Sunday, May 5, 2013

Benoît Mandelbrot's contribution to Finance

  Other mathematicians of probability like Kolmogorov may be more academic or progressive but Mandelbrot was unique he proved that mathematicians actually understand randomness, he is called by Nassim Nicholas Taleb as 'poet of randomness'. Before Nassim Taleb, Black Swans were dealt by him in a philosophical and aesthetic way.Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.In 1960s Mandelbrot presented his ideas on prices of commodity and stock prices and made a contribution on mathematics of randomness in economic theory.Mandelbrot also knew the pitfalls in Louis Bachelier's model.Mandelbrot linked randomness to geometry and made randomness a more natural science.If stock markets were Gaussian then stock market crashed would have happen once in a Billion years. Mandelbrot's randomness methods make the statistics methods look useless. 



Benoît Mandelbrot , the late Sterling Professor of Mathematical Sciences at Yale University

The first formal model for security price changes was put forward by Louis Bachelier (1900). His price difference process in essence sets out the mathematics of Brownian Motion before Einstein and Wiener rediscovered his results in 1905 and 1923 in the context of physical particles, and in particular generates a Normal (i.e. Gaussian) distribution where variance increases proportionally with time. A crucial assumption of Bachelier’s approach is that successive price changes are independent. His dissertation, which was awarded only a “mention honorable” rather than the “mention très honorable” that was essential for recognition in the academic world, remained unknown to the financial world until M.F. M. Osborne , who made no reference to Bachelier’s work, rediscovered Brownian Motion as a plausible model for security price changes.

But in 1963 the famous mathematician Mandelbrot produced a paper pointing out that the tails of security price distributions are far fatter than those of normal distributions (what he called the “Noah effect” in reference to the deluge in biblical times) and recommending instead a class of independent and identically distributed “alpha-stable” Paretian distributions with infinite variance. Towards the end of the paper Mandelbrot observes that the independence assumption in his suggested model does not fully reflect reality in that “on closer inspection … large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes.” Mandelbrot later called this the “Joseph effect” in reference to the biblical account of seven years of plentiful harvests in Egypt followed by seven years of famine. Such a sequence of events would have had an exceptionally low probability of taking place if harvest yields in successive years were independent. While considering how best to model this dependence effect, Mandelbrot came across the work of Hurst (1951, 1955) which dealt with a very strong dependence in natural events such as river flows (particularly in the case of the Nile) from one year to another and developed the Hurst exponent H as a robust statistical measure of dependence. Mandelbrot’s new model of Fractional Brownian Motion, which is described in detail in Mandelbrot & van Ness (1968), is defined by an equation which incorporates the Hurst exponent H. Many financial economists, particularly Cootner (1964), were highly critical of Mandelbrot’s work, mainly because – if he was correct about normal distributions being seriously inconsistent with reality – most of their earlier statistical work, particularly in tests of the Capital Asset Pricing Model and the Efficient Market Hypothesis, would be invalid. Indeed, in his seminal review work on stockmarket efficiency, Fama (1970) describes how non-normal stable distributions of precisely the type advocated by Mandelbrot are more realistic than standard distributions .

Partly because of estimation problems with alpha-stable Paretian distributions and the mathematical complexity of Fractional Brownian Motion, and partly because of the conclusion in  Andrew Lo's work (1991) that standard distributions might give an adequate representation of reality, Mandelbrot’s two suggested new models failed to make a major impact on finance theory, and he essentially left the financial scene to pursue other interests such as fractal geometry. However, in his “Fractal Geometry of Nature”, Mandelbrot (1982) commented on what he regarded as the “suicidal” statistical methodologies that were standard in finance theory: “Faced with a statistical test that rejects the Brownian hypothesis that price changes are Gaussian, the economist can try one modification after another until the test is fooled. A popular fix is censorship, hypocritically called ‘rejection of outliers’. One distinguishes the ordinary ‘small’ price changes from the large changes that defeat Alexander’s filters. The former are viewed as random and Gaussian, and treasures of ingenuity are devoted to them .The latter are handled separately, as ‘nonstochastic’."

Shortly after the “Noah effect” manifested itself with extreme severity in the collapse of Long-Term Capital Management, Mandelbrot (1999) produced a brief article, the cover story of the February 1999 issue of “Scientific American”, in which he used nautical analogies to highlight the foolhardy nature of standard risk models that assumed independent normal distributions. He also pointed out that a more realistic depiction of market fluctuations, namely Fractional Brownian Motion in multifractal trading time, already existed.

Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness. Mandelbrot designed the mathematical object called "Mandelbrot set" and later worked on shapes and fractals of maths and also worked on Chaos Theory. These objects play an important role on aesthetics , music , architecture , poetry , gestures and tones are derived from fractals . Mandelbrot's book "Fractal Geometry of Nature" it made a fame in arts , visual arts and every artistic circle. Many artists used to call Mandelbrot "The Rock Star of Mathematics". Mandelbrot became famous because of the number of applications of mathematics in our society.



Mandelbrot used his fractal theory to explain the presence of extreme events in Wall Street.


In fact he was one of the pioneers in studying the variation of financial prices even before Bchelier's Brownian model became widely accepted in academia and Mandelbrot also knew the pitfalls in Bachelier's model.For this reason many call him as the "father of Quantitative Finance".Mandelbrot has been best known since the early 1960s as one of the pioneers in studying the variation of financial prices.He pointed out that two features of Bachelier's model are unacceptable (in 1960s when Bachelier's model got accepted by academia and financial world). These flaws were based on power-law distributions and so Mandelbrot scaled these both by fractal theory and thus correcting the errors and flaws.Since then scaling by use of fractal theory has become important in finance and as well as in Physics.In fact Nassim Nicholas Taleb's "Black Swan Theory" is inspired by work of Mandelbrot as Mandelbrot was much concerned about high-risk rare events (Black Swans).Nassim and Mandelbrot collaborated in  research projects related to risk and randomness.



Mandelbrot's contribution in finance fall into three main stages:

He was the first to stress the essential importance, even in a first approximation, of large variations that may occur as sudden price discontinuities. The Brownian model is unjustified in neglecting them. They are not “outliers” one can safely disregard or study separately. To the contrary, their distribution is much more important than that of the "background noise" constituted by the small changes of Brownian motion. He followed this critique in by showing in 1963 that the big discontinuities and the small "noise" fall on a single power-law distribution and represented them by a scenario based on Levy stable distributions. He and Taylor introduced in 1967 the new notion of intrinsic "trading time." In recent years, fractal trading time and his 1963 model have gained wide acceptance.

Secondly, Mandelbrot tackled the fact that the “background noise” of small price changes is of variable “volatility.” This feature was ordinarily viewed as a symptom of non-stationality that must be studied separately. To the contrary, Mandelbrot interpreted this variability as indicating that price changes are far from being statistically independent. In fact, for all practical purposes, their interdependence should be viewed as continuing to an infinitely long term. In particular, it is not limited to the short term that is studied by Markov processes and more recently ARCH and its variants. In fact, it too follows a power-law side of dependence. He followed this critique and illustrated long-dependence by introducing in 1965  a process called fractional Brownian motion which has become very widely used.

Thirdly, he introduced the new notion of multifractality that combines long power-law tails and long power-law dependence. Early on, his work was motivated by the context of turbulence, but he immediately observed and pointed out that in 1972 the same ideas also apply to finance. After a long hiatus while he was developing other aspects of fractal geometry, he returned to finance in the mid-1990s and developed the multifractal scenario theory in detail in his 1997 book "Fractals and Scaling in Finance". The concept of scaling invariance used by Mandelbrot started by being perceived as suspect, because at that time other fields did not use it. However the period after 1972 also saw the growth of a new subfield of statistical physics concerned with “criticality.” The concepts used in that field are similar to those Mandelbrot had been using in finance.In "The Misbehavior of Markets", another popular book by Mandelbrot,he argues that the Gaussian models for financial risk used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk.


Sunday, February 24, 2013

The 2008 Financial Crisis.


The 2008 Financial Crisis.

This article is compiled from Rand Corporation article, articles from Wall Street Journal and my own research.


The crisis was faced in 2007-2008 and it occurred in U.S and with effects all over the world , still in 2013 we have not recovered. Since the Great depression of U.S the Government and Financial industry of U.S boomed consistently for about 40 years, till 1980s.In 1980s U.S had few big companies who controlled a lot of capital and in the term of Bill Clinton regulations were relieved to advantage of the big companies and investors so risking society's safety and benefits. Later in 1990s the Financial Services Authority in Washington relieved laws for big investment banks i.e UBS, Goldman Sachs and Lehman Brothers which is a criminal act in a legal terms.

Citi Bank was financing the Iran's weapon supply and also took part in sales of Mexico's Cocaine market for high profits. The rich investors started to invest in U.S because of lack of proper regulations and high interest rates. Government made laws lenient to attract investors from all over the world and thus increasing the systematic risk in U.S Financial market. Then came the era of hedge funds and derivatives, derivatives is the riskiest financial instrument. Hedging techniques and the models for derivatives were being practiced in almost all major banks .The models were mathematical and complex, even the use of High Frequency Computing in trading floor was practiced. Graduates with PhD in Mathematics , Physics or Computer Science were being hired to handle these complex derivatives. Many Noble Prize winners started their own Hedge Funds and attracted big investors to make profits, the models were so complex that only a few can grasp the understanding. Derivatives traders were using mathematical models to manage the derivatives and securities. These quants were playing with high risk/high return financial derivatives and were at that time confident of the models being used by them.

Nouriel Roubini, the first economist to warn against the crisis and is against the use of derivatives.

 The other big cause was the imprudent mortgage lending, as house prices boomed in later 1990s many people were unable to afford homes and so they required home loans and Government facilitated them by home loan scheme. Many banks started giving loans to these people and later investment banks came to give mortgage loans. These big banks preferred sub-prime loans for higher interest rates. Sub-prime loans increased 10 times from 1996-2006 , Lehman Brothers were biggest gainers from such mortgage loans. In fact  mortgage loans are a sort of derivative which is traded in financial system and it is prone to both systematic and default risk. Rating agencies started to give even the financially weaker companies a AAA credit rating so they can attract mortgage loans customers. So sub-prime loans were packed as AAA bonds. Credit rating agencies gave AAA ratings to numerous issues of sub-prime mortgage-backed securities .

The big companies and investment banks who controlled the most of world's money were involved in such derivatives and mortgage loans. Risk management was poorly handled by companies and firms separated analysis of market risk and credit risk This division did not work for complex structured products like derivatives. Complexity of financial instruments at the heart of crisis had two effects: 1. investors were unable to make independent judgements on merits of investments, 2.regulations were baffled.  Investors do not always make optimal choices as they suffer from bounded rationality and limited self-control and even regulators did not help them to manage complexity through better disclosure.

Quants ruled Wall Street in 1990s and 2000s

The computer models used were bad , expectations of the performance of complex structured products linked to mortgages were based on only a few decades worth of data. In the case of sub-prime loans. As complex instruments like derivatives were new and so not much data was available as well for forecasting.Low interest rates and abundant capital forced investors to borrow funds to boost the return of their capital, excessive leverage magnified the impact of housing downturn. Excessive leverage leads to mispricing of risk and credit bubble. Even the regulation laws were relaxed for excessive leverage. New laws in 1990s and 2000s i.e. GLBA and the CFMA permitted financial institutions to engage in unregulated risky transactions on vast scale. Over the counter derivatives which are largely unregulated  began to get exercised extensively. CDOs ,an investment-grade security backed by a pool of bonds, loans and other assets. CDOs do not specialize in one type of debt but are often non-mortgage loans or bonds. CDOs are sort of derivatives used by banks for profits , they are actually derivative based loans. The last cause was use of Gaussian curve for risky and low probability instruments in financial industry i.e if you use Bell curve in top stocks,derivatives and genetic measures, extreme events if calculated by Bell curve may cause disaster. Head -tail on a coin is a random walk (left or right or win or loose) is a mediocre event so we use Gaussian curve.This idea was popularized by Nassim Nicholas Taleb who stated this as "Black Swan theory". He suggested that markets are non-linear , random and highly complex and chaotic and so the events with low probability and high impact i.e Black Swan events are not calculated by Gaussian curve.These events are outliers with massive impact and economic advisers were not aware of this and so were unable to forecast these events.

Hedge funds did not contribute to the crisis itself but they contributed to high systematic risk (though sysmematic risk is known as risk caused by randomness in financial parameters i.e interest rates, inflation but in generic term it is the possibility of failure of whole financial system i.e a Crisis or a Black Swan event) in financial system.Many top financial engineers attracted investors and spent millions of dollars for high return, but due to the volume of hedge funds in terms of dollar value these caused highest systematic risk  in financial system as compared to any other financial instrument. So, when system fails due to causes like mortgage lending , hedge funds increase the effect of the damage i.e. hedge fund act as a catalyst when crisis happens but do not itself contribute to crisis. Hedge funds make market parameters like interest rates and inflation go random due to their high return behavior, also weak regulatory laws for hedge funds made them even more riskier financial instrument. Hedge fund managers were greedy and traded in high risk/high return funds and thus making financial system more complex and more prone to crisis.John Paulson made billions in trading complex derivatives and mortgage loans.Many people blamed hedge funds for crisis but that is not true , they only increased the damage of crisis because of their systematic risk but never were the cause of crisis.

No regulator had comprehensive jurisdiction over all systemically important financial instructions, the Federal Reserve Bank had role of systematic risk regulator by default but lacked authority to oversee investment banks , hedge funds, nonbank derivative dealers. Even the rating agencies did the worst by giving good rating to risky securities and borrowers, AAA rated firms increased from few hundred to thousands in few years(early 2000s). All in all this crisis was the biggest the world has seen so far in terms of damage, many academics and economic advisors like Nouriel Roubini ,Peter Schiff, Raghuram Rajan , Nassim Nicholas Taleb, Kenneth Rogoff, George Soros,Robert J. Shiller and Andrew Lo warned against crisis and wrote extensively against the deregulation of Financial acts and excessive use of derivatives. At the same time there were advisers like Larry Summers and Frederic Mishkin favored deregulation of financial system and favored use of derivatives and risky financial instruments and even were advisers to such instruments and hedge funds for their own profits. 

Henry Paulson

 Henry Paulson the U.S Treasury Governor along with Ben Bernanke were unable to understand or predict this crisis and gave his precautions and new regulations after four month of crisis. Many CEOs of large companies including Lehman Brothers, Goldman Sachs  went away with billions of dollars inside their pocket after damaging the whole financial system, the U.S Government and the world. Many other Harvard economists and other top university economists made millions by consulting to firms and promoting the deregulation of the financial system. This also contributed to the corruption of economics studies in universities and business schools.

Larry Summers

Computational models need to be more precise and forecasting should be done by taking sufficient amount of data, and if data is not available i.e in case of derivatives then unreliable forecasting should not be made and even the financial instrument with high risk and low data should be banned from financial system.Black-Scholes equation has no flaw,though many went against it and it should be used within its underlying assumptions and limitations.Mathematical models do give accurate results and a bit of error is always involved and only more research can decrease it but the use of Gaussian curve in calculation of complex derivatives and other financial instruments should be banned and other relevant models and distributions should be used(this is a new research field for Probability Theory experts i.e to advance the work of Benoit Mandelbrot). Oliver Blanchard, the chief economist of IMF stated that the Crisis will last for a decade and argues that for supply-side reasons(after the Crisis), government spending should go towards financial infrastructure projects, not cash handouts.

Oliver Blanchard