne month chart:No patterns apparent

25SlideIKnowFirst.comStock Market and Quantum MechanicsThree days chartGranularity: SingletransactionsNo patterns apparentStock market exhibits deterministic chaos,making the short-term movements of pricesextremely impossible to predict.

26SlideIKnowFirst.comStock Market and Quantum MechanicsHow is the Stock Market like Quantum Mechanics?A single photon (a particle of light) or an electronbehave like a particle (quantum), an assembly ofthem behaves like a wave.Discrete quantum levels of electrons in an atom.Market is an assembly of individual transactions(quanta). Together they show similar patterns.• Both are Probabilistic• Discrete levels• Granularity: unpredictable on microscopicscale, but predictable on large scale.

27SlideIKnowFirst.comChaos ExampleWhat is the pattern?

28SlideIKnowFirst.comFinancial BubblesBig BubbleiPhone 5 release

29SlideIKnowFirst.comChaos Theory and Financial BubblesFeedback mechanisms:Positive feedback amplifies trends (bubbleformation):→Crowd mentality, chain reaction: chasing one stock,→Go with the trend!

30SlideIKnowFirst.comChaos Theory and Financial BubblesFeedback mechanisms:Negative feedback limits the trend→bubble bursting,→range-bound trading→“price got too high: sell!,→“price got too low: buy!”

31SlideIKnowFirst.comChaos ExampleNegative FeedbackPositive Feedback

32SlideIKnowFirst.comApple Inc. AAPL Bubble CrashFinancial Bubble Detection

33SlideIKnowFirst.comChaos Theory and Financial BubblesRandomness is also part of the market:Occurs when the market is indecisive→Low volume→Increased volatility→Randomness is stronger near turning pointsHidden variables: Albert Einstein VS. Niels Bohr:"I am convinced God does not play dice"→A trader placed big order at the low liquidity time→Computer error→Market manipulation (HFT program trading, etc)Increased volatility (randomness) is a warning: stay awayfrom the market or adjust your tactics

34SlideIKnowFirst.comWhat are financial bubbles?Stock market is a bubble machine on alltime scale levels. Big bubbles can lastyears.A bubble is a very basic part of themarket and can’t be eliminated, but if itis recognized it could be exploited!Part of the price discovery process inthe presence of uncertainty.To get to the “fair” market price the pricehas to “overshoot” in both directions.The market constantly behaves like a drunk driver

35SlideIKnowFirst.comThe Key to the MarketMarkets are chaotic, they alternate between threeregimes: positive feedback, negative feedback, andrandomness.The three regimes could be present simultaneously atdifferent time scales The one who can recognize these regimes has the key tothe market.The “buy” or “sell” decision depends on what regime youthink the market is at now, and at what time scale!

36SlideIKnowFirst.comThe Key to the Market“For every complex problem there is an answerthat is clear, simple and wrong.”→– H. L. Mencken

37SlideIKnowFirst.comThe Key to the MarketMarket fallacies:“The trend is your friend —(until that nasty bend at the end)”.“Buy low, sell high”.The “buy” or “sell” decision dependson what regime you think the market isat now, and at what time scale!

38SlideIKnowFirst.comThe Key to the Market"I took economics courses in Harvard Collegefor four years, and everything I was taughtwas wrong."Franklin D. Roosevelt (1882 –1945)"The established theory has collapsed but wehavent actually got a proper understandingof how financial markets operate”.George SorosWorld Economic Forum in Davos 2013

39SlideIKnowFirst.comI Know First Algorithmic SystemI Know First algorithmic system was developedto discover the laws of the market that could showwhich way the market is going.

40SlideIKnowFirst.comI Know First - Israeli Forecasting Technologies is an Israeli start up company. Our mainproduct is a financial market Forecasting system that predicts daily more than 200 markets:Stocks, Currencies & Commodities.The company-I Know first-Introduction• The system is a predictivemodel based on artificialIntelligence (AI) andMachine Learning (ML),and incorporatingelements of ArtificialNeural Networks andGenetic Algorithms, builtwith insights of ChaosTheory and self-similarity,the Fractals.• I Know Firsttracks andpredicts the flowof money fromone market orinvestmentchannel toanotherI Know FirstPredicts 200investment channelsdailyTracks theflow of money ArtificialIntelligence(AI) andMachineLearning (ML),ArtificialNeuralNetworksGeneticAlgorithms

41SlideIKnowFirst.comI Know First Algorithmic SystemI Know First Algorithmic System is based on the realizationthat a stock value is a function of many factors which interactin a non-linear way and affect the future trajectory of the stockcreating waves in prices.Being completely empirical, the I Know First self learningalgorithms analyze the inputs and rank them according to theirsignificance in predicting the target stock price.Then they create multiple models, and test them automaticallyon the historical data.The robustness of the model is measured by how it performsin different market circumstances.The best predicting models are kept and the rest are rejected.Such refinement has continued daily as the new market datais added to the historical pool.

42SlideIKnowFirst.comI Know First Algorithmic SystemAdaptable and balanced algorithm:→Empiricaldoes not depend on any human assumptionsself-learning→learns new patterns daily,→adapts to new reality,→but still follows the general historical rules.

43SlideIKnowFirst.comI Know First Algorithmic SystemThe stocks, indexes, commodities and currenciesrepresent most of the liquid forms of investment“What if” scenarios:→Effect of raising or lowering interest rates on themarkets and on real estate prices.→Effect of currency exchange rates→Effect of higher or lower oil prices→Effect of price of oil on oil company stocks→More….A Model of the World Economy!

44SlideIKnowFirst.comI Know First Algorithmic System Many inputs from different sources go into each algorithmicforecast. Up to 15 years of data go into each model. Powerful computers process and learn the data, and createforecast.

45SlideIKnowFirst.comI Know First SystemrunningCycleThe Product- I Know First SystemBasic PrincipleDaily stock DataGet the daily marketupdate, and add it tothe 15-years databaseRun a learning & prediction cycle with new combineddata (Takes about 8 hours per cycle, it runs non-stoparound the clock).15 yearsstockdatabaseReporting Module-Software as aservice (SaaS)The daily prediction for each stock/currency/ commodity is produced forthe following periods :3 days, 7 days,14 days, 30 days, 90 days & 365 daysAgenTeamIQSHIPLearning &PredictionCycleLearning &PredictionCycleGenerateResults”procedureGenerateResults”procedurePredictionsPredictionsDatabase3 Daysprediction7 Daysprediction14 Daysprediction30 Daysprediction90 Daysprediction365 Daysprediction

46SlideIKnowFirst.comI Know First Algorithmic SystemFeature Tech.analysisI Know FirstAlgorithm Self learning Adaptable Learns new patterns daily Looks at many different stocks,indexes, etc Signals at different time horizons Quantitative Predictability indicator Artificial Intelligence Neural Networks Genetic algorithmsNoNoNoNoNoNoNoNoNoNoNo Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes YesThe system is self learning, which sets it apartfrom the technical analysis.

47SlideIKnowFirst.comI Know First Algorithmic System-Performance

48SlideIKnowFirst.comI Know First Algorithmic System-Performance

49SlideIKnowFirst.comThe Product- I Know First SystemBasic PrincipleGTI CHKP X F VNQ16.69 14.58 13.90 13.42 12.620.622 0.528 0.676 0.655 0.679GT KEY Platinum ^HSI ORBK11.82 11.80 11.09 9.75 9.450.709 0.209 0.591 0.707 0.685AMD BA IFN WAG DD9.20 8.77 7.87 7.50 6.760.363 0.553 0.575 0.484 0.576DIS CSCO VUG UN POT5.68 5.56 5.34 5.31 4.760.592 0.642 0.694 0.585 0.551^VIX ZRAN CSX VPL VXF4.71 4.55 4.41 4.26 4.010.526 0.469 0.643 0.66 0.629WFR ^TA100 AAUKY VDE BAC3.86 3.86 3.77 3.73 3.680.557 0.731 0.634 0.618 0.463VTV VGK S&P500 INTC HAS3.64 3.61 3.38 3.22 2.900.605 0.696 0.628 0.595 0.439DOX WFC RTLX LMT ADM2.81 2.79 2.36 2.02 1.700.448 0.269 0.456 0.39 0.265EWG TEVA EWC NICE WMT1.54 1.25 0.68 0.56 0.510.59 0.487 0.373 0.411 0.125EWA JNJ KO VWO T0.34 0.26 0.24 0.21 0.190.741 0.406 0.491 0.707 0.028K BMY AIP NIS/EUR NIS/GBP0.09 0.07 0.03 -0.09 -0.090.402 0.048 0.614 0.272 0.476TRP NIS/$US CVS FCX CAT-0.18 -0.22 -0.29 -0.29 -0.350.536 0.485 0.381 0.29 0.583ENB NSC JY/$US ESLT TKF-0.38 -0.51 -0.77 -0.78 -2.210.625 0.479 0.203 0.594 0.418OTEX AAPL Crude Oil IRL ISRL-2.32 -3.19 -5.09 -7.09 -11.640.459 0.644 0.668 0.47 0.398WDC DOW-12.53 -43.430.692 0.602Negative Negative Positive Postive++ ++SignalSymbolPredictabilityVWO0.210.707• For each stock/ currency/commodity the followingdata is calculated by thesystem:•Predictability - The"strength" of theprediction•Signal- the movementdirection(increase/decrease)•The daily prediction isproduced for thefollowing periods :3days, 7 days, 14 days,30 days, 90 days & 365days

50SlideIKnowFirst.comThe Product- I Know First SystemBasic PrincipleGTI CHKP X F VNQ16.69 14.58 13.90 13.42 12.620.622 0.528 0.676 0.655 0.679GT KEY Platinum ^HSI ORBK11.82 11.80 11.09 9.75 9.450.709 0.209 0.591 0.707 0.685AMD BA IFN WAG DD9.20 8.77 7.87 7.50 6.760.363 0.553 0.575 0.484 0.576DIS CSCO VUG UN POT5.68 5.56 5.34 5.31 4.760.592 0.642 0.694 0.585 0.551^VIX ZRAN CSX VPL VXF4.71 4.55 4.41 4.26 4.010.526 0.469 0.643 0.66 0.629WFR ^TA100 AAUKY VDE BAC3.86 3.86 3.77 3.73 3.680.557 0.731 0.634 0.618 0.463VTV VGK S&P500 INTC HAS3.64 3.61 3.38 3.22 2.900.605 0.696 0.628 0.595 0.439DOX WFC RTLX LMT ADM2.81 2.79 2.36 2.02 1.700.448 0.269 0.456 0.39 0.265EWG TEVA EWC NICE WMT1.54 1.25 0.68 0.56 0.510.59 0.487 0.373 0.411 0.125EWA JNJ KO VWO T0.34 0.26 0.24 0.21 0.190.741 0.406 0.491 0.707 0.028K BMY AIP NIS/EUR NIS/GBP0.09 0.07 0.03 -0.09 -0.090.402 0.048 0.614 0.272 0.476TRP NIS/$US CVS FCX CAT-0.18 -0.22 -0.29 -0.29 -0.350.536 0.485 0.381 0.29 0.583ENB NSC JY/$US ESLT TKF-0.38 -0.51 -0.77 -0.78 -2.210.625 0.479 0.203 0.594 0.418OTEX AAPL Crude Oil IRL ISRL-2.32 -3.19 -5.09 -7.09 -11.640.459 0.644 0.668 0.47 0.398WDC DOW-12.53 -43.430.692 0.602Negative Negative Positive Postive++ ++SignalSymbolPredictabilityVWO0.210.707Two components tothe stock action→ Stock-specific action.→ Stock action related togeneral market action. A stock is a part of themarket and responds tothe general market news. The forecast tableshows how the stockforecast is positionedrelatively to other stocksforecast.

51SlideIKnowFirst.comThe Product- I Know First SystemBasic PrincipleGTI CHKP X F VNQ16.69 14.58 13.90 13.42 12.620.622 0.528 0.676 0.655 0.679GT KEY Platinum ^HSI ORBK11.82 11.80 11.09 9.75 9.450.709 0.209 0.591 0.707 0.685AMD BA IFN WAG DD9.20 8.77 7.87 7.50 6.760.363 0.553 0.575 0.484 0.576DIS CSCO VUG UN POT5.68 5.56 5.34 5.31 4.760.592 0.642 0.694 0.585 0.551^VIX ZRAN CSX VPL VXF4.71 4.55 4.41 4.26 4.010.526 0.469 0.643 0.66 0.629WFR ^TA100 AAUKY VDE BAC3.86 3.86 3.77 3.73 3.680.557 0.731 0.634 0.618 0.463VTV VGK S&P500 INTC HAS3.64 3.61 3.38 3.22 2.900.605 0.696 0.628 0.595 0.439DOX WFC RTLX LMT ADM2.81 2.79 2.36 2.02 1.700.448 0.269 0.456 0.39 0.265EWG TEVA EWC NICE WMT1.54 1.25 0.68 0.56 0.510.59 0.487 0.373 0.411 0.125EWA JNJ KO VWO T0.34 0.26 0.24 0.21 0.190.741 0.406 0.491 0.707 0.028K BMY AIP NIS/EUR NIS/GBP0.09 0.07 0.03 -0.09 -0.090.402 0.048 0.614 0.272 0.476TRP NIS/$US CVS FCX CAT-0.18 -0.22 -0.29 -0.29 -0.350.536 0.485 0.381 0.29 0.583ENB NSC JY/$US ESLT TKF-0.38 -0.51 -0.77 -0.78 -2.210.625 0.479 0.203 0.594 0.418OTEX AAPL Crude Oil IRL ISRL-2.32 -3.19 -5.09 -7.09 -11.640.459 0.644 0.668 0.47 0.398WDC DOW-12.53 -43.430.692 0.602Negative Negative Positive Postive++ ++SignalSymbolPredictabilityVWO0.210.707Customized forecastChoosing stocks fromparticular industry intothe table composition

52SlideIKnowFirst.comThe Product- I Know First SystemShort term predictions: 3 days, 7 days, 14 daysExample

53SlideIKnowFirst.comThe Product- I Know First SystemLong term predictions: 30 days, 90 days & 365 days

54SlideIKnowFirst.comI Know First Algorithmic SystemMore uses for algorithms:Forecasting demand for products and servicesModeling complex chemical processesGlobal climate trendsAgricultural forecasting: crops, water demandMore….Send us requests for your forecasting needsiknowfirst@iknowfirst.com

55SlideIKnowFirst.comWhat are the stocks for 2013?

56SlideIKnowFirst.comDecember performance

57SlideIKnowFirst.comDecember 2012 performance-aggressive

58SlideIKnowFirst.comRecent performance

59SlideIKnowFirst.comOctober 2012 performance

60SlideIKnowFirst.comGoogle Stock Forecast: Case Study http://iknowfirst.com/Google-stock-forecast-case-study

61SlideIKnowFirst.comGoogle Stock Forecast: Case Study

62SlideIKnowFirst.comGoogle Stock Forecast: Case Study

63SlideIKnowFirst.comSignal Analysis of a Group of StocksThe following chart shows a combined signal of allstocks in the system, calculated for each of thesix time ranges.http://iknowfirst.com/S-P-500-May-12-2013Send us requests for your forecasting needsiknowfirst@iknowfirst.com

64SlideIKnowFirst.comSignal analysis of a group of stocks

65SlideIKnowFirst.comSignal analysis of a group of stocksThe stocks and indices in the I Know Firstsystem are a good representation of abroader market.The forecast for the plurality of stocks in theI Know First system can serve as a proxy forthe S&P500 forecast.

66SlideIKnowFirst.comPredictability: Measuring ChaosPredictability is the indicator that tells predictablechaos from randomness.Just like the market rises and falls in waves, sodoes the predictability. And the waves are notsynchronous.The focus shifts between gold, stocks, oil, bonds.Some become more predictable, while the othersretreat to randomness.

67SlideIKnowFirst.comPredictability: Measuring ChaosBy monitoring predictability one can get advancewarning that the market paradigm change is inprogress.

68SlideIKnowFirst.comWhich stocks can be predicted Most of the major stocks in the S&P 500 index arepredictable to some extent. Start-Ups are Unpredictable→ Investor hopes: ‘This is going to be the new Google, the newFacebook.’→ Some start-ups make it, some don’t — nobody knows inadvance But the main reason our algorithms can’t forecast start-ups: no history. No way to predict future moves.

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80SlideIKnowFirst.comRisk ManagementSo, is it a trading system that could make moneyconsistently?Most of the time yes, with the right riskmanagement strategy and a bit of luck!Luck factorWe don’t know all risk factors.

81SlideIKnowFirst.comRisk Management●There are:●“known knowns; there are things we know that weknow.●known unknowns; that is to say there are thingsthat, we now know we dont know.●But there are also unknown unknowns – there arethings we do not know we dont know”.●—Donald Rumsfeld●United States Secretary of Defense

82SlideIKnowFirst.comRisk Management: Normal DistributionNormaldistributionapplies tomanyrandomevents,but it is not arule in themarkets, butrather anexception.

83SlideIKnowFirst.comRisk Management: Fat Tailed DistributionFat Taileddistribution is verycommon in themarkets.Large swings (3 to 6standard deviationsfrom the mean)Far more frequentthan the normaldistributionPower Law

84SlideIKnowFirst.comThe Danger of Fat TailsThe uncertainty about price distribution makes“rational” decision making impossible.One could be right about the market direction, butlose money or miss an opportunity.

85SlideIKnowFirst.comThe Danger of Fat TailsRisk management:allocation of capital traditional allocation model)allocation of risk.

86SlideIKnowFirst.comRisk Management

87SlideIKnowFirst.comStrategies for SuccessWatch the signals daily, but act only on strong ones.To minimize risk stay out of the market until you see agreat opportunity: a strong signal, extreme price.When predictability is high, invest on strong signals.When predictability goes down, expect a storm.When the signal disappears or weakens, reduce yourexposure.For a stable portfolio invest in non-correlated securities.→Caveat: during times of global financial crisis all assets becomepositively correlated, because they all move (down) together.

88SlideIKnowFirst.comRisk Management: Fat Tailed DistributionWhat happened to Apple (AAPL) in the lastminute of trading Friday, January 23, 2013

89SlideIKnowFirst.comRisk Management: Fat Tailed DistributionAAPL FlashDumpHigh-frequencytradingalgorithms.How elsecould 800,000shares worthnearly $300million be soldin 17-secondintervals?:

90SlideIKnowFirst.comWhat Causes Fat Tails●Extreme risks of "high consequence", but of lowprobability. The risks of●terrorist attack, major earthquakes, accidents●hurricanes, a volcanic-ash cloud grounding allflights for a continent,●HFT trading algorithmsthe frequency and impact of totally unexpected events isgenerally underestimated

91SlideIKnowFirst.comSelf-Similarity: Fractals.Chaotic systems and fractals.Fractals are objects which are "self-similar" in thesense that the individual parts are related to thewhole. Mandelbrot.→The detail looks just about the same as the whole. Market patterns are the same on all time scales,except the shortest times (quanta).

92SlideIKnowFirst.comFractals and the Power Laws The main attribute of power laws that makes them interesting is theirscale invariance. Given a relation f(x) = ax^k, scaling the argument xby a constant factor c causes only a proportionate scaling of thefunction itself. That is,→ f(c x) = a(c x)^k = c^{k}f(x) is proportional to f(x). That is, scaling by a constant c simply multiplies the original power-law relation by the constant c^k. Thus, it follows that all power lawswith a particular scaling exponent are equivalent up to constantfactors, since each is simply a scaled version of the others. Thisbehavior is what produces the linear relationship when logarithms aretaken of both f(x) and x, and the straight-line on the log-log plot is often called the signature of apower law.

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