(Dt. Sunday, May 04, 2008)

Update 2: A Marginalistic Interpretation of the GARCH model

In this second update to my research proposal, I would like to discuss a marginalistic interpretation of the GARCH model for estimating volatility. I had originally sent out my research proposal, 'A New Perspective on the Role of Markets in an Economy', in the third week of March 2008 (a copy of this proposal is attached to this e-mail). In my research proposal, I had mooted the idea of considering the duration of interaction between the buyer and the seller in a market, especially when the value, frequency and volume of trading are high. I had also sent out an initial update to my research proposal in the second week of April 2008 (a copy of this update is also attached to this e-mail). In this initial update, I had focused completely on the ongoing housing mortgage crisis, especially on four issues which were not given adequate coverage in the media. In this second update that I am now sending out, I would like to first clarify certain aspects of the discussion between Professor Robert Engle and Professor Joseph Stiglitz on April 25, 2008 during the Squawk Box program on CNBC television channel. Specifically, I would like to address two fundamental issues raised by Professor Stiglitz in that discussion, about the role of the mathematical models that Wall Street uses for estimating risk. After this clarification, I would like to explain how my idea for studying the duration of interaction between the buyer and the seller fits in with what Professors Engle and Stiglitz were saying.This second update is organized in three sections: I. Volatility Models, II. Professor Joseph Stiglitz's Observations on Volatility Models, III. Duration of Interaction.

I. Volatility Models

When one tries to understand the role in modern economics played by mathematical models for estimating time-varying volatility, it is important to realize that some of these models, like GARCH, provide an elegant and useful conceptual synthesis of several different streams of economic thought. Firstly, empirical observation indicates reliably that in the trading of an asset in the market, a sustained increase in the asset price is directly related to periods of reduced volatility, and conversely, periods of great volatility are directly associated with significant drops in the asset price. This empirically supported knowledge helps to capture the behavioral psychology of the individual players in the market in a quantitative manner. Now, other things being equal, the inflation-adjusted value of an income producing asset is expected to increase as time goes by, due to several inter-related factors like technological advancement, productivity gains, time value of money, increased return of profits commensurate with the systematic risk, welfare-enhancing trade, division of labor, overall growth in the economy, etc. However, if the individual players are worried about certain information that is unfavorable (in the short run) to the ownership of the asset, they would take actions which in the aggregate would reflect their anxieties as increased volatility, and also lead to a sustained period of reduction in the asset price (in the short run). Similarly, if the players in the market felt confident and certain that the short-term future would influence an income producing asset either favorably or not at all, then the price of this asset would continue to go up (for the reasons mentioned above) over the next few weeks or so. In this way, the volatility models provide a good approximation for the influence of information on the price of an asset, an influence which works its way indirectly through the psychology of the individual players in the market.

Secondly, these volatility models incorporate the Efficient Market Hypothesis, a major stream of economic thought in the 20th century. Now, if one analyzes further, the role of volatility models in predicting asset prices, the question naturally arises as to how exactly does information influence asset prices. The Efficient Market Hypothesis specifies, among other criteria, that the market should not entertain sustained and predictable arbitrage opportunities. Clearly, for this property of the market to hold, it would have to be the case that whatever information that first came to light in the past, and was relevant to the value of the asset being traded, was not hoarded by only a few players in the market, but was available to all the market participants without delay. Moreover, the assumption that markets functioned efficiently also implies that each of the individual players, once given the relevant information, are capable of rationally analyzing its influence on the asset and proceed to initiate buying or selling of the asset, which would lead to changes in the price of the asset accordingly. Thus, in an efficient market, at any given point of time, the current value of the asset has already incorporated all past information that is relevant to the asset, and any imminent change of the asset price at that point of time only depends on any new information that has a bearing on the asset's value. The mechanism by which volatility models incorporate the Efficient Market Hypothesis is called a Time Series. Time Series is a mathematical construction that is useful for recording the state of the market at various moments of time, in chronological order. The state of the market can be described by different variables whose values may depend on the elapsed time. In particular, the volatility at a given instant can be modeled to depend on the volatility at previous instances.

Thirdly, these volatility models can be fine-tuned to simulate the market's adherence to the theory of Marginal Utility. As an aside, I note that Marginal Utility Theory is the crown jewel among the achievements of 19th century economics. I have explained in my previous update how Marginal Utility Theory enables the market mechanism to retain its relevance during periods of great social and technological change by the creation and destruction of wealth at breathtaking pace. This theory values a class of similar assets by asking what the utility of one additional unit of the asset would be. Thus this theory focuses relentlessly on the present -- supply and demand determine the price of an asset instantaneously. The volatility models simulate this marginalistic aspect of asset valuation by specifying that the influence of past volatilities on current volatility is weighted in such a way that the influence decays quickly with time (at exponential rates). In fact, once the rate of decay is specified (to be exponential), then statistical techniques like maximal likelihood estimation can be used to find the weight coefficients that fit the data best.

Fourthly, volatility models possess great predictive power largely due to some major developments of the 20th century like statistical techniques, powerful computing systems, real-time news, and the wide availability of split-second economic data from reliable sources. But, there is an additional factor involved here. The ability of volatility models to predict future market behavior is also greatly influenced by the very nature of economic information and the news mechanisms that currently exists for transmitting this information to the public. We discuss this last point in detail later. Now, as an illustrative example, let us consider the fact that the average of prices of future contracts on the trading of a certain commodity can be used as an estimator of the future spot price. That is, the (informed) opinions of those market participants who have entered into future contracts is taken as an indicator of the price of the commodity in the future. It may be argued that this method also allows new information to work its way into the prices of assets through the behavioral psychology of individual players in the market. However, without further enhancements, this method is rather crude and unrefined, when compared with the volatility models. The volatility models can take into account massive amounts of data, are more computation intensive, and they provide a much more sophisticated mathematical framework for capturing the influence of new information on the price of an asset.

The obvious advantage of mathematical sophistication is that it allows for inferences and decisions to be made on a mathematical basis, minimizing the effects of any collusion or blind belief in ideology and fantasy among the market participants. Thus the volatility models help to track the behavior of the individual players, as well as to influence their behavior in such a way that they exhibit rational and fact-based responses to new information. There is another more subtle advantage to employing mathematical sophistication, as I will explain now. Mathematical sophistication, especially when buttressed with the ability of computers to analyze massive amounts of data, provides such a comprehensive grip on current reality, in such a logically sound manner that it allows for one to focus totally on new information. However, economic information, by its very nature arrives, not all at once, but in installments, through channels that themselves take time to digest and interpret the information. To mention recent examples, the housing mortgage crisis, the collapse of Bear Stearns, the impact of rising oil prices, the world-wide food shortage, global warming, slowdown in economic growth and rise of unemployment, unprecedented levels of income inequality, economic and political globalization, huge US trade deficits, rise of China and US foreign debt are all phenomena whose impact on the American economy have been unfolding on the public consciousness over long periods ranging from a few weeks or several months to a decade. In fact, one may say, almost surely, that at any given point of time, a significant portion of information on a certain macroeconomic phenomenon has already arrived. It is this fact that makes the volatility models so much more powerful in predicting the future behavior of the markets. If these models were just abstract theoretical contraptions, they would not have so much credibility and effectiveness as they do now. But, since economic news, especially new information on macroeconomic phenomena, unfolds in installments, these volatility models are able to adjust dynamically to any crisis that is currently in the process of coming to light. Thus the predictions they provide on future prices retain a lot of relevance since these predictions are not just statistical guesses or averages of public opinion or abstract logical inferences. They are all of these, but they are also rooted in the reality of recent events. It is mainly in this respect that Professor Engle and Professor Stiglitz seemed to interpret the market mechanism differently in their discussion on CNBC.

II. Professor Joseph Stiglitz's observations on the volatility models

Professor Stiglitz referred to two issues in answering the question why volatility models did not work effectively with collateralized debt obligations (CDOs) and why they could not prevent the current mortgage crisis: (1) business school graduates who are employed on Wall Street thought that they were creating totally new financial instruments that were going to change the world, but they were actually using data on the world from before they had changed it. This created a logical inconsistency in what they were doing. (2) the investment firms on Wall Street were all using similar models. In his response, Professor Engle said that the main driver of volatility and correlation is the news, not the way the financial instruments are used. Moreover, he felt that the main reason for the current crisis is macroeconomic uncertainty, and in any case, the economy is not going to do too badly but would soon recover as people become more comfortable with the future. The transcripts of the 14-minute discussion on CNBC is attached to this e-mail, and it might be useful for the reader to look at the transcripts, and read at least the first page, at this point. The video recording is also available at CNBC's website (search for Stiglitz or Engle at http://www.cnbc.com).

There seems to be a fundamental dichotomy regarding what constitutes new information in a market. On the one hand, in order to adhere to the Efficient Market Hypothesis, one needs to think that the market framework consists only of those entities that are actively participating in the trading. For example, one may assume that the Wall Street investment firms, the Securities & Exchange Commission (SEC), the pension funds, the hedge funds and other financial institutions, the brokers, the stock exchange and all the Wall Street boutique firms that provide the supporting infrastructure like software, accounting, etc, would constitute the financial markets. In this view, the house-owner or the neighborhood bank that initiated the mortgage contract would not be considered to be part of the market, neither would be a government employee in Australia whose pension fund has purchased mortgage securities in America. Thus any information about foreclosures or defaults on mortgages or the news about Fed's rate cuts or the economy's performance or the earnings announcements of publicly traded companies are all assumed to come from outside the market framework and would be treated as new information. This appears to be the perspective of Professor Engle.

On the other hand, in the modern world, any developed country is a shareholder nation, and the functioning of the markets is at least of fiduciary interest to every citizen. Moreover, at a time of rapid globalization under the aegis of market capitalism, the scale and applicability of the market mechanism is being vastly expanded. It would be quite short-sighted to specify a definition of the market that enforces strict boundaries and excludes large sections of the population. Note that if one adopts this expanded view of the market, then the Efficient Market Hypothesis almost certainly would not hold, because most participants would only spend part of their time to follow the happenings in the market. This would imply that new information does not spread without delay and does not affect prices instantly. In addition, this viewpoint would obligate the financial institutions to actively engage in finding out accurate data on the prevalence of foreclosures, say, if these firms were trading in mortgage securities, and keep themselves up-to-date on a daily basis. This appears to be the perspective of Professor Joseph Stiglitz.

As it happened, it was not the Wall Street investment firms but two academics, Professor Robert Shiller and his former doctoral student Dr. Andrew Case, who set up the Case - Shiller Housing Price Index. At a time when computers are ubiquitous and the income of every household is well-tabulated, it is quite laughable that creditworthiness for mortgages is being determined by a catch-phrase like 'sub-prime'. The definition of sub-prime borrower as one who has not paid the 20% down payment on his/her house has no other logical basis than that sticking to this definition would make it possible to sell the mortgage to Fannie Mae and Freddie Mac, because these government institutions would not deal in mortgages that don't have the 20% down payment. For a whole year, the media, presumably under the influence of the major players in the market, has bandied sub-prime borrowing as the reason for the current housing mortgage crisis. Another example is the rating of mortgage securities. With the assignment of ratings like 'AAA' to mortgage securities, the range of ratings is quite restricted, and the whole rating system simply functioned like an 'old boys club'. Even credit card companies provide a credit score, a numerical score ranging from 300 to 900, which can help the consumer keep track of his/her credit worthiness in a somewhat more continuous manner. In contrast, with the crude rating system for mortgage securities, significant pressure would be brought to ensure 'stickiness' of ratings. For instance, nobody would want their 'AAA' tranches downgraded at all, and one could have expected in advance that there would be disproportionate and undue influence to coerce the rating agencies to keep giving out high ratings. The mortgage crisis is going on for nearly 15 months now, and the financial institutions are taking a long time to get their act together. It seems fair to say that Wall Street investment firms hardly spent any resources on their own to keep track of the mortgage situation around the country, even though they are engaging in trades worth billions of dollars on these mortgage securities.

Thus when Professor Engle says the fact that the Wall Street firms were all using similar models is not the crucial factor, he means that the theory governing volatility models was derived on a sound logical basis and it had synthesized several streams of economic thought, as explained in Section I. How could the fact that all the Wall Street firms were using similar volatility models be problematic? After all if ten different people said the same truth, does it invalidate the truth? Whereas Professor Stiglitz points out that the models would have worked differently for the Wall Street firms, if the firms had each ventured out to gather real-time data on the mortgage situation as it existed in the real economy (on Main Street). In that case, the firms would have had different estimates for the probability of defaults on mortgage loans, but those estimates would have all been much more realistic. Moreover, if the firms had done their own individual assessment of the mortgage situation, instead of exhibiting herd behavior, presumably they would not have had to take write-downs in their balance sheets running in tens of billions of dollars all at the same time. One may note that such massive write-downs of the investment banks which accumulated to over 100 billion dollars during the few weeks at the end of the first quarter to the beginning of the second quarter in 2008 led to serious concerns about economy-wide credit freeze and paralysis of the day-to-day functioning of the real economy .

As for the first issue raised by Professor Stiglitz, the business school graduates took the narrow view for the definition of the market. Hence, they believed that while working from within the center of the market, namely Wall Street, they were devising totally new financial instruments that were going to change the outside world. In particular, they didn't need to take stock of the world 'outside' the markets, as they understood the term 'market' to be. Thus they believed that they were bringing some sort of a new financial revolution to the outside world. However, the prediction of price movements that their volatility models foretold depended on the data on past volatilities which in turn directly depended on the way that the market participants had reacted to the news flowing in from the 'outside world'. The end result is that the fundamental question is not whether Wall Street has been creating a new financial order for the 21st century. The most pressing question has a much more modest scale -- are the mathematical models used on Wall Street able to incorporate information efficiently, without bias and without delay?

III. Duration of Interaction

Is the recent fire-sale of Bear Stearns, a Wall Street investment bank, the modern re-enactment of the tribal ritual of human sacrifice from ancient times? Stricken with fear, panic and paranoia, did the Wall Street firms decide to sacrifice one of their own to propitiate Mammon, the false God? I must ask the reader to excuse me for making an analogy that is so bloody and ugly. But, I must point out that, in recent decades, anthropologists have come to associate deep significance to the tribal ritual of human sacrifice as a clue to understanding ancient cultures. My own argument here is that the sale of Bear Stearns illustrates that the only instruments that were available to the financial community to ward off the mortgage crisis, if only temporarily, were so blunt and crude, that comparisons with barbaric rituals of times past are inevitable. Is it a coincidence that only a few weeks later, the financial newspapers and TV Channels are proclaiming business-as-usual, buoyed by the release of less-than-dismal growth and unemployment figures for the first quarter of 2008? Why would the necessity for waving magic wands, such as the fire-sale of an 80-year old financial institution, arise if the sophisticated mathematical models that Wall Street used were behaving as they were supposed to?

Perhaps in the final analysis, Professor Engle is right that the American economy is not going to do too badly this year, and that the financial community would learn from their mistakes in the recent crisis and recover in the near future to perform successfully. In fact, if one goes with the narrow definition of the market as explained in Section II, then recently available empirical evidence already indicates that this is true. However, one might insist that the market mechanism needs to evolve so that it can be applied to solve much bigger problems in a rapidly globalizing world. One could note that there are problems like the environment or housing that are of a much larger scale than the market mechanism has successfully handled so far. After all, in his recent book, 'Making Globalization Work', Professor Stiglitz argues quite convincingly that the existing legal and economic institutional framework in capitalist countries, when applied on the international scale, have led to dire consequences. For example, the poorest region in the world, sub-Saharan Africa, has been worse off as a result of liberalization of international trade. If one takes this latter view that the market needs to address larger challenges, then the market mechanism, as understood currently, has glaring shortcomings.

The example in Section I that compared the relative performance of future contracts versus volatility models for predicting asset prices showed that the volatility models facilitate the influence of information on prices much more quickly and efficiently. However, if the value, frequency and volume of trading are very high, then the speed with which volatility models incorporate information is nowhere near the levels required for the stable functioning of the markets. Financial markets began to trade intensely in housing mortgages in this decade. The scale of this adventure is in the trillions of dollars. This is an order of magnitude larger than the earlier trading of company stocks and treasury securities. Over the last one year, one could have clearly noted how this enlarged role for the financial markets brought untoward influence on the media. The mortgage crisis is going on for nearly 15 months now. It is simply amazing that the perspective of the house-owner is hardly being addressed in the media. The house-owner has been portrayed repeatedly as a gullible ignoramus who got smooth-talked into buying complicated financial instruments. This has resulted in a situation where most of the relief that the government authorities have announced so far have gone largely to extricate the financial institutions from the mess they got themselves into. In fact, the house-owner has a more legitimate case because both the parties that initiated the mortgage contract, the house-owner and the neighborhood bank, are equally liable for the appraisal of the value of the house at the time of purchase of the mortgage. However, the neighborhood bank is long out of the picture, having sold off the mortgage to other parties, but the house-owner is still being held to the value of the mortgage.

In sticking to a narrow definition of the markets, the volatility models adjust themselves only according to the new information that is being delivered in the News channels, because this is the channel by which market participants get the information and proceed to influence the asset price according to the effect the information has on their individual behavioral psychology. If those News channels are unduly influenced to favor one section of the population, as mentioned above, then these models would go along with it. Thus the mathematical models on Wall Street do not incorporate new information without bias. Also, there is plenty of delay in these models incorporating the information that is relevant to the assets that they trade. By enforcing a strict boundary for the notion of a market, the financial institutions had freed themselves of the obligation to directly collect data on the securities that they were trading. For a small sized market, this might have worked fine. But when the value of trade is in the billions of dollars, it is simply foolish to wait, as long as a year, for the information to arrive through the News channels.

Now, one might wonder why the financial institutions on Wall Street chose to accept such delays and bias. Perhaps the answer is that having subscribed to the philosophy that markets should exchange objects instantaneously, they were forced to accept the fact that trades worth billions of dollars were going on at an instant without careful consideration as to the risks involved. So the financial firms felt that they needed to invest all of their resources and all of their attention on their trading activities on Wall Street. Making efforts to build supporting infrastructure for these transactions, like gathering secondary data on the mortgage defaults around the country, could be seen as a loss of focus. When trades are happening at such blinding intensity, keeping an eye on the outside world could be seen as a sign of weakness -- 'taking the eye off the ball'. This herd mentality was what resulted in Bear Stearns' experience of 'live by the sword, die by the sword'. The individual players in the market need to understand that when the value of the trade is in the billions of dollars, then they need to do their homework before going ahead with the trade. Thus the duration of interaction between the buyer and the seller is an important aspect of the market. The financial markets that trade in an instantaneous manner must be seen as special cases of the market mechanism where the duration of interaction, in the limiting case, is infinitesimal. But, for larger applications of the market mechanism, non-trivial duration of interaction would have to be a serious consideration.

Another important issue regarding the duration of interaction can be inferred from the current crisis in the financial markets. The marketplace is influenced to a large extent by the practices of the modern company organization. Normally, this is for the good, since it allows for the creation of wealth through risk-taking. More generally, the government, the media and the industry play direct roles in the functioning of markets. However, all these organizations suppress individual initiatives to a significant degree. This necessarily encourages herd behavior although these organizations may not be the root cause of it. In comparison, the university is a place that allows by far the most freedom of inquiry for the individual mind. If one were convinced that tribal orgies of bloodletting are to be avoided in the modern marketplace, then one would, I suppose, wonder how rational thought and scientific inquiry can be given more room in the investment decisions that happen in the marketplace. One obvious way would be to involve the resources of the university more closely in the functioning of the marketplace. Thus, to encourage thought and consideration, one should consider the duration of interaction in a transaction to be an important issue.