This post is an edited version of my responses in a Webinar panel-discussion organized by Wiley-Finance and FinCAD. The freely available Webcast is linked in the post, and contains responses from the other participants — Paul Wilmott and Espen Huag. An expanded version of this post may later appear as an article in the Wilmott Magazine.

What is Risk?

When we use the word Risk in normal conversation, it has a negative connotation — risk of getting hit by a car, for instance; but not the risk of winning a lottery. In finance, risk is both positive and negative. At times, you want the exposure to a certain kind of risk to counterbalance some other exposure; at times, you are looking for the returns associated with a certain risk. Risk, in this context, is almost identical to the mathematical concept of probability.

But even in finance, you have one kind of risk that is always negative — it is Operational Risk. My professional interest right now is in minimizing the operational risk associated with trading and computational platforms.

How do you measure Risk?

Measuring risk ultimately boils down to estimating the probability of a loss as a function of something — typically the intensity of the loss and time. So it’s like asking — What’s the probability of losing a million dollars or two million dollars tomorrow or the day after?

The question whether we can measure risk is another way of asking whether we can figure out this probability function. In certain cases, we believe we can — in Market Risk, for instance, we have very good models for this function. Credit Risk is different story — although we thought we could measure it, we learned the hard way that we probably could not.

The question how effective the measure is, is, in my view, like asking ourselves, “What do we do with a probability number?” If I do a fancy calculation and tell you that you have 27.3% probability of losing one million tomorrow, what do you do with that piece of information? Probability has a reasonable meaning only a statistical sense, in high-frequency events or large ensembles. Risk events, almost by definition, are low-frequency events and a probability number may have only limited practical use. But as a pricing tool, accurate probability is great, especially when you price instruments with deep market liquidity.

Innovation in Risk Management.

Innovation in Risk comes in two flavors — one is on the risk taking side, which is in pricing, warehousing risk and so on. On this front, we do it well, or at least we think we are doing it well, and innovation in pricing and modeling is active. The flip side of it is, of course, risk management. Here, I think innovation lags actually behind catastrophic events. Once we have a financial crisis, for instance, we do a post-mortem, figure out what went wrong and try to implement safety guards. But the next failure, of course, is going to come from some other, totally, unexpected angle.

What is the role of Risk Management in a bank?

Risk taking and risk management are two aspects of a bank’s day-to-day business. These two aspects seem in conflict with each other, but the conflict is no accident. It is through fine-tuning this conflict that a bank implements its risk appetite. It is like a dynamic equilibrium that can be tweaked as desired.

What is the role of vendors?

In my experience, vendors seem to influence the processes rather than the methodologies of risk management, and indeed of modeling. A vended system, however customizable it may be, comes with its own assumptions about the workflow, lifecycle management etc. The processes built around the system will have to adapt to these assumptions. This is not a bad thing. At the very least, popular vended systems serve to standardize risk management practices.

Featuring Paul Wilmott, Espen Haug and Manoj Thulasidas

PLEASE JOIN US FOR THIS FREE WEBINAR PRESENTED BY FINCAD AND WILEY GLOBAL FINANCE

How do you identify, measure and model risk, and more importantly, what changes need to be implemented to improve the long-term profitability and sustainability of our financial institutions? Take a unique opportunity to join globally recognised and respected experts in the field, Paul Wilmott, Espen Haug and Manoj Thulasidas in a free, one hour online roundtable discussion to debate the key issues and to find answers to questions to improve financial risk modelling.

Join our experts as they address these fundamental financial risk questions:

• What is risk?
• How do we measure and quantify risk in quantitative finance? Is this effective?
• Is it possible to model risk?
• Define innovation in risk management. Where does it take place? Where should it take place?
• How do new ideas see the light of day? How are they applied to the industry, and how should they be applied?
• How is risk management implemented in modern investment banking? Is there a better way?

Our panel of internationally respected experts include Dr Paul Wilmott, founder of the prestigious Certificate in Quantitative Finance (CQF) and Wilmott.com, Editor-in-Chief of Wilmott Magazine, and author of highly acclaimed books including the best-selling Paul Wilmott On Quantitative Finance; Dr Espen Gaarder Haug who has more than 20 years of experience in Derivatives research and trading and is author of The Complete Guide of Option Pricing Formulas and Derivatives: Models on Models; and Dr Manoj Thulasidas, a physicist-turned-quant who works as a senior quantitative professional at Standard Chartered Bank in Singapore and is author of Principles of Quantitative Development.

This debate will be critical for all chief risk officers, credit and market risk managers, asset liability managers, financial engineers, front office traders, risk analysts, quants and academics.

Physics also can provide beautiful semantic contexts to the empty formalisms of advanced mathematics. Look at Minkowski space and Riemannian geometry, for instance, and how Einstein turned them into descriptions of our perceived reality. In addition to providing semantics to mathematical formalism, science also promotes a worldview based on critical thinking and a ferociously scrupulous scientific integrity. It is an attitude of examining one’s conclusions, assumptions and hypotheses mercilessly to convince oneself that nothing has been overlooked. Nowhere is this nitpicking obsession more evident than in experimental physics. Physicists report their measurements with two sets of errors — a statistical error representing the fact that they have made only a finite number of observations, and a systematic error that is supposed to account for the inaccuracies in methodology, assumptions etc.

We may find it interesting to look at the counterpart of this scientific integrity in our neck of the woods — quantitative finance, which decorates the syntactical edifice of stochastic calculus with dollar-and-cents semantics, of a kind that ends up in annual reports and generates performance bonuses. One might even say that it has a profound impact on the global economy as a whole. Given this impact, how do we assign errors and confidence levels to our results? To illustrate it with an example, when a trading system reports the P/L of a trade as, say, seven million, is it $7,000,000 +/- $5,000,000 or is it $7,000, 000 +/- $5000? The latter, clearly, holds more value for the financial institution and should be rewarded more than the former. We are aware of it. We estimate the errors in terms of the volatility and sensitivities of the returns and apply P/L reserves. But how do we handle other systematic errors? How do we measure the impact of our assumptions on market liquidity, information symmetry etc., and assign dollar values to the resulting errors? If we had been scrupulous about error propagations of this, perhaps the financial crisis of 2008 would not have come about.

Although mathematicians are, in general, free of such critical self-doubts as physicists — precisely because of a total disconnect between their syntactical wizardry and its semantic contexts, in my opinion — there are some who take the validity of their assumptions almost too seriously. I remember this professor of mine who taught us mathematical induction. After proving some minor theorem using it on the blackboard (yes it was before the era of whiteboards), he asked us whether he had proved it. We said, sure, he had done it right front of us. He then said, “Ah, but you should ask yourselves if mathematical induction is right.” If I think of him as a great mathematician, it is perhaps only because of the common romantic fancy of ours that glorifies our past teachers. But I am fairly certain that the recognition of the possible fallacy in my glorification is a direct result of the seeds he planted with his statement.

My professor may have taken this self-doubt business too far; it is perhaps not healthy or practical to question the very backdrop of our rationality and logic. What is more important is to ensure the sanity of the results we arrive at, employing the formidable syntactical machinery at our disposal. The only way to maintain an attitude of healthy self-doubt and the consequent sanity checks is to jealously guard the connection between the patterns of reality and the formalisms in mathematics. And that, in my opinion, would be the right way to develop a love for math as well.

My point is that it is my love for math that lets me do a large number of things fairly well. As a writer, for instance, I have done rather well. But I attribute my success to a certain mathematical ability rather than literary talent.  I would never start a book with something like, “It was the best of times, it was the worst of times.” As an opening sentence, by all the mathematical rules of writing I have formulated for myself, this one just doesn’t measure up. Yet we all know that Dickens’s opening, following no rules of mine, is perhaps the best in English literature. I will probably cook up something similar someday because I see how it summarizes the book, and highlights the disparity between the haves and the have-nots mirrored in the contrasting lead characters and so on. In other words, I see how it works and may assimilate it into my cookbook of rules (if I can ever figure out how), and the process of assimilation is mathematical in nature, especially when it is a conscious effort. Similar fuzzy rule-based approaches can help you be a reasonably clever artist, employee, manager or anything that you set your sights on, which is why I once bragged to my wife that I could learn Indian classical music despite the fact that I am practically tone-deaf.

So loving math is a probably a good thing, in spite of its apparent disadvantage vis-a-vis cheerleaders. But I am yet to address my central theme — how do we actively encourage and develop a love for math among the next generation? I am not talking about making people good at math; I’m not concerned with teaching techniques per se. I think Singapore already does a good job with that. But to get people to like math the same way they like, say, their music or cars or cigarettes or football takes a bit more imagination. I think we can accomplish it by keeping the underlying patterns on the foreground. So instead of telling my children that 1/4 is bigger than 1/6 because 4 is smaller than 6, I say to them, “You order one pizza for some kids. Do you think each will get more if we had four kids or six kids sharing it?”

From my earlier example on geographic distances and degrees, I fancy my daughter will one day figure out that each degree (or about 100km — corrected by 5% and 6%) means four minutes of jet lag. She might even wonder why 60 appears in degrees and minutes and seconds, and learn something about number system basis and so on. Mathematics really does lead to a richer perspective on life. All it takes on our part is perhaps only to share the pleasure of enjoying this richness. At least, that’s my hope.

I was always among people who liked math. I remember my high school days where one of my friends would do the long multiplication and division during physics experiments, while I would team up with another friend to look up logarithms and try to beat the first dude, who almost always won. It didn’t really matter who won; the mere fact that we would device games like that as teenagers perhaps portended a cheerleader-less future. As it turned out, the long-multiplication guy grew up to be a highly placed banker in the Middle East, no doubt thanks to his talents not of the cheerleader-phobic, math-phelic kind.

When I moved to IIT, this mathematical geekiness reached a whole new level. Even among the general geekiness that permeated the IIT air, I remember a couple of guys who stood out. There was “Devious” who also had the dubious honor of introducing me to my virgin Kingfisher, and “Pain” would drawl a very pained “Obviously Yaar!” when we, the lesser geeks, failed to readily follow a his particular line of mathematical acrobatics.

All of us had a love for math. But, where did it come from? And how in the world would I make it a general educational tool? Imparting the love math to one kid is not too difficult; you just make it fun. The other day when I was driving around with my daughter, she described some shape (actually the bump on her grandmother’s forehead) as half-a-ball. I told her that it was actually a hemisphere. Then I highlighted to her that we were going to the southern hemisphere (New Zealand) for our vacation the next day, on the other side of the globe compared to Europe, which was why it was summer there. And finally, I told her Singapore was on the equator. My daughter likes to correct people, so she said, no, it wasn’t. I told her that we were about 0.8 degrees to the north of the equator (I hope I was right), and saw my opening. I asked her what the circumference of a circle was, and told her that the radius of the earth was about 6000km, and worked out that we were about 80km to the north of the equator, which was nothing compared to 36,000km great circle around the earth. Then we worked out that we made a 5% approximation on the value of pi, so the correct number was about 84km. I could have told her we made another 6% approximation on the radius, the number would be more like 90km. It was fun for her to work out these things. I fancy her love for math has been augmented a bit.

Looking back at these writings of mine, I feel as though I may have been a bit unfair on us. I did try to blunt my accusations of avarice (and perhaps decadence) by pointing out that it was the general air of insatiable greed of the era that we live in that spawned the obscenities and the likes of Madoff. But I did concede the existence of a higher level of greed (or, more to the point, a more sated kind of greed) among us bankers and quantitative professionals. I am not recanting my words in this piece now, but I want to point out another aspect, a justification if not an absolution.

Why would I want to defend bonuses and other excesses when another wave of public hatred is washing over the global corporations, thanks to the potentially unstoppable oil spill? Well, I guess I am a sucker for lost causes, much like Rhett Butler, as our quant way of tranquil life with insane bonuses is all but gone with the wind now. Unlike Mr. Butler, however, I have to battle and debunk my own arguments presented here previously.

One of the arguments that I wanted to poke holes in was the fair compensation angle. It was argued in our circles that the fat paycheck was merely an adequate compensation for the long hours of hard work that people in our line of work put in. I quashed it, I think, by pointing out other thankless professions where people work harder and longer with no rewards to write home about. Hard work has no correlation with what one is entitled to. The second argument that I made fun of was the ubiquitous “talent” angle. At the height of the financial crisis, it was easy to laugh off the talent argument. Besides, there was little demand for the talent and a lot of supply, so that the basic principle of economics could apply, as our cover story shows in this issue.

Of all the arguments for large compensation packages, the most convincing one was the profit-sharing one. When the top talents take huge risks and generate profit, they need to be given a fair share of the loot. Otherwise, where is the incentive to generate even more profits? This argument lost a bit of its bite when the negative profits (by which I indeed mean losses) needed to be subsidized. This whole saga reminded me of something that Scott Adams once said of risk takers. He said that risk takers, by definition, often fail. So do morons. In practice, it is hard to tell them apart. Should the morons reap handsome rewards? That is the question.

Having said all this in my previous articles, now it is time to find some arguments in our defense. I left out one important argument in my previous columns because it did not support my general thesis — that the generous bonuses were not all that justifiable. Now that I have switched allegiance to the lost cause, allow me to present it as forcefully as I can. In order to see compensation packages and performance bonuses in a different light, we first look at any traditional brick-and-mortar company. Let’s consider a hardware manufacturer, for instance. Suppose this hardware shop of ours does extremely well one year. What does it do with the profit? Sure, the shareholders take a healthy bite out of it in terms of dividends. The employees get decent bonuses, hopefully. But what do we do to ensure continued profitability?

We could perhaps see employee bonuses as an investment in future profitability. But the real investment in this case is much more physical and tangible than that. We could invest in hardware manufacturing machinery and technology improving the productivity for years to come. We could even invest in research and development, if we subscribe to a longer temporal horizon.

Looking along these lines, we might ask ourselves what the corresponding investment would be for a financial institution. How exactly do we reinvest so that we can reap benefits in the future?

We can think of better buildings, computer and software technologies etc. But given the scale of the profits involved, and the cost and benefit of these incremental improvements, these investments don’t measure up. Somehow, the impact of these tiny investments is not as impressive in the performance of a financial institution compared to a brick-and-mortar company. The reason behind this phenomenon is that the “hardware” we are dealing with (in the case of a financial institution) is really human resources — people — you and me. So the only sensible reinvestment option is in people.

So we come to the next question — how do we invest in people? We could use any number of euphemistic epithets, but at the end of the day, it is the bottom line that counts. We invest in people by rewarding them. Monetarily. Money talks. We can dress it up by saying that we are rewarding performance, sharing profits, retaining talents etc. But ultimately, it all boils down to ensuring future productivity, much like our hardware shop buying a fancy new piece of equipment.

Now the last question has to be asked. Who is doing the investing? Who benefits when the productivity (whether current or future) goes up? The answer may seem too obvious at first glance — it is clearly the shareholders, the owners of the financial institution who will benefit. But nothing is black and white in the murky world of global finance. The shareholders are not merely a bunch of people holding a piece of paper attesting their ownership. There are institutional investors, who mostly work for other financial institutions. They are people who move large pots of money from pension funds and bank deposits and such. In other words, it is the common man’s nest egg, whether or not explicitly linked to equities, that buys and sells the shares of large public companies. And it is the common man who benefits from the productivity improvements brought about by investments such as technology purchases or bonus payouts. At least, that is the theory.

This distributed ownership, the hallmark of capitalism, raises some interesting questions, I think. When a large oil company drills an unstoppable hole in the seabed, we find it easy to direct our ire at its executives, looking at their swanky jets and other unconscionable luxuries they allow themselves. Aren’t we conveniently forgetting the fact that all of us own a piece of the company? When the elected government of a democratic nation declares war on another country and kills a million people (speaking hypothetically, of course), should the culpa be confined to the presidents and generals, or should it percolate down to the masses that directly or indirectly delegated and entrusted their collective power?

More to the point, when a bank doles out huge bonuses, isn’t it a reflection of what all of us demand in return for our little investments? Viewed in this light, is it wrong that the taxpayers ultimately had to pick up the tab when everything went south? I rest my case.

There is only a thin blurry line between the schemes that thrive on other people’s greed and confidence jobs. If you can come up with a scheme that makes money for others, and stay legal (if not moral), then you will make yourself very rich. We see it most directly in the finance and investment industry, but it is much more widespread than that. We can see that even education, traditionally considered a higher pursuit, is indeed an investment against future earnings. Viewed in that light, you will understand the correlation between the tuition fees at various schools and the salaries their graduates command.

When I started writing this column, I thought I was making up this new field called the Philosophy of Money (which, hopefully, somebody would name after me), but then I read up something on the philosophy of mind by John Searle. It turned out that there was nothing patentable in this idea, nor any cash to be made, sadly. Money comes under the umbrella of objective social realities that are quite unreal. In his exposition of the construction of social reality, Searle points out that when they give us a piece of paper and say that it is legal tender, they are actually constructing money by that statement. It is not a statement about its attribute or characteristics (like “This is a glass of water”) so much as a statement of intentionality that makes something what it is (like “You are my hero”). The difference between my being a hero (perhaps only to my six-year-old) and money being money is that the latter is socially accepted, and it is as objective a reality as any.

I conclude this article with the nagging suspicion that I may not have argued my point well enough. I started it with the premise that money is an unreal meta-thing, and wound up asserting its objective reality. This ambivalence of mine may be a reflection of our collective love-hate relationship with money – perhaps not such a bad way to end this column after all.

Put it that way, investment sounds like a positive concept, which it is, in our current mode of thinking. We can easily make it a negative thing by portraying the demand for the investment value of money as greed. It then follows that all of us are greedy, and that it is our greed that fuels the insane compensation packages of top-level executives. Greed also fuels fraud – ponzi and pyramid schemes.

Indeed, any kind of strong feeling that you have can be bought and sold for personal gain of others. It may be your genuine sympathy for the Tsunami or earthquake victims, your voyeuristic disgust at the peccadilloes of golf icons or presidents, charitable feeling toward kidney patients of whatever. And the way money is made out of your feelings may not be obvious at all. Watching the news five minutes longer than usual because of a natural disaster may bring extra fortune to the network’s coffers. But of all the human frailties one can make money out of, the easiest is greed, I think. Well, I may be wrong; it may actually be that frailty that engendered the oldest profession. But I would think that the profession based on the lucrative frailty of greed wasn’t all that far behind.

If we want to exploit other people’s greed, the first thing to ask ourselves is this: why do we want money, given that it is a meta-entity? I know, we all need money to live. But I am not talking about the need part. Assuming the need part is taken care of, we still want more of it. Why? Say you are a billionaire. Why would you want another billion? I think the answer lies in something philosophical, something of an existential angst, although those with their billions would the last ones to admit it. The reason behind this deep-rooted need for more is a quest for a validation, or a justification for our existence, and a meaning and purpose for our life. It is all part of that metaphorical holy grail. I know, it sounds a bit nutty, but what else could it be? The Des Cartes of our time would say, “I have loads of money, therefore I am!”

To be fair, this column won’t give you any get-rich-quick, sure-fire schemes or strategies. What it will tell you is why and how some people make money, and hopefully uncover some new insights. You may be able to put some of these insights to work and make yourself rich – if that’s where you think your happiness lies.

By now, it is clear to most people that they cannot become filthy rich by working for somebody else. In fact, that statement is not quite accurate. CEOs and top executives all work for the shareholders of the companies that employ them, but are filthy rich. At least, some of them are. But, in general, it is true that you cannot make serious money working in a company, statistically speaking.

Working for yourself – if you are very lucky and extremely talented – you may make a bundle. When we hear the word “rich,” the people that come to mind tend to be

1. entrepreneurs/industrialists/software moguls – like Bill Gates, Richard Branson etc.,
2. celebrities – actors, writers etc.,
3. investment professionals – Warren Buffet, for instance, and
4. fraudsters of the Madoff school.

There is a common thread that runs across all these categories of rich people, and the endeavors that make them their money. It is the notion of scalability. To understand it well, let’s look at why there is a limit to how much money you can make as a professional. Let’s say you are a very successful, highly-skilled professional – say a brain surgeon. You charge $10k a surgery, of which you perform one a day. So you make about $2.5 million a year. Serious money, no doubt. How do you scale it up though? By working twice as long and charging more, may be you can make $5 million or $10 million. But there is a limit you won’t be able to go beyond.

The limit comes about because the fundamental economic transaction involves selling your time. Although your time may be highly-skilled and expensive, you have only 24 hours of it in a day to sell. That is your limit.

Now take the example of, say, John Grisham. He spends his time researching and writing his best-selling books. In that sense, he sells his time as well. But the big difference is that he sells it to many people. And the number of people he sells his product to may have an exponential dependence on its quality and, therefore, the time he spends on it.

We can see a similar pattern in software products like Windows XP, performances by artists, sports events, movies and so on. One performance or accomplishment is sold countless times. With a slight stretch of imagination, we can say that entrepreneurs are also selling their time (that they spend setting up their businesses) multiple times (to customers, clients, passengers etc.) All these money-spinners work hard to develop some kind of exponential volume-dependence on the quality of their products or the time they spend on them. This is the only way to address the scalability issue that comes about due to the paucity of time.

Investment professionals (bankers) do it too. They develop new products and ideas that they can sell to the masses. In addition, they make use of a different aspect of money that we touched upon in an earlier column. You see, money has a transactional value. It plays the role of a medium facilitating economic exchanges. In financial transactions, however, money becomes the entity that is being transacted. Financial systems essentially move money from savings and transforms it into capital. Thus money takes on an investment value, in addition to its intrinsic transactional value. This investment value is the basis of interest.

The first indication of the meta-ness of money comes from the fact that it has a value only when we assign it a value. It doesn’t possess an intrinsic value that, for instance, water does. If you are thirsty, you find that water has enormous intrinsic value. Of course, if you have money, you can buy water (or Perrier, if you want to be sophisticated), and quench your thirst.

But we may find ourselves in situations where we may not be able to buy things with money. Stranded in a desert, for instance, dying of thirst, we may not be able to buy water despite our sky-high credit limits or the hundreds of dollars we may have in our wallet. One reason for this inability of ours is obvious – we may be alone. The basic transactional value of money evaporates when we have nobody to transact with.

The second dimension of the meta-ness of money is economical. It is illustrated in the well-worn supply-and-demand principle, assuming transactional liquidity (which is a term I just cooked up to sound erudite, I confess). I mean to say, even if we have willing sellers of water in the desert, they may see that we are dying for it and jack up the price – just because we are willing and able to pay. This apparent ripping off on the part of the devious vendors of water (perfectly legal, by the way) is possible only if the commodity in question is in plentiful supply. We need commodity liquidity, as it were.

It is when the liquidity dries up that the fun begins. The last drop of water in a desert has infinite intrinsic value. This effect may look similar to the afore-mentioned supply-and-demand phenomenon, but it really is different. The intrinsic value dominates everything else, much like the strong force over short distances in particle physics. And this domination is the flipside of the law of diminishing marginal utility in economics.

The thing that looks a bit bizarre about money is that it seems to run counter to the law of diminishing marginal utility. The more money you have, the more you want it. Now, why is that? It is especially strange given its lack of intrinsic value. Great financial minds could not figure it out, but came up with pithy and memorable statements like, “Greed, for lack of a better word, is good.” Although that particular genius was only fictional, he does epitomize much of the thinking in the modern corporate and financial world. Good or bad, let’s assume that greed is an essential part of human nature and look at what we can do with it. Note that I want to do something “with” it, not “about” it – an important distinction. I, intrepid columnist that I am, want to show you how to use other people’s greed to make more money.

We know that this assumption is not quite right. If it was, there wouldn’t be any transient arbitrage opportunities. But even at a more fundamental level, the assumption has shaky justification. The reason that the market is efficient is that the practitioners take advantage of every little arbitrage opportunity. In other words, the markets are efficient because they are not so efficient at some transient level.

Mark Joshi, in his well-respected book, “The Concepts and Practice of Mathematical Finance,” points out that Warren Buffet made a bundle of money by refusing to accept the assumption of market efficiency. In fact, the weak form of market efficiency comes about because there are thousands of Buffet wannabes who keep their eyes glued to the ticker tapes, waiting for that elusive mispricing to show up.

Given that the quant careers, and literally trillions of dollars, are built on the strength of this assumption, we have to ask this fundamental question. Is it wise to trust this assumption? Are there limits to it?

Let’s take an analogy from physics. I have this glass of water on my desk now. Still water, in the absence of any turbulence, has a flat surface. We all know why – gravity and surface tension and all that. But we also know that the molecules in water are in random motion, in accordance with the same Brownian process that we readily adopted in our quant world. One possible random configuration is that half the molecules move, say, to the left, and the other half to the right (so that the net momentum is zero).

If that happens, the glass on my desk will break and it will make a terrible mess. But we haven’t heard of such spontaneous messes (from someone other than our kids, that is.)

The question then is, can we accept the assumption on the predictability of the surface of water although we know that the underlying motion is irregular and random? (I am trying to make a rather contrived analogy to the assumption on market efficiency despite the transient irregularities.) The answer is a definite yes. Of course, we take the flatness of liquid surfaces for granted in everything from the useless lift-pumps and siphons of our grade school physics books all the way to dams and hydro-electric projects.

So what am I quibbling about? Why do I harp on the possibility of uncertain foundations? I have two reasons. One is the question of scale. In our example of surface flatness vs. random motion, we looked at a very large collection, where, through the central limit theorem and statistical mechanics, we expect nothing but regular behavior. If I was studying, for instance, how an individual virus propagates through the blood stream, I shouldn’t make any assumptions on the regularity in the behavior of water molecules. This matter of scale applies to quantitative finance as well. Are we operating at the right scale to ignore the shakiness of the market efficiency assumption?

The second reason for mistrusting the pricing models is a far more insidious one. Let me see if I can present it rather dramatically using my example of the tumbler of water. Suppose we make a model for the flatness of the water surface, and the tiny ripples on it as perturbations or something. Then we proceed to use this model to extract tiny amounts of energy from the ripples.

The fact that we are using the model impacts the flatness or the nature of the ripples, affecting the underlying assumptions of the model. Now, imagine that a large number of people are using the same model to extract as much energy as they can from this glass of water. My hunch is that it will create large scale oscillations, perhaps generating configurations that do indeed break the glass and make a mess. Discounting the fact that this hunch has its root more in the financial mess that spontaneously materialized rather than any solid physics argument, we can still see that large fluctuations do indeed seem to increase the energy that can be extracted. Similarly, large fluctuations (and the black swans) may indeed be a side effect of modeling.

Coupled with this perfectionism, there is a curious tendency among academic researchers toward creating a “wow” factor with their results, as opposed to finance professionals who are quite content with the “wow” factor in their bonuses. This subtle mismatch generates interesting manifestations. Academics who make the mid-career switch to finance tend to work either alone or in small groups, trying to perfect an impressive prototype. Banking professionals, on the other hand, try to leverage on each other (at times taking credit for other people’s work) and roll out potentially incomplete solutions as early as possible. The intellectual need for a “wow” may be a factor holding back at least some quant deliverables.

Given that the investment value is also measured and returned in terms of money, we get the notion of compound interest and “putting money to work.” Those who have money demand returns based on the investment risk they are willing to assume. And the role of modern financial system becomes one of balancing this risk-reward equation.

We should keep in mind that this signification of money as investment entity is indeed a philosophical choice that we have made over the past few centuries. Other choices do exist — Islamic banking springs to mind, although its practice has be diluted by the more widely held view of money as possessing an investment value. It is fascinating to study the history and philosophy of money, but it is a topic that calls for a full-length book on its own right. Understanding money at its most fundamental level may in fact enhance our productivity — which is again measured in terms of the bottom line, consistent with the philosophy of money that enjoys currency.

Another point about the current bonus scheme is that it already conceals in it the same time segmentation that I ridiculed in my earlier post. True, the time segmentation is by the year, rather than by the month. If a trader or an executive does well in one year, he reaps the rewards as huge bonus. If he messes up the next year, sure, he doesn’t get any bonus, but he still has his basic salary till the time he is let go. It is like a free call option implied in all high-flying banking jobs.

Such free call options exist in all our time-segmented views of life. If you are a fraudulent, Ponzi-scheme billionaire, all you have to do is to escape detection till you die. The bane of capitalism is that fraud is a sin only when discovered, and until then, you enjoy a rich life. This time element paves the way for another slippery slope towards fraud and corruption. Again, it is something like a call option with unlimited upside and a downside that is somehow floored, both in duration and intensity.

There must be a happy equilibrium between these two slippery slopes — one toward dysfunctional socialism, and the other toward cannibalistic corruption. It looks to me like the whole financial system was precariously perched on a meta-stable equilibrium between these two. It just slipped on to one of the slopes last year, and we are all trying to rope it back on to the perching point. In my romantic fancy, I imagine a happier and more stable equilibrium existed thirty or forty years ago. Was it in the opposing economic ideals of the cold war? Or was it in the welfare state concepts of Europe, where governments firmly controlled the commanding heights of their economies? If so, can we expect China (or India, or Latin America) to bring about a much needed counterweight?

Sections

• Bonus Plans of Mice and Men
• Hard Work
• Talent and Intelligence
• Talent Retention
• Profit Sharing
• Slippery Slopes

The last argument for bonus incentives we will look at is this one in terms of profit (and therefore shareholder value) generation. Well, shareholder value in the current financial turmoil has taken such a beating that no sane bank executive would present it as an argument. What is left then is a rather narrow definition of profit. Here it gets tricky. The profits for most financial institutes were abysmal. The argument from the AIG executive is that he and his team had nothing to do with the loss making activities, and they should receive the promised bonus. They distance themselves from the debacle and carve out their tiny niche that didn’t contribute to it. Such segmentation, although it sounds like a logical stance, is not quite right. To see its fallacy, let’s try a time segmentation. Let’s say a trader did extremely well for a few months making huge profits, and messed up during the rest of the year ending up with an overall loss. Now, suppose he argues, “Well, I did well for January, March and August. Give me my 300% for those months.” Nobody is going to buy that argument. I think what applies to time should also apply to space (sorry, business units or asset classes, I mean). If the firm performs poorly, perhaps all bonuses should disappear.

As we will see in the last post of the series, this argument for and against hefty incentives is a tricky one with some surprising implications.

Sections

• Bonus Plans of Mice and Men
• Hard Work
• Talent and Intelligence
• Talent Retention
• Profit Sharing
• Slippery Slopes

The next argument in favour of hefty bonuses presents incentives as a means of retaining the afore-mentioned talent. Looking at the state of affairs of the financial markets, the general public may understandably quip, “What talent?” and wonder why anybody would want to retain it. That implied criticism notwithstanding, talent retention is a good argument.

As a friend of mine illustrated it with an example, suppose you have a great restaurant thanks mainly to a superlative chef. Everything is going honky dory. Then, out of the blue, an idiot cook of yours burns down the whole establishment. You, of course, sack the cook’s rear end, but would perhaps like to retain the chef on your payroll so that you have a chance of making it big again once the dust settles. True, you don’t have a restaurant to run, but you don’t want your competitor to get his hands on your ace chef. Good argument. My friend further conceded that once you took public funding, the equation changed. You probably no longer had any say over payables, because the money was not yours.

I think the equation changes for another reason as well. When all the restaurants in town are pretty much burned down, where is your precious chef going to go? Perhaps it doesn’t take huge bonuses to retain him now.

Sections

• Bonus Plans of Mice and Men
• Hard Work
• Talent and Intelligence
• Talent Retention
• Profit Sharing
• Slippery Slopes

But, I am threading on real thin ice when I compare, however obliquely, senior executives to cabbies and slum dogs. They are (the executives, that is) clearly a lot more talented, which brings me to the famous talent argument for bonuses. What is this talent thing? Is it intelligence and articulation? I once met a taxi driver in Bangalore who was fluent in more than a dozen languages as disparate as English and Arabic. I discovered his hidden talent by accident when he cracked up at something my father said to me — a private joke in our vernacular, which I have seldom found a non-native speaker attempt. I couldn’t help thinking then — given another place and another time, this cabbie would have been a professor in linguistics or something. Talent may be a necessary condition for success (and bonus), but it certainly is not a sufficient one. Even among slum dogs, we might find ample talent, if the Oscar-winning movie is anything to go by. Although, the protagonist in the movie does make his million dollar bonus, but it was only fiction.

In real life, however, lucky accidents of circumstances play a more critical role than talent in putting us on the right side of the income divide. To me, it seems silly to claim a right to the rewards based on any perception of talent or intelligence. Heck, intelligence itself, however we define it, is nothing but a happy genetic accident.

Sections

• Bonus Plans of Mice and Men
• Hard Work
• Talent and Intelligence
• Talent Retention
• Profit Sharing
• Slippery Slopes

I have met taxi drivers in Singapore who ply the streets hour after owl-shift hour before they can break even. Apparently the rentals the cabbies have to pay are quite high, and they end up working consistently longer than most executives. Farther beyond our moral horizon, human slum dogs forage garbage dumps for scraps they can eat or sell. Back-breaking labour, I imagine. Long hours, terrible working conditions, and hard-hard work — but no bonus.

It looks to me as though hard work has very little correlation with what one is entitled to. We have to look elsewhere to find justifications to what we consider our due.

Sections

• Bonus Plans of Mice and Men
• Hard Work
• Talent and Intelligence
• Talent Retention
• Profit Sharing
• Slippery Slopes

This philosophical clarity inspires analyses (and columns, of course) that are at times self-serving and at times soul-searching. We now worry about the moral rectitude behind the insane bonus expectations of yesteryears, for instance. The case in point is Jake DeSantis, the AIG executive vice president who resigned rather publicly on the New York Times, and donated his relatively modest bonus of a million dollars to charity. The reasons behind the resignation are interesting, and fodder to this series of posts.

Before I go any further, let me state it outright. I am going to try to shred his arguments the best I can. I am sure I would have sung a totally different tune if they had given me a million dollar bonus. Or if anybody had the temerity to suggest that I part with my own bonus, paltry as it may seem in comparison. I will keep that possibility beyond the scope of this column, ignoring the moral inconsistency others might maliciously perceive therein. I will talk only about other people’s bonuses. After all, we are best in dealing with other people’s money. And it is always easier to risk and sacrifice something that doesn’t belong to us.

Sections

• Bonus Plans of Mice and Men
• Hard Work
• Talent and Intelligence
• Talent Retention
• Profit Sharing
• Slippery Slopes

The binomial tree pricing model works by assuming that the price of an underlying asset can only move up or down by constant factors and during a small time interval . Stringing together many such time intervals, we make up the expiration time of the derivative instrument we are trying to price. The derivative defined as a function of the price of the underlying at any point in time.

Figure 1. Binomial tree pricing model. On the X axis, labeled i, we have the time steps. The Y axis represents the price of the underlying, labeled j. The only difference from the standard binomial tree is that we have let j be both positive and negative, which is mathematically natural, and hence simplifies the notation in a functional language.

We can visualize the binomial tree as shown in Fig. 1. At time , we have the asset price . At (with the maturity ). we have two possible asset values and , where we have chosen . In general, at time , at the asset price node level , we have

By choosing the sizes of the up and down price movements the same, we have created a recombinant binomial tree, which is why we have only price nodes at any time step . In order to price the derivative, we have to assign risk-neutral probabilities to the up and down price movements. The risk-neutral probability for an upward movement of is denoted by . With these notations, we can write down the fair value of an American call option (of expiry , underlying asset price , strike price , risk free interest rate , asset price volatility and number of time steps in the binomial tree ) using the binomial tree pricing model as follows:

where denotes the fair value of the option at any the node in time and in price (referring to Fig. 1).

At maturity, and , where we exercise the option if it is in the money, which is what the first Max function denotes. The last term in the express above represents the risk neutral backward propagation of the option price from the time layer at to . At each node, if the option price is less than the intrinsic value, we exercise the option, which is the second Max function.

The common choice for the upward price movement depends on the volatility of the underlying asset. and the downward movement is chosen to be the same to ensure that we have a recombinant tree. For risk neutrality, we have the probability defined as:

For the purpose of illustrating how it translates to the functional programming language of Haskell, let us put all these equations together once more.

where

Now, let us look at the code in Haskell.

optionPrice t s0 k r sigma n = f 0 0
    where
      f i j =
          if i == n
          then max ((s i j) - k) 0
          else max ((s i j) - k)
                    (exp(-r*dt) * (p * f(i+1)(j+1) +
                    (1-p) * f(i+1)(j-1)))
      s i j = s0 * u**j
      u = exp(sigma * sqrt dt)
      d = 1 / u
      dt = t / n
      p = (exp(r*dt)-d) / (u-d)

As we can see, it is a near-verbatim rendition of the mathematical statements, nothing more. This code snippet actually runs as it is, and produces the result.

*Main> optionPrice 1 100 110 0.05 0.3 20
10.10369526959085

Looking at the remarkable similarity between the mathematical equations and the code in Haskell, we can understand why mathematicians love the idea of functional programming. This particular implementation of the binomial pricing model may not be the most computationally efficient, but it certainly is one of great elegance and brevity.

While a functional programming language may not be appropriate for a full-fledged implementation of a trading platform, many of its underlying principles, such as type abstractions and strict purity, may prove invaluable in programs we use in quantitative finance where heavy mathematics and number crunching are involved. The mathematical rigor enables us to employ complex functional manipulations at the program level. The religious adherence to the notion of statelessness in functional programming has another great benefit. It helps in parallel and grid enabling the computations with almost no extra work.

Sections

• Zeros and Ones
• Paradigms All the Way
• Magic of Object Oriented Languages
• Functional Programming
• A New Kind of Binomial Tree