**INTRODUCTION**

Statistical principles and tools as used in Black -Scholes assumptions which remains corner stone of risk management world wide from petty traders to large hedge funds and wall street banks may not really be so robust bull work against risk of seismic proportions as so often touted.In the era of Algos and high frequency trading ,outliers and black swans are too common ,wide spread and frequent than formerly assumed and threaten to wipe out trading accounts in few seconds. Comfort and confidence of statistical techniques are greatly shaken.In fact one may be mistakenly picking pennies right in front of a giant road roller –A calamity

In one of my articles on positive feed back systems ,I have taken example of Anglo-Irish bank to demonstrate how positive feed back system driven bubble led to collapse of the bank.In the present which deals with Randomness hypothesis,,I have chosen Cyprus bank to demonstrate pitfalls of statistical model of risk management .Statistical principles such as central limit theorem, The lawof averages or power law and statistical tools such as normal distribution ,standard deviation, variance and volatility may fail to manage risk and instead may drive by default to the brink of a cliff..

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FAILURE OF RANDOM SYSTEM

Well, the big Cypriot banks made a really bad investment. They lent money to Greece. When the Greek economy took a dive, the Cypriot banks took a bigger gamble, buying up Greek government bonds in the hopes of a bailout. Now they’re broke. The banks owe more money than they have. In fact they owe more money than the country’s GDPThe only place Cyprus could find large amounts of cash were in people’s personal savings accounts. Fearing the government was coming for their cash, Cypriots ran to the bank. The government declared a national holiday and closed the banks, so everyone ran to the ATM outside the bank. There were restriction on how much cash could be withdrawn, but that didn’t stop people from waiting in long lines to get what they could of their money.

Above example demonstrates the Bank was driven to the brink of collapse on account of outliers in global financial eco system when enormity of risk overwhelmed the robust risk management apparatus of the bank..It is a black swan event ,an out lier.Their risk distribution model which always managed random risk also failed and bank collapsed under the load of defaults.

**WHAT IS STATISTICAL MODEL AS APPLIED TO OPTIONS TRADING ?**

Statistical models such as central limit theorem normal distribution ,standard deviation etc are widely applied to market or stock returns which are random time series.Volatility of data provides the risk character as well as return potential of a stock.These tools find applications in optimization of risk and return profile of portfolios successfully.Is this idea of risk sufficient although necessary?Is assuming risk as identical components and hence predictable always valid?Is their no interaction between component parts?Is the whole going to have different character than the component parts?Are the components going to be random and independent while whole could still be build up or coupling phenomenon?

The statistical analysis helps measure the risk.Find the volatility of component parts and then extrapolate it for future component to the right of the chart.and now you know your risk in advance.The process ignores outliers.Black swans as per this model is just an insight which just make you cautious but does not protect you from being a victim.

**COMMON USAGE OF STATISTICS IN VOLATILITY BASED STOCK TRADING**

When you take a leveraged position, you are not simply speculating on the direction of the market, you also are making a market timing decision and a position on volatility. You limit how far the market can go against you before you must bail by impact of volatility.Regarding the profit/loss ratio, if price moves in our direction by two standard deviations, we take profits and initiate an opposite trade. If the market moves against us by one standard deviation, we take a loss and initiate a trade in the opposite direction. Our strategy is designed to manage our profits and losses to achieve our goals.. The goal of this strategy is that if we win, we’ll make two standard deviations. If we lose, we’ll lose one standard deviation. (A standard deviation is a statistical measure of price movement, or volatility, and virtually all charting programs offer it.)This is principle of trading systems based on statistical properties of time series data and concepts of standard deviation or volatility and edge Regarding the profit/loss ratio, if price moves in our direction by two standard deviations, we take profits and initiate an opposite trade. If the market moves against us by one standard deviation, we take a loss and initiate a trade in the opposite direction. Our strategy is designed to manage our profits and losses to achieve our goals. The initial coin flip simply gets us into the market in an objective manner. The goal of this strategy is that if we win, we’ll make two standard deviations. If we lose, we’ll lose one standard deviation. (A standard deviation is a statistical measure of price movement, or volatility, and virtually all charting programs offer it.)

**EXAMPLES OF RANDOMNESS**

1.Annual distribution of rain fall in a country over a certain period.

2.frequency distribution of road accidents for insurance purposes over a period.

3.Frequency or monthly distribution of TV sales of a company or a dealer over a year .

4.Distribution of insurance claims over a certain period.

In all above examples ,data generated are random variables having little or no correlation to either preceding or succeeding data.They represent simple time series.Therefore they have to be arranged to bring out order or a pattern having consistency and repeatability to make some kind of sense out of random time series which then can lend itself some kind of interpretation for the purposes of forecasting. The model distribution for daily sales of shop is no different from distribution of daily returns of a stock or Index as both share similar characteristics and dynamics.The distribution or the order pattern not only provides clues as to future sales or returns but more significantly reveal the amount of risk involved and implications of viability of business as this pattern is expected to repeat in future too. Time series data or random data are independent and heterogeneous just .like rolling of dice.These property of random series is called order parameter or sequencing behavior of data.

**COMMON RANDOM EVENTS**

1.Rolling of dice

2.Playing Roulette

3.Tossing of coin

**WHAT IS ORDER OR DISORDER IN OUR DAILY LIFE?**

Imagine big stores such as shopping mall where huge amount of items are delivered by trucks and which are simply dumped on the floor randomly and in a state of disorder.Subsequently Those items are placed in order in racks as per a certain strategy as may be suitable for the store.Exactly similar way random data generated can be arranged to a statistical order called Bell curve shape .This is statistical property of a random time series as per central limit theorem.When huge amount of materials are delivered in a store,store keeper then get on to arrange every thing to order to ensure good house keeping and efficient tractability. This is achieved by segregating total delivery into homogenous components and arranging them into some kind of order or sequence each time.This is how disorders are handled in day to day business of life to bring a sense of rationality ,consistency and efficiency.Similar approach is applied in statistics to handle disorderly and random data series..From market perspective which is more relevant here,returns of the stocks take the shape of bell curve over a period and this central tendency tend to repeat .Then standard deviation of the curve gives the volatility of the stock which implies the risk involved.And mean stands for expected return.In fact statistical methods lend itself to multiplicity of valuable data interpretation in terms of risk and reward in future and occupies predominant place in finance and business.Risk is quantified in advance and controlled and it ceases to surprise and traders then.Risk using statistical treatment of data can be successfully used to anticipate, plan and control which is other wise commonly known as Risk management.

**WHY PATTERN OF DISTRIBUTION IS ASSUMED TO REPEAT IN FUTURE?**

The assumption of repetition of pattern such as normal distribution also called standard deviation for returns of a stock or index and volatility is based on central limit theorem and power law.The central limit theorem was originally conceived for typical random events such as tossing of coin or rolling of dice.There is rationale behind applying principle to a time series random data such as daily returns of the stock or index.To make it simple ,one is to imagine daily returns of a stock over a week as five faces of a dice.In other words we are rolling here a five faced dice with different numbers spotted on its five faces.Well now from first principle ,central limit applies to stock returns and associated conclusions of the theorem also then apply.

**CAVEAT EMPTOR**

However the statistical models based on randomness although handy and fool proof and find wide application in option pricing and option trading also has serious pitfall .Bell curves have rare sighted sinister sister called BLACK SWANS.Some times they tend to appear frequently .They are commonly known as fat tails ,long tails or outliers and spell doom for traders.Random systems and normal distribution too fail. Goose that lays golden eggs too die ultimately.Statistical high ways with head lights and road markings also has puzzles that leads to no where or dead end.

KEEPING PERSPECTIVE WITH BLACK SWANS

Black swans are not rare but common reality.They tend to happen more than probability predictions of Black sholes.Well Gaussian curve is not exactly fallacy but should be traded with caution.Fat tails if not accounted for can destroy one’s edge.Many people tend to play because they either don’t believe or underestimate black swan.Remember Ranbaxy once tanked by over 20% or so in couple of days..

**ORGANISATIONS OF BLACK SWANS**

Organization or incidences of black swan events are different from smaller siblings Smaller siblings are heterogeneous and independent. But black swan events are product of a long build up process or coupling.which can not be predicted with statistical model of risk. These bifurcations take engineers, practitioners and students by surprise, because of the ubiquitous tendency to extrapolate new behavior from past ones. Such inferences are fundamentally mistaken at phase transitions, since the new collective organization is in general completely different from the previous one. It is also wrongly considered as unrelated.It needs different model and hypothesis to be understood.Above concept can be better understood with drawing parallel to earth quakes.

**Gutenberg-Richter law and characteristic earthquakes.**

As mentioned above, earthquakes can be thought of as relaxation events of coupled heterogeneous faults, each fault acting as a threshold oscillator of relaxation under the influence of an overall slow tectonic loading

: distribution of earthquake magnitudes . . The characteristic earthquakes would be associated with the clusters, which are visibly OUTLIERS in the chart above. Extrapolation clearly does not work at the extreme case.

**COEXISTANCE OF RANDOM AND POSITIVE FEED BACK SYSTEMS**

Tendency of mean reversal ,reversal on overbought or oversold conditions .cyclical behavior or channeling of stocks are consistent with law of averages or principle of linear regression etc.Extrapolation of behavior of random components are valid to predict behavior of whole.On the contrary ,when random components evolve into a trend or multi week rally during break outs when volatility tends to be very high. Overbought or oversold condition get extended ,so does the price.Price candles tend to tag sides of Bollinger band or standard deviation channel.This is combination stage where extrapolation does not really work.

Taking one example.Future price depends on what happened in the past and also depends on what news occurs in future.Depending on the outcome of election the market could have moved up or down.This is uncertainty element and is not there in chart.This is because past price behavior is not useful to predict the result of election.So past price behavior is not going to predict what happens to market when election results are announced..

Whenever there is little or no news markets can still move around considerably.Many market moves in absence of those characteristic binomial events ,have repeated patterns that can be exploited most of times .When there is no major news and there is plenty of speculation over future news ,past market works best in those kind of scenarios.

**CONCLUSION**

Stock market crashes are momentous financial events that are fascinating to academics and practitioners alike. According to the standard academic textbook worldview that markets are efficient, only the revelation of a dramatic piece of information can cause a crash, yet in reality even the most thorough post-mortem analyses are typically inconclusive as to what this piece of information might have been. For traders and investors, the fear of a crash is a perpetual source of stress, and the onset of the event itself always ruins the lives of some of them. Most approaches to explain crashes search for possible mechanisms or effects that operate at very short time scales (hours, days or weeks at most). Other researchers have suggested market crashes may have endogenous origins . Associated with these questions is the problem of determining if there exist qualifying signatures in the statistical properties of time series of price returns that make crashes, and more generally large losses, different from the rest of the population? . Financial markets constitute one among many other systems exhibiting a complex organization and dynamics with similar behavior. stock markets are both efficient and unpredictable. The main concepts that are needed to understand stock markets are imitation, herding, self-organized cooperativity and positive feedbacks, leading to the development of endogenous instabilities. According to this theory, local effects such as interest raises, new tax laws, new regulations and so on, invoked as the cause of the burst of a given bubble leading to a crash, are only one of the triggering factors but not the fundamental cause of the bubble collapse. It is proposed that the true origin of a bubble and of its collapse lies in the unsustainable pace of stock market price growth based on self-reinforcing over-optimistic anticipation. As a speculative bubble develops, it becomes more and more unstable and very susceptible.