Tuesday, March 15, 2011

Mathematical Models & Future Predictions 3

Here is the last installment on our series of the proper use of mathematical models.

Since in most scientific circles, the limitations of mathematical models are rarely discussed, we attempt to balance the picture.  To many these mathematical models are magical "black boxes" that can predict future events with dead certainty.  But to those who design them, the limitations are legion.

The Failure of Logic:

A model that considered strange attractors:

Economic Black Swan Effect: One can trace the history of the 2007-2008 Market meltdown and the fall of Wall Street giants like Lehman Brothers, Bear Sterns and Washington Mutual, to two Nobel prize winners Robert Merton and Myron Scholes with Fischer Black propounded the theory of the Black-Scholes Option-pricing model. The model assumed that the price of heavily traded assets follow a geometric Brownian motion with constant drift and limited volatility. In relation to stock option, the model incorporates the constant price variation of the stock, the time value of money, the option's strike price and the time to the option's expiry.

The Black-Scholes option theory became the darling of Wall Street. It created derivatives and the CBOE (Chicago Board of Option Exchange) enjoyed a tremendous rise in contracts and therefore revenues. In 1973 the Exchange traded 911 contracts and in 2007 the volume of contracts reached one trillion.  

If you cannot see the embedded video here is the link: http://bit.ly/7Itit7.

Myron Scholes and Robert C. Merton started the now infamous LTCM (Long Term Capital Management) Hedge fund The fund was based on absolute-return trading strategies that included fixed income arbitrage using convergence trades exploiting asymmetries of the US, Japanese and European government bond prices, combined with high leverage. The latter strategy exposed the fund management to a 25 to 1 debt to equity ratio in its later stages. The meteoric monetary gains of 40% in 12 months using this exploitation lasted until the East Asian Financial Crises of 1997 followed by the Salomon Brothers withdrawal from arbitrage and the Russian Financial Crises in 1998 sealed the insolvency of the mathematically modeled LTCM Hedge Fund. Investors ran from the Japanese and European Bonds to the US Treasuries for safety and in so doing erased and reversed the asymmetries against the proposed models used by the LTCM managers leading to huge losses in less than 4 months.

Yet there are those who still think that this method is viable.  Fixed income abritrage works by taking advantages of financial disasters.  This is a full explanation by Bob Treue who is still following this model.  If you cannot see the embedded video here is the link: http://youtu.be/yvHG1suYDKY.

"Reliance on models based on incorrect axioms has clear and large effects."
Jean-Philippe Bouchard

Keeping things in context, the Black-Scholes model was a simplified model using laws of thermodynamics to equate to the changes in the price of shares. The gentle trending above and below the mean was the premise of the stock movement rather than the violent seesaw effect that happens in the real world. The concept of slow movements of share prices in short term trading strategies was given credence to behavior within the model rather then the real world events that can whipsaw and rubberneck in a trending market.

The computers with buying and selling software programs that can hit the market at record speed can create havoc with the share prices and are predetermined by programs created by the quantitative financial analysts or quants. The market is a living, breathing mechanism based on the desires expressed singly by individuals or in large packets by the computing devices and governed ultimately by humans. This was not parameter used in the Black-Scholes model.

In keeping with the economic modeling methods the quants then decided to use the derivatives market to develop products like the CDOs or Collateralized Debt Obligation notes, the CDSs or Credit Default Swaps bundled securities. The concept was simple, take a large number of mortgages of different risks, package them together and securitize them. The large bundled mortgages would be rated by an agency that was paid by the CDO creators and everyone would be content; the investor, comforted by the rating, the loan originator by the diversity of the portfolio and the seller by the pricing power of the product. The mortgage payments were allocated to the triple “A” investors first and then the rest. The equity investor in cases of default was left holding the useless IOU. These large caches of securitized mortgages made the banks and lending agencies hustle more, lending to individuals that they would not have touched in prior years due to financial risk, these individuals were now welcomed. These individuals included people who could not afford to pay the mortgages, or that did not hold any assets as collateral to the loans.

In their paper published in August 2010 The Failure of Models that Predict Failure: Distance, Incentives and Defaults, Authors: Uday Rajan, Amit Seru, Vikrant Vig, state that, 
As the level of securitization increases, lenders have an incentive to originate loans that rate high based on characteristics that are reported to investors, even if other unreported variables imply a lower borrower quality…To illustrate this effect, we show that a statistical default model estimated in a low securitization period breaks down in a high securitization period in a systematic manner: it under-predicts defaults among borrowers for whom soft information is more valuable. Regulations that rely on such models to assess default risk may therefore be undermined by the actions of market participants.
The house of cards fell, almost swallowed a country and plunged the world into a fiscal abyss.

The banks and Mortgage lending agencies, looking for a quick profit and unaware individuals looking for a picket fenced home, all participated in the “perfect storm.” The credit default market took advantage of desire and at one point was estimated at $55 trillion!

Using mathematical modeling has its benefits of dealing with an idea, however in a complex world where the variables are in the millions if not billions the chance of creating a model to satisfy all potential adversities is zero. In economics especially, real life and mathematic modeling are decoupled. Benoit Mandelbrot calculated that IF the Dow Jones Average followed a normal distribution it would have moved 3.4% on 58 days between 1916 and 2003. In fact it moved 1001 times. He also calculated that the market should have moved 4.5% on six days however in reality it moved 366 times. It should have moved 7% once in 300,000 years. The reality is in the 20th century it moved 48 times. In all the market had 25-standard-deviations several days in a row. The market sometimes moves in the extreme tail of its normal distribution curve. A perfect example of the tail wagging the dog!

So are Mathematical Models cursed to eventually fail? The simple answer is no. They work well in design details of the components, assemblage and when environs are taken into account. However when large-scale “real world” scenarios are attempted without complete relevant information, the asymmetries that remain unaccounted for in the modeling concept are not validated until the “real world” tests the product in its own environment.

Alas the hope of perfection will remain a miniscule too short. The emergence of chaos from Edward Lorenz’s minute strange attractor will always be a suspect for all experiments or real world experiences that go “clunk” in the night.

Co-Written by JediMedicine & PlusUltraTech


Anonymous said...

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Monica Anderson said...

Great article. You all might like my graph at http://artificial-intuition.com/possible.html that discusses the choice between Model Based (Reductionist) methods that provide long term reliable predictions in simple domains and Model Free Methods ("Intuition based methods") that provide reasonable short term predictions in complex domains. Each will outperform the other in some kinds of problem domains. In some domains we can use either, and in others ("the Absurd Region") neither will work. The entire website (about 6 pages) discusses the choice and tradeoff between Logic Based and Intuition Based methods for Artificial Intelligence. I didn't use the words "Reductionism" and "Holism" on that site on the advice of my focus group ("They juts confuse people") but I have since then (on my blog and in my videos) used those terms with abandon.