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We are, of course, all well aware of the recent and current difficulties in the job market foe mathematicians. They are chronicled and lamented in almost every issue of the AMS Notices; see [1] for the latest discussion and in-depth analysis. In the "old days", say prior to the last 5 years, the defense industry served as an alternate source of technical employment, but that area has faded lately along with other DoD downsizing.
Mixed in with this generally bleak outlook is a ray of sunshine in the form of a high ongoing need for technical sophisticates on Wall Street (= a metaphor for the investment industry). Mathematicians, physicists, computer scientists, EEs and AIs are all welcome, and not badly paid either! This type of opportunity is no longer a secret but it will still be a while before supply catches up with demand. It can be judged by the number of recruiting visits by Wall St. firms and by the growth in personal savings and investment activity, due in part to increasing use of self-directed retirement programs (e.g., 401(k), 403(b), IRA and Keogh plans). So, almost none of us can avoid thinking about and making investment decisions, even if, in the end, we simply farm out the detailed decision-making to a professional, such as a mutual fund manager. But what then do the pro's need to carry out their responsibilities in an ever more competitive business?
In the following paragraphs we will describe the nature of the investment business, and its academic relative, finance theory, from a mathematical perspective. However, no attempt is made to write a small textbook on the subject! Indeed, note that typical investment texts run to 1000 pages or so, e.g., [2]. We just want to get readers to the point of having their interest piqued, if that is going to happen. For such interested parties, lots of supplementary reading will be suggested. Let us admit, here at the beginning, that, for brevity's sake, there will be a certain degree of over-simplification in the following efforts to parse out these fields and their activities.
Players in the World of
Finance and Investment Management
We may say that Finance is an academic subfield of (micro)economics, whose concern is the working of capital markets and the pricing of capital assets. Members of this profession will be called PROFs. Mathematically inclined PROFs may try to invent models of the stock market, such as the Capital Asset Pricing Model (CAPM), due to Sharpe and Lintner in the mid-60s, or the Arbitrage Pricing Theory (APT) of Ross, a decade later. Or, they may try to model a specific kind of price behavior, such as that of a stock or an interest rate. There are at least two ways to proceed here: look at some data and try to fit a statistical model, or first propose an economic model and then see what sort of statistical model ensues. Of course, such a theoretically derived model would have to be tested against real data.
By contrast, Investment Management is concerned with the allocation of real money into various financial assets and instruments. Members of this profession are called practitioners or portfolio managers; we will call them PMs. These are the folks who run mutual funds, corporate and municipal pension funds, university endowments, insurance company reserves, hedge funds, etc. A subclass of PMs, as well as some private individuals, are called traders (TRs). They tend to differ from the PMs by moving and shifting money rapidly (sometimes by the day or even hour!) among assets. Normally such decisions are guided by computer programs, which are believed to capture some regularity in price fluctuations.
There is another kind of investment professional, of relatively recent creation, called financial engineers, or FEs. This area is actually the most likely landing place for a technical person entering the FIM business. Typically, FEs are charged with constructing and valuing novel financial instruments for eventual use by a PM. These instruments may be used for insurance, risk transfer, hedging, speculation, etc., and are seemingly without end. Here are just a few examples:
- Suppose that a PM wants to insure a portfolio against a market decline over some time period, but not lose out on any upside activity. Or, the goal may be to guarantee a minimal growth rate g (necessarily less than the risk-free rate r). Depending on how much decline is tolerable, or how much r exceeds g, what sort of instrument will deliver this kind of insurance to the PM, and what is a fair price to charge for the protection? Certain instruments of this nature are called protected equity notes (PENs)
- Consider a stock, say IBM, and an instrument that, at the end of some time period [0,T], pays $(A-X) if and only if the average price A of IBM over [0,T] exceeds a given amount X, otherwise nothing. Such an instrument is an Asian call option. How should it be priced? And, of course, a related question is...what is the use of such a thing, i.e., should there be a market for it at all?
- Consider again some stock and a debt instrument that pays a regular, say quarterly, dividend over [0,T], but whose value at maturity T depends (in some specified way) on the then-price of the stock. This is called an equity-linked security (ELKS) and again the issues are how to price it, and what is its investment role.
So now we have our basic cast of characters: PROFs, FEs, PMs, and TRs. The PROFs deal in theories of the markets and/or carry out empirical studies, usually to test some popular or pet theory. The FEs work at a less grandiose level, with specific instruments, but there is often a high degree of mathematical sophistication required. The PMS and TRs have the real responsibility of investing other people's money. In addition to their own specialized styles and techniques, they may avail themselves of the output of the PROFs and FEs. Finally, the TRs usually employ some ad hoc methodology; it may be quite sophisticated but need not be, although that is the trend nowadays. Often TRs are viewed with some disdain by the PROFs and PMs who, on theoretical grounds, tend to doubt the efficacy of trader efforts to predict market movements.
Note: hereafter FIM means "Finance and Investment Management".
Some Generalities about FIM
As has been the case with so many other disciplines, FIM has become increasingly quantitative. This seems inevitable in retrospect, given the immense amount of high frequency data produced by the markets on a daily basis, and the ever-increasing availability of computer power and computer-intensive methods for analyzing and modeling data. For the most part these latter have not initially been constructed with FIM in mind, but rather have first appeared in the fields of statistics, nonlinear physics, and artificial intelligence. Prime examples are Bayesian and max entropy formalisms, simulation techniques, artificial neural networks, chaotic dynamical systems, pattern recognition, evolutionary computation, and data mining. (In fact, know-ledge discovery via machine learning and data mining of large databases are coming to serve as alternatives to macroeconomic analysis [3].) The high end of FIM, academic finance theory, has also benefited from the use of advanced notions in pure math, as original economic intuitions have been abstracted and rigorized. See further discussion in next section.
In addition to its increasingly quantitative nature, FIM is a vast, dynamic, and controversial field. Its modern form has been under sustained development for about 4 decades now. An excellent non-technical survey of the major achievements has recently been given by Bernstein [4]. In addition, there is the slightly more technical classic by Malkiel [5], now in its 6th edition. The field is vast because, to slightly paraphrase the bank robber Willy Sutton, "That's where the money is". And this is ever more the case, as assets under institutional management continue to swell, due in part, as already noted, to stepped-up retirement savings. The field is dynamic because of the increase in information transmission and processing speed, the introduction of ever more new financial products and strategies (e.g., "exotic" options, their pricing, and use in risk control), and the constant attempted application of the new mathematical/statistical techniques, such as those mentioned above, to gain a trading edge. Let's note too that many of these so-called exotic options trade over-the-counter (OTC), so that constant observable market prices do not exist; consequently it is all the more important to have math models for proper pricing.
Finally, the field of FIM is controversial, by its very nature. First, what is that nature? Investment science, whatever it may be, can be seen as bearing some similarities to natural sciences, such as physics and biology; for developments of such a view, see [6,7]. But it is probably more important to emphasize the dissimilarities. For example, finance is not inherently experimental. Data cannot be arbitrarily generated to conduct experiments and test hypotheses. Also, what is the purpose of working in FIM? Is it truth, as in the natural sciences? No, it is rather success. This term can have various connotations in FIM. It can mean establishing a theorem that formalizes some economic intuition, or garnering some insight into market operation from a study of some data. It can also mean implementation of a new investment strategy which somehow outperforms, although it is tricky (but important!) to understand just what this should mean, and how it can be recognized. Generally the evaluation of investment performance by PMs and TRs is difficult due to the amount of data required to arrive at statistically significant decisions. Since this is how such people earn a living, i.e., based on their performance records, separating investment skill from luck in a reasonable period of time is an ongoing challenge.
We can remark in passing on some similarity between empirical work in finance and in public health. Both involve samples from certain populations, and in each case there is the issue of how large and how representative is the sample. In finance, the sample typically contains data from a fixed time period, within which certain economic and political conditions hold, so that the relevance of conclusions drawn to another (future) time period is problematic.
There is another dissimilarity between FIM and the natural sciences, and that is the role of psychology. Market activity is often influenced by assorted human foibles, such as failure to engage in rational or coherent decision making, to learn from experience, and to overcome the common emotional instincts of hope, greed, fear, panic(!), overconfidence, and herd following. All these aspects of FIM have come to be bundled and studied in the field of Behavioral Finance. Although this is not really an area of applied mathematics, its depth and practical relevance in investment activity bears a message. On the one hand, it alerts us to the limits of a purely mathematical approach to the markets. Yet on the other hand it suggests that such an approach may be the best we can hope for, as it may save us from our otherwise emotional and erratic selves.
Another aspect of the financial markets is that they function as a large complex adaptive nonlinear system, in which participants interact based on their interpretations of observations, and on their goals. Unlike the assumptions of classical economic models, which enforce some kind of homogeneity on market players, in real life, this is not remotely valid. As we know, some TRs work in the short term, while some PMs have a virtual eternal time frame (because of the institutions that employ them). Also, some players utilize serious data processing, such as Bayesian learning, while others react to emotions and "hot tips". An appreciation of this view leads to alternate theoretical market models and also to the possibility of simulated or artificial market models that may capture some well-known but mysterious real behavior, such as volatility clustering and crashes. A description of some recent efforts in this direction is given in [8,9].
Now, to return to the notion of FIM as controversial, the idea is that the field consists of assorted models and prescriptions for action. Some of these will be discussed in the next section. As in the sciences, all models are approximations, derived either theoretically from some axioms or numerically via a curve-fitting procedure. But unlike science there is no natural law. Nevertheless, there is no lack of theoretical models in mathematical economics and finance, no lack of advice on how to make decisions in the face of uncertainty or how specifically how to structure investment portfolios, and certainly no lack of market data models. There are many approaches and recipes, but no agreement. How to define and measure risk, or personal utility, or goodness-of-fit and its tradeoff with model complexity, how much data from which time period to use in model validation, how (and whether!) to forecast, are all questions without unique correct answers. Market students and participants simply have too wide a range of life experience, goals, risk attitudes, time horizons, in-formational access, etc. to permit any easy agreement. Of course, it is the resultant inevitable differences of opinions and purposes that make a market exist at all!
Mathematicians in FIM
Let's now look at the role of mathematicians in this business. We'll concentrate on the non-academic side as it's unlikely that most mathematicians will either want to, or be able to, become finance professors, at least without extensive further schooling. For completeness we might note that there is also a small amount of activity by math professors in mathematical finance; for a sample of such research see the article [10] and/or peruse the journal Mathematical Finance. This latter activity tends to focus on relatively arcane matters, such as extending previous theorems in finance to more abstract settings, and seems to be more math than finance.
But let's briefly examine what (some of) the PROFs do. Typical output is a series of gradually more rigorous theorems concerning what happens in an idealized economy, such as a capital market. "Idealized" means simply that certain specific definitions and assumptions are made about its operation and the behavior of agents acting within it. Normally these assumptions are intended to formalize some intuition gleaned from economic theory and/or experience, although sometimes they are made for convenience, i.e., from them something can be proved, such as an asset pricing model. The most famous example here is the Black/Scholes option pricing formula, which was derived back in 1973 [11], from an assumption about the stochastic nature (specifically, geometric Brownian motion with constant drift and volatility) of the underlying stock price, and several other idealized assumptions concerning the operation of the market (e.g., constant interest rate, no transactions costs, etc.). The resulting formula has survived and prospered for a quarter century, and spawned a huge industry of derivative pricing.1 Generally, of course, as in all sciences, theoretically derived models have to be tested against empirical data.
The mathematics of modern academic finance is usually a mix of advanced probability theory, such as continuous-time stochastic processes and integration, and functional analysis. The latter comes into play when defining the space of assets, which are available to investors for purchase or sale, and which are usually modeled as (sub)spaces of random variables on some probability space (W ,S,P). The underlying set W is thought of as the "states of the world", i.e., all conceivable relevant outcomes at some future time, after the investment decision is taken. Of course, the exact specification of W is something of an art, and heavily affects the ensuing mathematics. Quantizing future reality to the point that W is finite is the simplest case, and permits interesting conclusions just from linear algebraic dualities, such as Farkas' lemma or, more abstractly, the Bipolar Theorem. An intermediate case is when W is an interval, say [0,¥ ), representing possible prices of an asset at some fixed future time T. An advanced case is when W is the space of continuous functions on some interval [0,T], with fixed value at 0, representing possible price trajectories of an asset, and P is the corresponding Wiener measure. Anyway, the assets, thought of as payoffs on W , are next modeled as belonging to some function space X over W . This could be one of the Banach spaces Lp(W ) for some choice of p ³ 1, or perhaps the Banach lattice C(W ), if W is taken to be a compact metric space. Depending on this choice of asset space, and the specific problem under study, various standard and not-so-standard extension and separation theorems (Hahn-Banach, Krein-Rutman, Kantorovich, Kreps-Yan, etc.) and duality theorems are invoked. A very common procedure, which opens the door to all this mathematical power, is some form of a no-arbitrage assumption. This is a way of getting into the analysis the intuition that in a viable economy there should not exist costless ways to make a riskless profit. A common goal of all this work is to see how to price some asset in X in terms of some other assets, the prototypical case being that of Black & Scholes, wherein a call option is priced in terms of a stock and a (riskless) bond.
Let's move on to the FEs. Typically these work for large investment houses or trading companies, in concert with PMs or TRs. They too use advanced math, perhaps even more so than the PROFs. In addition to what the latter use, FEs might utilize PDE models, hoping for analytic solutions but willing to resort to numerical methods, and/or simulation procedures. In the numerics there is often the need to trade off speed and accuracy. Generally an FE is trying to construct a trading product from some design specification submitted by a PM or TR. With modern techniques there is a drive to automate the process into what might be thought of as "financial CAD".
Part of the value of a mathematician (or even a physicist!) in the FE business is just their general training as problem organizers and solvers. They can think in a disciplined manner and work in reference to fundamental principles. This may not quite be the case with graduates of CS departments or business schools, although it is admittedly risky to make such generalizations.
There may be a bit of culture shock in transitioning from math/physics to the FE profession. There is not so much emphasis or even encouragement to publish. The aim, as already mentioned, is some kind of operational success, and this is often a proprietary matter. That is, an FE's accomplishments are intended to benefit his company and not usually other FEs. There is also often the practical problem of "selling" one's solution of some problem to a manager who may be technically less advanced and/or who may be concerned about maintaining control of his domain. For some discussion of personal experiences in transitioning from a science to a finance career, see [12]; also see [13], written by one who successfully made this transition, which illustrates some principles of financial modeling.
Finally, we want to consider the doers, as opposed to the thinkers, i.e., the PMs and TRs. These are the people who have to translate all the theory and models into investment or trading strategies. But they face a further difficulty, that of competitive pressure and consequent emotional stress. All institutional money managers are subject to strict oversight at least on a quarterly basis, while the thousands of mutual fund managers, of both open and closed end funds, have their changes in net asset value publicly posted on a daily basis. So it is in the interest of both the manager and the sponsor to carefully agree on just what the fund's goal is, and how it will be determined if that goal is being achieved. Usually such judgments are made relative to something. This "something" can be a benchmark, such as an index, or else a performance universe, i.e., a collection of other funds which are alternative possible investments, and about which it is agreed that they are following similar strategies and, in particular, share a similar risk level. There are some mathematical problems in this evaluation process of attempting to separate skill from luck, due to the need to reach a judgment in less time, say 1-5 years, than permits a statistically significant decision.
By contrast, the TRs usually operate in a somewhat different environment, although it too creates pressures. Typically a TR is trying to achieve some high level of absolute, as opposed to relative performance. TRs may work for their own account, for a hedge fund or investment club, or for a large brokerage or investment bank. They are flexible and always looking for an edge in short-term forecasting of certain markets. Today's TRs are highly computerized and usually employ some sort of advanced mathematical technology, such as nonlinear statistical models (artificial neural networks - backprop or RBF, dynamical systems and time-delay embedding), wavelet theory, pattern recognition, whatever might work. (Here's just one example, of interest to PMs too: high dimensional pattern recognition can be used to decide which attributes of a bunch of stocks are currently being rewarded by the market. Of course, everything here has to be defined. Thus we might say that "rewarded" means being in the top 10% of performance during the last quarter. We might also say that an "attribute" is some observable such as company size or p/e ratio, and we might hope that our PR algorithm could pick out profitable ranges of at least some attributes.) But it's not just a case of better and faster methods; they have to be adapted to the trading profits goal. For example, in tracking some tradable security, is it enough to be able to predict its future direction with some level of accuracy? Is getting up-or-down correct, say 55% of the time, good enough? (in fact, 60% has been achieved in some markets.) How confident in this sort of result does the TR have to be before real money will be risked on the next prediction? And how much money? That is, in addition to the prediction problem, there is the money management problem, which relates to the theory of optimal gambling, hence more mathematics [14].
As a brief aside, let's try to clarify the distinction between gambling and investment. Generally, gambling involves the seeking or creation of risk, in the hope of an unlikely but large payoff, while investors are concerned with the control of risk, perhaps by diversification or transfer (insurance). The relation between TRs and gambling just mentioned above is that a TR presumably has made an assessment of the payoff probabilities of whatever method is being employed, say by back testing or by real-time experience, and the problem is how much of an existing stake is to be "bet" on each signal given by the trading method. This situation bears some analogy to that of betting on a horse race, where there are several horses with known odds. It is frequently the case that the correct fraction of current wealth to bet is a rather sharp maximum, so that even small deviations from this can cause a serious drop in long term expected gain. A nice application of applied probability!
Let's take a further look at the PM business. Their basic problem, viewed abstractly, is that of decision making under uncertainty. They begin with some cash, and usually an ongoing but random cash flow, and have to decide how to allocate it into various asset classes and (then into) securities, of which there is a huge number. These securities can be viewed (by a mathematician!) as random variables, i.e., they generate future uncertain payoffs, although some possibilities such as Treasury bills are effectively deterministic. When, by whatever method, the available funds have been allocated, the PM has a bunch of securities collectively called a portfolio, and the question arises as to how two such portfolios can be compared, with the intent to see if an optimization is eventually possible. A great part of the problem is that the probability distributions of the various securities are not known, and can at best be rather roughly estimated. (In fact, this problem is intrinsic to the data: the signal-to-noise ratio in typical monthly data is on the order of -5dB, and in daily data is on the order of -15dB, although there's a range here, as stocks have widely varying volatilities.) So now we can begin to glimpse several mathematical tasks: estimating the future price distribution of some security from some historical data (how much?) plus whatever other information is deemed relevant, estimating the joint distribution, deciding on a portfolio comparison, and then doing some optimization to pick out a good (optimal?) portfolio. Whether or not this appears easy, it isn't! The most straightforward part is the optimization. Even here there are difficulties in that the underlying problem, once it has been formulated, is usually large, so that there are issues of time and accuracy. Still worse is the fact that optimization is too precise for the accuracy with which the problem parameters can be known. So there is the matter of solution robustness.
As another brief aside we should mention the powerful role of Bayesian statistics in assisting the PM with the estimation problem. Particularly we have in mind the use of the predictive distribution and of empirical Bayes methods. For example, in parametric modeling of a price distribution, rather than use the likelihood function with estimated parameters, rather use the predictive distribution, which results from integrating the likelihood against the posterior parameter distribution. This permits use of any prior information deemed relevant. In this context empirical Bayes can be used to produce informative priors [15,16].
The issue of portfolio comparison and selection is fascinating. It is one thread in a huge tapestry of decision- making- under- uncertainty studies that have occupied economists and psychologists for the past 50 years, since [17]. Actually, earlier efforts trace back another couple of hundred years, to the initial realization that decisions involve more than just the opportunity for gain, they involve the risk of loss. From this sprang the idea of a utility function and eventually the principle of maximizing expected utility as a decision paradigm. This in turn has been under various assaults recently, with assorted defenses and modifications proposed. The principle itself is actually a theorem, just as are many other bits of FIM such as CAPM and Bayes' rule; hence the issue is how realistic and descriptive are the axioms and hypotheses invoked to arrive at the conclusion. Since these always involve human behavior and understanding, they are endlessly controversial. (If experimental subjects fail to act as the paradigm predicts, are they reasoning in a faulty manner and need to be corrected, or is there some failure in the axioms to fully capture human nature?)
Anyway, many quantitatively inclined PMs who take the view of securities as random variables, consider in particular just the first two moments of the corresponding return distributions, with the intuition that higher means and lower variances are desirable. This leads to a methodology known as Modern Portfolio Theory (MPT), originated by Markowitz [18]2, wherein one tries to construct efficient portfolios, i.e., those that have the highest possible mean for a given level of variance. But MPT cannot tell a PM what level of variance, viewed as risk, is appropriate; for that the PM needs some form of decision theory, at the least a personal utility function. A simple, yet common, form of utility function
U(r) = E(r) - a var(r) - b var(TE),
where r = portfolio return, relative to some time period, var(r) is its variance, and TE is the tracking error which is the difference between r and a benchmark return. The positive constants a and b reflect the relative importance of the two types of variability (but how are they to be chosen?). In classical MPT the TE term is omitted. There might also be a (negative) cost term in U. It should be noted that there is more to MPT than just the search for efficient portfolios. There is also the background of an overall Efficient Market and the CAPM, and the use of a "beta coefficient" to capture the idea of risk of a security relative to that of the overall market.
The task of keeping TE small is interesting. Suppose the benchmark is the S&P500 index; this is actually quite common. So, there are 500 stocks to choose from. Suppose further that there are size restrictions on the possible portfolios, say to N of these stocks, where N might be on the order of 20-50. Which subset of N do you choose? Even with N = 20 the search space has order of magnitude 10**35! And what fraction of the eventual portfolio does each of the selected stocks constitute? Is there reason to deviate from equal fractions? (Note that the S&P is not an equally weighted index).
Now let's consider what comes next, after the PM has somehow acquired an initial portfolio. The various security prices and prospects will bounce around, and new opportunities will appear, such as IPOs, as well as (hopefully!) new cash. At what point is something new added to the portfolio, and what is discarded to make room? Remember that each transaction has an associated cost. So the addition has to be some degree better than the worst current member.
It might be worth pointing out an analogy here with other kinds of assignment problems, such as those occurring in the military. There a commander has a number of assets, such as attack plus support aircraft, to be used against a list of targets of varying degrees of importance, and with various levels of defense. The aircraft therefore have varying likelihoods of success against the targets, and the problem is to make an assignment in some optimal fashion, where "optimal" has to be defined. This is the static version; a dynamic version occurs when there is an element of time involved, e.g., targets have to be attacked in a certain order and/or new targets can appear after an initial assignment has been made.
Summary and
Further DiscussionWe have partitioned the field of FIM into four areas where mathematics has significant impact - models, valuation, allocation, and evaluation, and the players into four groups - PROFs, FEs, PMs, and TRs. In this short survey we have omitted other branches of finance, such as corporate and managerial, and not really done justice to behavioral finance, although that can get pretty mathematical too (c.f. use of Choquet capacities and integrals in "Prospect Theory", a modern-day modification of expected utility theory).
We have several times alluded to controversies yet not mentioned one of long standing, that between the fundamentalists and the technicians. We have taken the view of the latter, people who view the markets mathematically and statistically. They see a stock, for example, as a stretch of price action, a random variable of returns determined by the actions of others over a holding period. By contrast, the fundamental view is that a stock represents ownership in a business, in an industry group, in an economy,..., which of course it does. In this view a stock's worth can be assessed by various accounting procedures, visits with company personnel, and macroeconomic analysis. And indeed, this is how many PMs believe and act. Naturally there is merit and validity to both sides. Our sympathies lie with the technical view for many reasons, besides the obvious one of having a mathematical world-view. A general argument is that any one individual is bound by information constraints (how much can be known by a single person?), and by all sorts of personal biases and foibles. But market activity reflects the knowledge and decisions of all PMs/TRs, and so nulls out individual shortcomings, hence reflects the true underlying fundamental picture. A paradoxical aspect here is that this view only works because of the large number of market participants who practice fundamental analysis. Other arguments against the fundamentalists are given in [5]. Granting these, we are left to decide between the technical PMs and the TRs, in the sense that, oversimplifying somewhat again, the former see the market as essentially random and hence practice a kind of stochastic risk control, while the latter see pockets of short-term predictability, and try to act on these quickly with carefully determined fractions of their trading stake. Note that even the PMs with a random market view expect to be successful over a long term because a) a growing economy pulls most businesses along with it and b) generally accepted statistical models of stock prices, such as geometric Brownian motion, with drift coefficient n , imply a future expected price greater than today's price, specifically, E(S(T)) = S(0)*exp(m T), where S(t) = stock price at time t, m = n + s2/2, and s2 is the local volatility (note > 0 even if drift = 0).
Mathematicians should keep in mind two familiar progressions which are reflected in common finance models. These are
- in measure theory, indicator functions à simple functions à measurable functions; and
- in probability theory, Bernoulli trials à binomial distribution à Central Limit Theorem à normal distribution à multivariate normal distribution à Functional CLT (a.k.a. Invariance Principle) à Brownian motion.
Progression a) can be used to build a unified view of option pricing models, starting with binary options, those which pay $1 if a certain event occurs and $0 otherwise. These are modeled by indicator functions on an appropriate state space, and off we go. Progression b) can be used to lead up to the statistical stock price model mentioned above, beginning with the simplest possible model, that where over the next time instant, the price either increases by a fixed percentage, or declines by a fixed percent, each with a given probability.
It is fascinating that Brownian motion, a "central object in mathematics" [19], and of course in probability theory, with all sorts of unexpected connections to complex and harmonic analysis, potential theory, PDEs, etc. [20], should also serve as the central (but not only!) model of stock price movements. More precisely, it models the fundamental uncertainty in the changes of log prices, an idea first conceived discretely and empirically by Osborne [21]. The move to the full continuous time model came gradually over the next 15 years. The subject is now known as Continuous Time Finance [22, 23] and is mainstream. One of the great virtues of the use of Brownian motion models is simply that so much information has accumulated about this topic, so that regardless of its validity as a literal description of reality, formulas and approximations can be derived, and simulations performed. In truth, the suitability of Brownian motion as a price model breaks down as the time interval à 0, because of well-known pathologies of the Brownian sample paths, such as almost-sure nowhere differentiability, unbounded variation, and hence infinite arc length on any finite interval. But in practice, the time interval between price observations is not arbitrarily short. Just how short depends on intended usage. For some purposes monthly or even quarterly data is adequate, and so is the normal model. For others, weekly, daily, or even hourly data may be relevant, and then other statistical models may be more accurate. Of course, this basic modeling issue is controversial, especially for high frequency data.
The word "simulation" was just mentioned. This is a powerful methodology in FIM to which mathematicians can contribute. There are three levels of issues here. First, what model is being used, Brownian motion or other? Second, given the model, how are we to simulate from it, i.e., how can a string of pseudo-random numbers be created on a computer that will pass some goodness-of-fit (relative to the selected model) and independence tests? What tests, and what level of satisfaction of these tests, are deemed sufficient to "bless" the simulation process? And third, what are the intended applications, on what problems do we expect a simulation approach to shed light?
Well, there are two likely areas that can benefit: derivative pricing and portfolio and trading rule studies. The first will be of interest to FEs, the second to PMs and TRs. A derivative is a security that derives its price from that of another, called the underlying. Standard put and call options on a stock or index, and commodity futures, are the simplest examples, but nowadays there are innumerable further examples. But the point is simply that once a model has been selected for the underlying, sample paths from it can be generated and the corresponding derivative value computed. A discounted average of these then gives an estimate of its current value. In practice this may not be quite so simple. There are issues with sampling (how many points determine a path?), how accurate is the simulation (variance reduction), how many terms for the average, etc.? Also, speed may be an issue, if the valuation needs to be done updated frequently for trading purposes. If so, there will be an accuracy/speed tradeoff.
The other area of simulation application is analogous to common every day applications such as flight simulators. These let prospective pilots try out an airplane's controls before risking a real flight, and generally build up an appreciation of what to expect in various circumstances. Back in the world of FIM it's possible for computer simulations to serve a similar purpose. Some specific examples are observing the time-evolution of portfolios constructed via particular rules, in particular, wealth effect build-up, the worth of time-diversification strategies, and the effect of differing choices of utility functions. Also, instead of being limited to specific empirical time histories on which to test strategies, simulation can be used to create large amounts of "statistically similar" data (a term that requires definition!), and the strategies can then be exposed to all this data for further evaluation. A final application is the assessment of various parameter estimation schemes as inputs to portfolio optimizers, it having already been noted that this is a notorious problem for PMs.
One final comment about the specification of basic probability models in finance. We noted earlier the use of an abstract "state space" to represent uncertain future events, and assets as (payoff) functions on the space. There are several kinds of complexity here which, as they are varied, create a hierarchy of models of varying degrees of difficulty and realism. This means in particular that mathematicians of varying backgrounds and levels of training may all have an ability to contribute. What can be varied are first, the nature of the state space itself, starting with the simplest case, a finite set, and moving on from there, as already noted; second, the number of trading periods, starting with one, then a finite number, and finally ending up in continuous time; third, the number of different values an asset have assume at the end of any one of these trading periods, starting with two, which leads to the binomial price model, and moving on from there, to trinomial models and eventually to the Brownian motion level; fourth, the number of different assets trading in the model economy, starting with one, and possibly getting very large, for some asymptotics, as in the APT; and finally, some models consider the number of agents or traders in the economy.
Recommendations
Having arrived at this point you should now have some intuition about the FIM business. New books and edited conference proceedings on math finance are appearing regularly now. Here are a few more examples, just in case you haven't found enough to do with the preceding references. For general treatments see Pliska [24] for discrete time and Karatzas [25] for continuous time. For derivative pricing, where much of the mathematical "action" is, see Baxter-Rennie [26], Neftci, [27], Wilmott et al. [28], and Chriss [29].
You may have noted other articles in the math literature, such as the two-part survey [30] and the discussion of option models in a recent Notices [31]. You may also have become aware that many AMS and/or SIAM meetings have math finance sections. For example, the upcoming 1998 SIAM Annual Meeting in Toronto has math finance as one of its "meeting themes", and includes a plenary presentation on derivative pricing. So, by investigating the references of this report, and by attending a meeting or two, you can get up to speed pretty quickly.
If you're still interested in a career move, consider your training and mathematical specialties, and just what you enjoy doing. As you should recognize by now there are lots of potential opportunities. Do you enjoy heavy-duty math, like stochastic processes, PDEs, or numerical analysis? Do you like statistics and applied probability? Are you trained in AI techniques such as neural or Bayesian nets, pattern recognition, etc.? Or are you happy working with computers, say writing programs and searching databases? We've already noted how most of these disciplines might be put to use in FIM, so depending on your special interest, you now have a sense of where to seek an entry point. Since recruiting visits by banks, brokerage and investment houses are common nowadays on many campuses, you'll have opportunities to make contact and present a well-informed and focused impression. Good luck!
A last thought...lots of folks learn about the practicalities of investments, if not high finance, by doing it. A typical and usually pleasant method is to organize an investment club. So you might try formulating an initial investment philosophy, always subject to revision, and pool some time and capital with a few kindred spirits. The world is full of experiments of this nature with happy outcomes. As the "Beardstown Ladies" proved, you don't have to be a mathematician to become a successful and even famous investor, but we can hope that at least it doesn't hurt!
Footnotes
In fact, the 1997 Nobel prize in economics was awarded for this work. Also, recently Gulko [32], using the method of maximum entropy estimation, has rederived the classic option pricing formulas without the stringent assumptions on the stochastic behavior of the underlying, and on the associated trading.
This work shared the 1990 Nobel prize.
References
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Math. Ann. 300(1994), 463-520.
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Princeton Univ. Press, 1947.
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Bull. London Math. Soc. 29(1997), 257-279.
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[22] R. Merton, Continuous Time Finance, Oxford, 1990.
[23] D. Duffie, Dynamic Asset Pricing Theory, 2nd ed.,Princeton Univ. Press, 1996.
[24] S. Pliska, Introduction to Mathematical Finance, Blackwell, 1997.
[25] I. Karatzas, Lectures on the Mathematics of Finance, Amer. Math. Soc., 1997.
[26] M. Baxter and A. Rennie, Financial Calculus, Cambridge Univ. Press, 1996.
[27] S. Neftci, Introd. to the Mathematics of FinancialDerivatives, Academic Press, 1996.
[28] P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives,
Cambridge Univ. Press, 1995.
[29] N. Chriss, Black-Scholes and Beyond, Irwin, 1997.
[30] The mathemization of finance, SIAM News (July-Aug., 1994).
[31] J. Price, Optional mathematics is not optional, Not. AMS 43(1996), 964-971.
[32] L. Gulko, Dart boards and asset prices: introducing the entropy pricing theory,
in Advances in Econometrics, vol. 12, JAI Press, 1997, pp. 237-276.
December, 1997