Wednesday, January 25, 2012

Brief History of Mathematical Finance

Finance theory is the study of economic agents behavior allocating their resources across alternative financial instruments and in time in an uncertain environment. Mathematics provides tools to model and analyze that behavior in allocation and time, taking into account uncertainty.
1.Louis Bachelier’s 1900 math dissertation on the theory of speculation in the Paris markets marks the twin births of both the continuous time mathematics of stochastic processes and the continuous time economics of option pricing.
2.The most important development in terms of impact on practice was the Black-Scholes model for option pricing published in 1973.
3.Since 1973 the growth in sophistication about mathematical models and their adoption mirrored the extraordinary growth in financial innovation. Major developments in computing power made the numerical solution of complex models possible. The increases in computer power size made possible the formation of many new financial markets and substantial expansions in the size of existing ones.

Louis Bachelier

 Mathematical Ideas
 One sometime hears that “compound interest is the eighth wonder of the world”, or the “stock market is just a big casino”.These are colorful sayings, maybe based in happy or bitter experience, but each focuses on only one aspect of one financial instrument. The “time value of money” and uncertainty are the central elements that influence the value of financial instruments. When only the time aspect of finance is considered, the tools of calculus and differential equations are adequate. When only the uncertainty is considered, the tools of probability theory illuminate the possible outcomes. When time and uncertainty are considered together we begin the study of advanced mathematical finance.Finance is the study of economic agents’ behavior in allocating financial resources and risks across alternative financial instruments and in time in an uncertain environment. Familiar examples of financial instruments are bank accounts, loans,stocks, government bonds and corporate bonds. Many less familiar examples abound. Economic agents are units who buy and sell financial resources in a market, from individuals to banks, businesses, mutual funds and hedge funds. Each agent has many choices of where to buy, sell, invest and consume assets, each with advantages and disadvantages. Each agent must distribute their resources among the many possible investments with a goal in mind.
Advanced mathematical finance is often characterized as the study of the more sophisticated financial instruments called derivatives. A derivative is a financial agreement between two parties that depends on something that occurs in the future, such as the price or performance of an underlying asset. The underlying asset could be a stock, a bond, a currency, or a commodity. Derivatives have become one of the financial world’s most important risk-management tools. Finance is about shifting and distributing risk and derivatives are especially efficient for that purpose. Two such instruments are futures and options. Futures trading, a key practice in modern finance, probably originated in seventeenth century Japan, but the idea can be traced as far back as ancient Greece. Options were a feature of the “tulip mania” in seventeenth century Holland. Both futures and options are called “derivatives”. (For the mathematical reader, these are called derivatives not because they involve a rate of change, but because their value is derived from some underlying asset.) Modern derivatives differ from their predecessors in that they are usually specifically designed to objectify and price financial risk.

Derivatives come in many types. There are futures, agreements to trade something at a set price at a given dates; options, the right but not the obligation to buy or sell at a given price; forwards, like futures but traded directly between two parties instead of on exchanges; and swaps, exchanging flows of income from different investments to manage different risk exposure. For example, one party in a deal may want the potential of rising income from a loan with a floating interest rate, while the other might prefer the predictable payments ensured by a fixed interest rate. This elementary swap is known as a “plain vanilla swap”. More complex swaps mix the performance of multiple income streams with varieties of risk. Another more complex swap is a credit-default swap in which a seller receives a regular fee from the buyer in exchange for agreeing to cover losses arising from defaults on the underlying loans. These swaps are somewhat like insurance . These more complex swaps are the source of controversy since many people believe that they are responsible for the collapse or near-collapse of several large financial firms in late 2008. Derivatives can be based on pretty much anything as long as two parties are willing to trade risks and can agree on a price. Businesses use derivatives to shift risks to other firms, chiefly banks. About 95% of the world’s 500 biggest companies use derivatives. Derivatives with standardized terms are traded in markets called exchanges.
Derivatives tailored for specific purposes or risks are bought and sold “over the counter” from big banks. The “over the counter” market dwarfs the exchange trading. In November 2009, the Bank for International Settlements put the face value of over the counter derivatives at $604.6 trillion. Using face value is misleading, after off-setting claims are stripped out theresidual value is $3.7 trillion, still a large figure.Mathematical models in modern finance contain deep and beautiful applications of differential equations and probability theory.In spite of their complexity, mathematical models of modern financial instruments have had a direct and significant influence on finance practice.

Early History

The history of stochastic integration and the modelling of risky asset prices both beginwith Brownian motion, so let us begin there too. In 17th century put options were bought on tulip bulbs in Netherlands.The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copenhagen, who effectively created a model of Brownian motion while studying time series in 1880  the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900,and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter.

Though the origins of much of the mathematics in financial models traces to Louis Bachelier’s 1900 dissertation on the Theory of speculation in the Paris markets. This doctrate thesis was completed at the Sorbonne in 1900 under Henri Poincare, this work marks the twin births of both the continuous time mathematics of stochastic processes and the continuous time economics of option pricing. While analyzing option pricing,Bachelier provided two different derivations of the partial differential equation for the probability density for the Wiener process or Brownian motion. In one of the derivations, he works out what is now called the Chapman-Kolmogorov convolution probability integral. Along the way, Bachelier derived the method of reflection to solve for the probability function of a diffusion process with an absorbing barrier. Not a bad performance for a thesis on which the first reader, Henri PoincarĂ©,gave less than a top mark! After Bachelier, option pricing theory laid dormant in the economics literature for over half a century until economists and mathematicians renewed study of it in the late 1960s. Jarrow and Protter  speculate that this may have been because the Paris mathematical elite scorned economics as an application of mathematics.


Louis Bachelier’s 1900 dissertation on the Theory of speculation

Bachelier’s work was 5 years before Albert Einstein’s 1905 discovery of the same equations for his famous mathematical theory of Brownian motion. The editor of Annalen der Physik received Einstein’s paper on Brownian motion on May 11, 1905. The paper appeared later that year. Einstein proposed a model for the motion of small particles with diameters on the order of 0.001 mm suspended in a liquid. He predicted that the particles would undergo microscopically observable and statistically predictable motion. The English botanist Robert Brown had already reported such motion in 1827 while observing pollen grains in water with a microscope. The physical motion is now called Brownian motion in honor of Brown’s description.Einstein calculated a diffusion constant to govern the rate of motion of suspended particles. The paper was Einstein’s attempt to convince physicists of the molecular and atomic nature of matter. Surprisingly, even in 1905 the scientific community did not completely accept the atomic theory of matter. In 1908, the experimental physicist Jean-Baptiste Perrin conducted a series of experiments that empirically verified Einstein’s theory. Perrin thereby determined the physical constant known as Avogadro’s number for which he won the Nobel prize in 1926. Nevertheless, Einstein’s theory was very difficult to rigorously justify mathematically. Let us now turn to Einstein’s model. In modern terms, Einstein assumed that Brownian motion was a stochastic process with continuous paths, independent increments, and stationary Gaussian increments. He did not assume other reasonable properties (from the standpoint of physics), such as rectifiable paths. If he had assumed this last property, we now know his model would not have existed as a process. However, Einstein was unable to show that the process he proposed actually did exist as a mathematical object. This is understandable, since it was 1905, and the ideas of Borel and Lebesgue constructing measure theory were developed only during the first decade of the twentieth century.In a series of papers from 1918 to 1923, the mathematician Norbert Wiener constructed a mathematical model of Brownian motion. Wiener and others proved many surprising facts about his mathematical model of Brownian motion, research that continues today. In recognition of his work, his mathematical construction is often called the Wiener process.

 The next step in the groundwork for stochastic integration lay with A. N. Kolmogorov.Indeed, in 1931, two years before his famous book establishing a rigorous mathematical basis for Probability Theory using measure theory, Kolmogorov refers to and briefly explains Bachelier’s construction of Brownian motion ( pages 64, 102–103). It is this paper too in which he develops a large part of his theory of Markov processes. Most significantly, in this paper Kolmogorov showed that continuous Markov processes (diffusions) depend essentially on only two parameters:one for the speed of the drift and the other for the size of the purely random part (the diffusive component). He was then able to relate the probability distributions of the process to the solutions of partial differential equations, which he solved, and which are now known as “Kolmogorov’s equations.” He also made major contributions to the understanding of stochastic processes (involving random variables), and he advanced the knowledge of chains of linked probabilities. Shortly thereafter, he took an extended trip to Germany and France, and in 1933 laid out his probability theory in Foundations of the Theory of Probability. This work secured his reputation as the world's foremost expert in his field.

Andrey Kolmogrov, the father of Probability Theory

Paul Samuelson

We turn now to Kiyosi Ito, the father of stochastic integration,no doubt an attempt to establish a true stochastic differential to be used in the study of Markov processes was one of Ito’s primary motivations for studying stochastic integrals.His work, starting in the 1940s, built on the earlier breakthroughs of Einstein and Norbert Wiener. Mr. Ito’s mathematical framework for describing the evolution of random phenomena came to be known as the Ito Calculus.This random component is best modeled using a mathematics which can show the range of possible areas.Stochastic processes and Ito calculus make up most of the modern Financial Maths, it has important applications in Mathematical Finance and stochastic differential equations.Firstly, we are now dealing with random variables (more precisely, stochastic processes). Secondly, we are integrating with respect to a non-differentiable function (technically,stochastic processes).

Kiyosi Ito, maker of Ito Calculus

Growth of Mathematical Finance

Modern mathematical finance theory begins in the 1960s. In 1965 the economist Paul Samuelson published two papers that argue that stock prices fluctuate randomly. One explained the Samuelson and Fama efficient markets hypothesis that in a well-functioning and informed capital market, asset-price dynamics are described by a model in which the best estimate of an asset’s future price is the current price (possibly adjusted for a fair expected rate of return.). Under this hypothesis, attempts to use past price data or publicly available forecasts about economic fundamentals to predict security prices are doomed to failure. In the other paper with mathematician Henry McKean, Samuelson shows that a good model for stock price movements is geometric Brownian motion. The final precursor to the Black, Scholes and Merton option pricing formulaes can be found in the paper of Samuelson and Merton .Samuelson noted that Bachelier’s model failed to ensure that stock prices would always be positive, whereas geometric Brownian motion avoids this error.
The most important development in terms of practice was the 1973 Black-Scholes terms of practice was the 1973 Black-Scholes model for option pricing. The two economists Fischer Black and Myron Scholes (and simultaneously, and somewhat independently, the economist Robert Merton) deduced an equation that provided the first strictly quantitative model for calculating the prices of options. The key variable is the volatility of the underlying asset.  Myron Scholes published a paper with Fischer Black on 'Pricing of Options and Corporate Liabilities', incorporating suggestions from Merton Miller (of M&M Theory fame) and Eugene Fama (father of the Efficient Market Hypothesis). These equations standardized the pricing of derivatives in exclusively quantitative terms. The formal press release from the Royal Swedish Academy of Sciences announcing the 1997 Nobel Prize in Economics states that the honor was given “for a new method to determine the value of derivatives. Robert C. Merton and Myron S. Scholes have, in collaboration with the late Fischer Black developed a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society.”
 The Chicago Board Options Exchange (CBOE) began publicly trading options in the United States in April 1973, a month before the official publication of the Black-Scholes model. By 1975, traders on the CBOE were using the model to both price and hedge their options positions. In fact, Texas Instruments created a hand-held calculator specially programmed to produce Black-Scholes option prices and hedge ratios.The basic insight underlying the Black-Scholes model is that a dynamic portfolio trading strategy in the stock can replicate the returns from an option on that stock. This is called “hedging an option” and it is the most important idea underlying the Black-Scholes-Merton approach. Much of the rest of the book will explain what that insight means and how it can be applied and calculated.
The story of the development of the Black-Scholes-Merton option pricing model is that Black started working on this problem by himself in the late 1960s. His idea was to apply the capital asset pricing model to value the option in a continuous time setting. Using this idea, the option value satisfies a partial differential equation. Black could not find the solution to the equation. He then teamed up with Myron Scholes who had been thinking about similar problems. Together, they solved the partial differential equation using a combination of economic intuition and earlier pricing formulas.

Black and Scholes had solved stochastic partial differential equations to develop a formula for pricing European-type call options.The result was an equation that suggested how the price of a call option might be calculated as a function of a risk-free interest rate, the price variance of the asset on which the option was written, and the parameters of the option (strike price, term, and the market price of the underlying asset.)

At this time, Myron Scholes was at MIT. So was Robert Merton,who was applying his mathematical skills to various problems in finance.Merton showed Black and Scholes how to derive their differential equation differently. Merton was the first to call the solution the Black-Scholes option pricing formula. Merton’s derivation used the continuous time construction of a perfectly hedged portfolio involving the stock and the call option together with the notion that no arbitrage opportunities exist. This is the approach we will take. In the late 1970s and early 1980s mathematicians Harrison, Kreps and Pliska showed that a more abstract formulation of the solution as a mathematical model called a martingale provides greater generality.
By the 1980s, the adoption of finance theory models into practice was nearly immediate. Additionally, the mathematical models used in financial practice became as sophisticated as any found in academic financial research. There are several explanations for the different adoption rates of mathematical models into financial practice during the 1960s, 1970s and 1980s. Money and capital markets in the United States exhibited historically low volatility in the 1960s; the stock market rose steadily, interest rates were relatively stable, and exchange rates were fixed. Such simple markets provided little incentive for investors to adopt new financial technology. In sharp contrast, the 1970s experienced several events that led to market change and increasing volatility. The most important of these was the shift from fixed to floating currency exchange rates; the world oil price crisis resulting from the creation of the Middle East cartel; the decline of the United States stock market in 1973-1974 which was larger in real terms than any comparable period in the Great Depression; and double-digit inflation and interest rates in the United States. In this environment, the old rules of thumb and simple regression models were inadequate for making investment decisions and managing risk.

   During the 1970s, newly created derivative-security exchanges traded listed options on stocks, futures on major currencies and futures on U.S. Treasury bills and bonds. The success of these markets partly resulted from increased demand for managing risks in a volatile economic market. This success strongly affected the speed of adoption of quantitative financial models. For example, experienced traders in the over the counter market succeeded by using heuristic rules for valuing options and judging risk exposure. However these rules of thumb were inadequate for trading in the fast-paced exchange-listed options market with its smaller price spreads, larger trading volume and requirements for rapid trading decisions while monitoring prices in both the stock and options markets. In contrast, mathematical models like the Black-Scholes model were ideally suited for application in this new trading environment. The growth in sophisticated mathematical models and their adoption into financial practice accelerated during the 1980s in parallel with the extraordinary growth in financial innovation. A wave of de-regulation in the financial sector was an important factor driving innovation.

Quantum Mechanics in Financial Markets. 

The markets are non-linear, dynamic systems, subject to the rules of Chaos Theory. Market prices are highly random, with a short to intermediate term trend component. They are highly dependent on initial conditions. Markets also show qualities of fractals -- self-similar in the sense that the individual parts are related to the whole.Due to the non-Gaussian behavior of the markets the methods from Chaos Theory, Fractals and Quantum Physics are being used in Finance since 1980s.

One recent trend in the growing field of quantitative finance to apply techniques borrowed from quantum physics to Financial models. One example is Path Integrals, which were invented by Richard Feynman in 1948 , Feynman used path integrals along with the probability methods designed by Norbert Wiener to reformulate methods of Quantum Physics.The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition.Feynman suggested that when considering the Quantum Mechanics of a moving particle, every conceivable path can be assigned a certain complex(probability amplitude) number called the probability amplitude for that path.

Jan Dash a particle physicist later applied path integrals in Finance, the reason was simple ,the value of the financial derivative depends on the "path" followed by underlying asset.

 Methods of quantum mechanics for mathematical modelling of price dynamics of the financial market. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. On the one hand, our Bohmian model is a quantum-like model for the financial market, cf. with works of W. Segal, I.E. Segal, E. Haven, E.W. Piotrowski, J. Sladkowski. On the other hand, (since Bohmian mechanics provides for the possibility to describe individual price trajectories) it belongs to the domain of extended research on deterministic dynamics for financial assets. Our model emphasizes the complexity of the financial market: the traditional description of price dynamics is completed by Schrodinger's dynamics for the pilot wave of expectations of traders. This is a kind of socio-economic model for the financial market. 

Pricing an option is a complex mathematical problem which  involves diffusion processes such as Brownian motion.From this analysis, Black, Scholes and Merton were able to derive a general Partial Differential Equation for the value of a stock option, which turned out to look very similar to the heat-diffusion equation.

Richard Feynman

A path-integral description of the Black-Scholes model was recently developed by Belal Baaquie of the National University of Singapore and co-workers. From this, Baaquie and co-workers went on to devise a quantum-mechanical version of the Black-Scholes equation to describe the price of a simple, non-dividend-paying option.

 Conceptual breakthroughs in finance theory in the 1980s were fewer and less fundamental than in the 1960s and 1970s, but the research resources devoted to the development of mathematical models was considerably larger. Major developments in computing power, including the personal computer and increases in computer speed and memory enabled new financial markets and expansions in the size of existing ones. These same technologies made the numerical solution of complex models possible. They also speeded up the solution of existing models to allow virtually real-time calculations of prices and hedge ratios. 

 Ethical considerations
According to M. Poovey , new derivatives were developed specifically to take advantage of de-regulation. Poovey says that derivatives remain largely unregulated, for they are too large, too virtual, and too complex for industry oversight to police. In 1997-8 the Financial Accounting Standards Board (an industry standards organization whose mission is to establish and improve standards of financial accounting) did try to rewrite the rules governing the recording of derivatives, but in the long run they failed: in the 1999-2000 session of Congress, lobbyists for the accounting industry persuaded Congress to pass the Commodities Futures Modernization Act, which exempted or excluded “over the counter” derivatives from regulation by the Commodity Futures Trading Commission, the federal agency that monitors the futures exchanges. Currently,only banks and other financial institutions are required by law to reveal their derivatives positions. Enron, which never registered as a financial institution, was never required to disclose the extent of its derivatives trading.

   In 1995, the sector composed of finance, insurance, and real estate overtook the manufacturing sector in America’s gross domestic product. By the year 2000 this sector led manufacturing in profits. The Bank for International Settlements estimates that in 2001 the total value of derivative contracts traded approached one hundred trillion dollars, which is approximately the value of the total global manufacturing production for the last millennium. In fact, one reason that derivatives trades have to be electronic instead of involving exchanges of capital is that the sums being circulated exceed the total of the world’s physical currencies.In the past, mathematical models had a limited impact on finance practice.But since 1973 these models have become central in markets around the world. In the future, mathematical models are likely to have an indispensable role in the functioning of the global financial system including regulatory and accounting activities.

   We need to seriously question the assumptions that make models of derivatives work: the assumptions that the market follows probability models and the assumptions underneath the mathematical equations. But what if markets are too complex for mathematical models? What if irrational and completely unprecedented events do occur, and when they do – as we know they do – what if they affect markets in ways that no mathematical model can predict? What if the regularity that all mathematical models assume ignores social and cultural variables that are not subject to mathematical analysis? Or what if the mathematical models traders use to price futures actually influence the future in ways that the models cannot predict and the analysts cannot govern?
Any virtue can become a vice if taken to extreme, and just so with the application of mathematical models in finance practice. At times, the mathematics of the models becomes too interesting and we lose sight of the models’ ultimate purpose. Futures and derivatives trading depends on the belief that the stock market behaves in a statistically predictable way; in other words,that probability distributions accurately describe the market. The mathematics is precise, but the models are not, being only approximations to the complex, real world. The practitioner should apply the models only tentatively,assessing their limitations carefully in each application. The belief that the market is statistically predictable drives the mathematical refinement,and this belief inspires derivative trading to escalate in volume every year.
 Financial events since late 2008 show that many of the concerns of the previous paragraphs have occurred. In 2009, Congress and the Treasury Department considered new regulations on derivatives markets. Complex derivatives called credit default swaps appear to have been based on faulty assumptions that did not account for irrational and unprecedented events, as well as social and cultural variables that encouraged unsustainable borrowing and debt. Extremely large positions in derivatives which failed to account for unlikely events caused bankruptcy for financial firms such as Lehman Brothers and the collapse of insurance giants like AIG. The causes are complex, but some of the blame has been fixed on the complex mathematical models and the people who created them. This blame results from distrust of that which is not understood. Understanding the models is a  prerequisite for correcting the problems and creating a future which allows proper risk management.                                                                         

What is a Hedge Fund?

Ray Dalio ,founder of Bridgewater Associates,(world's richest Hedge fund manager with $120 billion under management)

 Hedge Funds

  A hedge fund is a fund that can take both long and short positions, use arbitrage, buy and sell undervalued securities, trade options or bonds, and invest in almost any opportunity in any market where it foresees impressive gains at reduced risk.The first thing to know about hedge funds is that the term hedge fund is not a legal term, but rather an industry term. What a hedge fund is, therefore, is subject to some amount of interpretation. Consider a few definitions. From Wall Street Words Houghton & Miflin 1997:
    A very specialized, volatility open-end investment company that permits the manager to use a variety of investment techniques usually prohibited in other types of funds. These techniques include borrowing money, selling short and using options. Hedge funds offer investors the possibility of extraordinary gains with above average risk. From Hedge Funds Demystified, Goldman Sachs & Co.:

    The term "hedge fund" includes a multitude of skill-based investment strategies with a broad range of risk and return objectives. A common element is the use of investment and risk management skills to seek positive returns regardless of market direction.

From the Hennessee Group LLC Web page:

A hedge fund is a "pool" of capital for accredited investors only and organized using the limited partnership legal structure... the general partner is usually the money manager and is likely to have a very high percentage of his/her own net worth invested in the fund.

As you can see, the definitions above focus on several aspects of investment companies known as hedge funds:
  • legal structure
  • investment strategy
  • investor pool
All three components are important in understanding hedge funds. To go further let's discuss first what hedge funds are and what some of their salient features are:
  • Hedge funds are legally limited partnerships.
  • Hedge funds are unregistered (i.e., unregistered with the SEC) investment companies. That is, they are not regulated by the SEC (more on this later).
  • Hedge funds can be users of a variety of investment strategies and products, including options, future, swaps and short selling.
  • Hedge funds often employ leverage, in that the amount of notional market exposure often exceeds the investment capital of the fund.
  • Hedge funds have limited liquidity. Typically investors can only get into funds on certain dates and can only get their money out of funds on certain dates.
Who invests in hedge funds?
  • Wealthy individuals. For an individual to invest in an unregistered investment company, the SEC must deem an individual to be an accredited investors (a defined by SEC rule 501 of Regulation D. The full text of this rule may be found at The rule includes the following points for an individual:
    • The individual must have at the time of investment a net worth (or joint net worth with spouse) exceeding $1,000,000, or
    • The individual must have individual income exceeding $200,000 in each of the two most recent years or joint income with spouse exceeding $300,000 in each of the two most recent years, and must have a "reasonable expectation" of reaching the same level in the coming year.
And the rule contains the following points for an organization, corporation or other such entity:
  • The organization must have total assets in excess of $5,000,000, or
  • The organization's owners must be accredited investors.
  • Endowments, e.g., the Wall Street Journal (December 8, 1998) and others report that the University of Pittsburgh made a $5 million investment in Long Term Capital
  • Pension Funds, e.g., Pension & Investments magazine regularly reports of pension fund activity investing in new hedge funds. For example:
    • Consolidated Paper's $650m pension fund added $10m in hedge fund investments in November
    • The University of Michigan hired four hedge fund managers to manage $100m of its $2.5bn pension fund, and P&I reports that University of Michigan has a total of about $300m invested with hedge funds.
  • Other Hedge Funds: Some hedge funds invest in other hedge funds, including offering Funds of Funds, that is, they strategically allocate their capital to other funds. Pension and Investments (November 30, 1998) reports that Grosvenor Capital Management invested $7m in Long Term Capital Management:
    • Grosvenor Capital Management, LP, a big, highly secretive hedge fund player, was stung by an estimated $7m investment in Long-Term Capital Management LP, and reportedly has been hit with redemptions following poor third-quarter performance.

Hedge Fund Industry
It is important to understand the magnitude of the hedge fund industry and the sizes of some of the key players in the industry. "Hedge Funds Demystified" estimates that the size of the whole industry is approximately $400bn, and that the investor pool is dominated by wealthy individuals (accredited investors), with pension fund interest increasing. "Hedge Funds Demystified" also notes that it is difficult to accurately assess the size of the industry, so this number should be read as mainly an indication of the order of magnitude. To get a sense of where this stands, consider the pension fund industry by contrast. Davis, in Pension Funds, reports that as of year end 1991, the US pension fund industry's assets were at least $2.9 trillion. In other countries, the number was less for two reasons: the pension fund industry contributes less assets as a percentage of GDP than the US (except Germany and Switzerland) and the US GDP is much larger than other countries. Nevertheless, the global pension fund industry (as of year end 1991) was estimated at approximately $4.2 trillion. The numbers since then have surely grown, but I currently do not have more up-to-date numbers.

Types of Hedge Funds

Hedge funds are generally classified according to the type of investment strategy they run. Below we review the major types of strategies, but refer members of the class to "Hedge Funds Demystified" for greater detail.

Market Neutral (or Relative Value) Funds

Market neutral funds attempt to produce return series that have no or low correlation with traditional markets such as the US equity or fixed income markets. Market neutral strategies are characterized less by what they invest in than by the nature of the returns. They often are highly quantitative in their portfolio construction process, and market themselves as an investment that can improve the overall risk/return structure of a portfolio of investments. Market neutral funds should not be confused with Long/Short investment strategies (see below). The key feature of market neutral funds are the low correlation between their returns and the traditional asset's.

Event Driven Funds

Event driven funds seek to make profitable investments by investing in a timely manner in securities that are presently affected by particular events. Such events include distressed debt investing, merger arbitrage (sometimes called risk arbitrage) and corporate spin-offs and restructuring.

Long/Short Funds

Funds employing long/short strategies generally invest in equity and fixed income securities taking directional bets on either an individual security, sector or country level. For example, a fund might do pairs trading, and buy stocks that they think will move up and sell stocks they think will move down. Or go long sectors they think will go up and short countries they think will go down. Long/Short strategies are not automatically market neutral. That is, a long/short strategy can have significant correlation with traditional markets, and surprisingly have seen large down turns in exactly the same times as major market downturns. For example, Pension & Investments reported on November 30, 1998:
Many long-short managers, which aim to profit from going long on stellar stocks and selling short equity albatrosses, typically use traditional stock valuation factors such as price-to-earnings and price-to-bok value ratios in their mathematical models to cull the winners from the losers. Unfortunately for them, when the market ran into turbulence in late July and August, investors sought safe haven in some of the largest-but expensive-stocks that these models had rejected as overpriced.Then, after the Federal Reserve Bank began easing interest rates in late September, investors rushed to buy small-capitalization, high-octane stocks that had been neglected in favor of large cap stocks for most of the year. ... As a result, some market long-short managers got hit with a double-whammy.

Tactical Trading

Quoting from "Hedge Funds Demystified":
    Tactical trading refers to strategies that speculate on the direction of market prices of currencies, commodities, equities and/or bonds. Managers typically are either systematic or discretionary. Systematic managers are primarily trend followers who rely on computer models based on technical analysis. Discretionary managers usually take a less quantitative approach and rely on both fundamental and technical analysis. This is the most volatile sector in terms of performance because many managers combine long and/or short positions with leverage to maximize returns...

The Hedge Fund Industry and Quantitative Methods

Quantitative methods have been successfully applied in the hedge fund industry to improve returns, and control risk. That said, there have been striking failures of seemingly quantitatively driven funds (such as Long-Term Capital). Some of the most quantitatively driven strategies occur in the Market Neutral/Relative Value Sector of the Hedge Fund World, so we will exam this sector in more detail by discussing some of the specific types of strategies they employ. The following is a list of important and quantitatively driven market neutral/relative value strategies. I refer you to "Hedge Funds Demystified" for a detailed description of each:
  • Fixed income arbitrage
  • Covertible bond arbitrage
  • Mortgage backed security arbitrage
  • Derivatives Arbitrage
  • Market Neutral Long/Short Equity Strategies

  Hedging Strategies

A wide range of hedging strategies are available to hedge funds. For example:
  • selling short - selling shares without owning them, hoping to buy them back at a future date at a lower price in the expectation that their price will drop.
  • using arbitrage - seeking to exploit pricing inefficiencies between related securities - for example, can be long convertible bonds and short the underlying issuers equity.
  • trading options or derivatives - contracts whose values are based on the performance of any underlying financial asset, index or other investment.
  • investing in anticipation of a specific event - merger transaction, hostile takeover, spin-off, exiting of bankruptcy proceedings, etc.
  • investing in deeply discounted securities - of companies about to enter or exit financial distress or bankruptcy, often below liquidation value.
  • Many of the strategies used by hedge funds benefit from being non-correlated to the direction of equity markets

Hedge Fund Returns

As hedge funds are often viewed as providing returns that are "cheap" relative to risk, their performance is usually evaluated on a risk-adjusted return basis. The common number that is quoted is the Sharpe Ratio which is the ratio of annualized excess returns to the annualized standard deviation of returns. The following repeats the data in Table 7 of "Hedge Funds Demystified" and gives an idea of the relative performance of hedge funds compared with some standard indexes over the period January 1993 - December 1997. The table represents returns on each Hedge Fund Sector, that is, the returns and standard deviations in each column represents the returns that were realized on an equal weighted investment portfolio of all the hedge funds in a given sector.

One important item that none of the definitions covered was hedge fund fee structures, which is, in my opinion, a key distinguishing feature of hedge funds versus in particular mutual funds. Hedge funds almost always have a fee structure that includes both a fixed fee and a management fee. The fixed fee usually ranges between 1 and 2% of assets under management and the management fee ranges between 20 and 25% of upside performance. As hedge funds are unregulated, these ranges are often exceeded, and can be as high as 5% fixed fee and 25% management fee. Hedge fund fees are often quoted in language such as "2 and 20" meaning 2% fixed fee and 20% management fee. There are two additional important points about hedge fund fees:
  • the benchmark
  • high water mark
The performance fee is sometimes calculated net of a benchmark. That is, the returns that fees are paid on are sometimes only those returns in excess of some benchmark. Sometimes the benchmark is a risk-free interest rate such as LIBOR (often called the cash benchmark, meaning performance fees are paid on the profit that would be made in excess of an investment in cash) and other times it is a market index such as the MSCI World Index or the S&P 500 index.

The biggest risk in pricing models is Assumption Risk. The trouble with bull and bear markets is that the price behavior and the width of bid-offer spreads can be quite different under the two regimes. If you are in roach motel assets getting out can become expensive. Bear Stearns was leveraged long CDOs of illiquid securities and "hedged" by shorting liquid ABX indices. As with similar problems in the past, the funds were long illiquid, short liquid. If a fund is leveraged and can only sell to a limited number of counterparties that KNOW it has a problem, getting out becomes difficult. Software for measuring risk doesn't help when risks are unmeasurable. And aren't you supposed to have proper risk management in place BEFORE you lose money not AFTER the fact?

You really have to know what you are doing when designing models of prepayment and mortgage default risks; nothing in the academic literature or public domain works. Credit is neither stochastic nor continuous and when it jumps it really jumps. I can count the number of good mortgage-backed securities hedge funds on one hand but I would need many more limbs for the traders who have been blown away by not having adequate trading AND quantitative abilities to manage ALL the exposures in this complex field. When a product is very thinly traded, indicative dealer prices are pretty useless. If a fund is investing in illiquid instruments the fund valuation needs to be marked to the real bid, in size. Mark to market is possible only when there is a market.

There is nothing inherently wrong with investing in "untraded" assets provided the risk-adjusted returns are sufficient to compensate. In bearish credit conditions ideally you usually want to be long the liquid and short the illiquid but weaker credit funds and less experienced managers do the opposite. Of course there have been skilled hedge funds in the areas of distressed debt and collateralised loans for a long time but their returns have justified the risks. But with some funds, even with apparently high absolute performance, often the excess RISK-ADJUSTED returns (the alpha!) was negative.

Just as with Long-Term Capital Management, being long the illiquid and short the liquid works well until the market reverses and then years of consistently positive months get given back in one massively negative month. Leverage, liquidity and valuation risks are ONLY worth taking if you are compensated for those risks and plainly this was not the case. This is where investors in a hedge fund need to look at whether the potential returns justify the potential risk. With good hedge funds it does but NOT with the many "hedge fund" journeyman.

With public equities, liquid bonds, fx and futures valuation is immediate, transparent, generally unarguable and there is plenty of alpha available in these liquid arenas IF you have the tools and expertise to find it. While liquidity is a variable even on an exchange you have access to the widest number of potential buyers and sellers. Venturing into illiquid areas raises the risk exponentially when there are much fewer counterparties to trade with. Leverage just exacerbates those problems. Funds investing in illiquid assets should be targeting MUCH higher performance than liquid funds as compensation for that extra risk. Yet some investors seems to compare them side by side without modelling the non-linear risks of gearing thinly traded securities.

What's even worse than a closet index fund? A leveraged closet index fund. And that is what most of these toxic waste CDO funds were in effect running. Making money in BAD conditions is what hedge fund clients pay the 2 and 20 for; long only funds are the ONLY products you need in good times. Having criticised some of John Bogle's thinking in my previous post let's make something clear; index equity and credit funds are the best investment IF (and only IF!) you think the asset class is going up. It is a waste of time and money to allocate to higher fee actively managed funds that simply fall apart when their underlying market falls apart.

Investors need to verify that a money management product purporting to be a hedge fund and charging hedge fund fees actually is one. Out of 10,000 funds that claim to be hedge funds, how many actually are hedge funds? The best estimate I have is maybe 25% tops. But of those how many are skilled? Perhaps 500-1000 at most. In other words probably only 10% of products that say they are hedge funds actually are GOOD hedge funds. Skill is rare by definition. While some investors might be discouraged by the bad news of 1/10 odds of picking a skilled fund, the good news is that they CAN be isolated in advance.

Identifying a good hedge fund is as rare a skill as being able to identify a good security. Some multi-manager products and weaker funds of funds have reduced their fees because they think picking hedge funds is easy! Most of them don't have the experience or analytical resources to decide what is and what is NOT a hedge fund, let alone trying to find the BEST ones. It is difficult but NOT impossible. Do "lower" fees help if an "advisor" puts you into a fund that drops 100%? There will always be semantically-challenged products that screw up which is why due diligence and alignment of interests are so important. Investors should select real hedge funds NOT leveraged beta products that SAY they are hedge funds. The industry needs to rid itself of non hedge funds who can't measure, manage or hedge their risks and ride beta when proper managers aim for alpha. Fortunately we can rely on the market to conduct these shakeouts over time. Unfortunately for some amnesiac investors it has been a long time since difficult credit conditions.

The colleagues of Ralph Cioffi may have liked the fund but how much personal cash did James Cayne have in? A necessary condition for a product to be considered a hedge fund is to verify senior management are eating their own cooking. In bull markets many unskilled traders make money; it is bear markets that tend to show who is good and who knows how to hedge. Many REAL hedge funds are MAKING MONEY out of these ongoing credit events. Marketing something is a hedge fund does not mean it is.

Andrew W. Lo ,Andrew is a Professor of Finance at MIT's Sloan School of Management and he is also the Director of Laboratory of Financial Engineering at Sloan is a leading authority on hedge funds.

Andrew W. Lo

Popular Misconception
The popular misconception is that all hedge funds are volatile -- that they all use global macro strategies and place large directional bets on stocks, currencies, bonds, commodities, and gold, while using lots of leverage. In reality, less than 5% of hedge funds are global macro funds. Most hedge funds use derivatives only for hedging or don't use derivatives at all, and many use no leverage.