“The underpinning of quantitative finance is arbitrage pricing theory. The fundamental assumption is that markets are efficient.”

—Response to the survey question: “How would you describe quantitative finance at a dinner party?” at wilmott.com

“We shape our tools, and afterwards our tools shape us.”

—Marshall McLuhan, Understanding Media: The Extensions of Man

Once the markets had a model for valuing derivatives there was no longer any excuse for not trading them. The market in options exploded. New financial instruments were created using the same kinds of mathematical model… new and increasingly complicated instruments. As the instruments got more complicated, so did the mathematical models. Where once there were traders in Savile Row suits drinking far too much at lunchtime, now there were geeks with badly fitting suits and PhDs. If you had a degree in mathematics or physics, then a job as one of those geeky quants became your goal.

A framework for valuing derivatives was all that was needed to ignite the fuse that led to the explosion in new and increasingly complex derivative contracts. The gullible might say that having a decent theoretical foundation for valuation and risk management allowed quants to create new instruments with known characteristics and whose risks could be understood, measured, and controlled. The cynical might say that having a foundation, any foundation, even the shakiest and dodgiest on sandy soil, over a defunct mine, at the edge of a cliff, in an earthquake zone, was perfect if all you needed was, in the language coined by the CIA, “plausible deniability” when the trade goes wrong. In this chapter we are going to look at some of the contracts that came into existence after the great derivatives-valuation breakthrough, as well as the new models that were created to value them. And we'll see how the brilliant idea of hedging was stretched to breaking point and beyond.

In the early 1970s, when the option-pricing work of Black, Scholes, and Merton was gaining recognition, the financial world was still reeling from the so-called Nixon Shock. In August 1971, Richard Nixon took the US dollar off the gold standard and ushered in a new era of floating currencies. As stock and commodity options began to be traded in force on the new Chicago Board Options Exchange (CBOE), they were soon joined by an even larger market: currency options. Chicago's International Money Market was initially set up in 1972 to sell currency futures, which are agreements to exchange set amounts of currencies at future dates, but it soon diversified into broader types of currency option. For the first time in the history of derivatives, theory and practice were in perfect alignment. Everything from yen/dollar swaps to pork bellies was up for the Black–Scholes treatment.

Standardization of contracts and trading on an exchange are important if participants are to trust the market, allowing large quantities to be traded with no risk of confusion about contract terms or risk of default. And a mathematical formula that everyone could agree on removed much of the mystery about these new derivatives. A trader only needed to tap some numbers into their Texas Instruments calculator to get a price on an option, courtesy of Black–Scholes. (This particular feature was announced by a half-page ad in the Wall Street Journal. Scholes, being an economist, approached the company for royalties, but was told the formula was in the public domain.¹)

Traders became so accustomed to using the Black–Scholes model that its parameters took on a life of their own. For example, suppose a call option on a stock was selling for a particular price. The model price for that option depends on the stock's volatility, whose value as discussed above is not completely certain or stable. But if you run the model, knowing already what the market price is, then you can infer what level of volatility is consistent with that price. Traders often found it convenient to quote this “implied volatility” along with the actual cost, because it acted as another playing-field leveler that could be compared for different contracts. In theory, the implied volatility should be the same for every option. In practice, Black Monday had taught traders that the Black–Scholes formula underpriced protection against extreme events, because of its baked-in assumption of “normal” behavior where such events are effectively impossible. Traders continued to use the formula, but tailored the volatility parameter according to the details of the contract. The implied volatility of an underlying therefore varied with things like strike price and exercise time. This was a warning sign that all was not well with this system, but at least it was convenient.

All of this convergence did not mean, though, that the derivative world became simpler. Instead, the scene was set for parallel stories of increasing complexity of products and their models. As soon as traders were comfortable with pricing the basic derivatives, they moved on to more sophisticated and exciting products.

Time to Exercise

One of the more straightforward, at least conceptually if not mathematically, variations to the terms of the basic call and put options is to change when you are allowed to exercise them. The options described here so far are what are known as “European.” They allow exercise only at expiration. “American” options allow exercise at any time prior to expiration. (The designations are centuries old and have nothing whatsoever to do with location.) It's clear that American options can't have a value less than an equivalent, ceteris paribus, European contract, since if you hold an American option you could just decide not to exercise it early. The freedom to exercise any time you like clearly adds value: the question is, how much?

Although this seems a simple enough question, mathematically it's not so straightforward. Even the great Fischer Black had problems solving it. Ed Thorp tells of a meeting with Black at a Chicago securities conference in May 1975: “I brought along my solution to the American put problem and had placed a folder of graphs on the table to show him. Then he said no one had solved the problem… I realized I had a fiduciary duty to my investors to keep our secrets, and quietly put my folder with the world's first American put curves back in my briefcase.”² Thorp put his obligations to his investors above mathematical glory – which shows another difference between physics and finance, and is a good example of how the best ideas in finance are often in no rush to surface.

To understand why the American option is difficult to value theoretically you should put yourself in the shoes of the person selling the option, the writer. We've seen that options are risky investments because of the unknown behavior of the underlying asset, which we model mathematically as a random walk. And we've seen how to hedge exposure to this behavior by continuously buying or selling the underlying asset in a clever way as it moves around in price. However, with American options there's another risk for the writer that is harder to model mathematically, and that's the risk inherent in the timing of the option's exercise. If the American option cannot be worth less than a European option, then that potential extra value must be linked to when the American option is exercised. And there's the rub: We don't know when the option will be exercised because that's in the control of the owner of the option. And therefore we don't know how much extra value to add. Somehow we have to model the behavior of the option holder. Three methods spring to mind.

The first is to assume that the option holder exercises at a random time. After all, in finance we like to model everything we can as random. Before we start postulating distributions for this exercise (do we toss a coin, roll a die, consult the I Ching?), there is a big snag. If we value the American option this way and word gets around, then buyers might find a strategy that gives the option greater value than this “average value.” We'll find people queuing up to buy American options from us, all exercising at the same time, and we'd lose a fortune.

An alternative, then, is to assume that the buyer acts in a rational way. He will exercise at a time that maximizes his expected utility. Or something. We've mentioned such ideas before. And as always with utility theories you have to figure out what the “utility function” of the holder is, assuming such a thing exists. If we were the holder then we could conceivably model our own utility function. However, we are in the position of the option writer, and it's not as if we could ask the buyer to complete a psychometric-evaluation questionnaire. So, we've no idea about his utility function. Then there's method three.

Assume that the option holder exercises at the time that maximizes the option's theoretical value. This is subtle. We don't believe that the option holder will exercise at this time, but the point is that he could do, which is the worst-case scenario that we need to protect against. And this is the recognized correct method.

To understand how this works, let's break it down into manageable pieces. The first piece is that we, the option seller, are going to be delta hedging to remove market risk, leaving us with only exercise-time risk. We don't know the option holder's plans, he might even change his mind, but to guarantee we don't lose any money on this deal we have to assume that the holder exercises in whatever way is worst for us. (Even though we hope he won't. It's not as if it's personal, is it?) And that is like saying that the option holder will exercise whenever it maximizes the theoretical value of the option. So, whatever that theoretical maximum is, that's what we sell the option for (plus our profit margin). If we were to sell for less than this highest value then the holder himself would start delta hedging to get rid of market risk, and then exercise at the optimal time. He'd have the opposite position to us but with the benefit of having bought the option cheap, from us, and would therefore make a risk-free profit at our expense.³

As well as a theoretical option value, this method results in information about early exercise, which can be represented as a plot with time and asset price as the axes and regions, which can be called “hold” and “exercise.” When the asset price moves into an area labeled “exercise,” that's when our model assumes that the holder will exercise the option.

Part of the subtlety in this idea is that the option holder will almost certainly exercise at some time other than that we have modeled. Maybe he decides to exercise to lock in a profit or cut his losses, or isn't paying attention. Chances are he just isn't going to exercise at exactly the time we have modeled.

Now here's a question for you, intelligent reader: How do we feel about him exercising at a time different from that we have modeled?

If you think we are disappointed, because this then means that our model is wrong, then go back and reread the above. Go to the bottom of the class.

If you think we are pleased then you are correct. After all, exercise at any time different from the optimal exercise time means that the exercise was suboptimal, and means that we are going to make a profit (above even the markup). Go to the top of the class, you are smarter than the Journal of Finance.⁴ Complexity usually works in favor of the person selling something, because it puts the onus on the buyer to figure out how to get maximum value from it (see mobile phone contracts).

It's worth mentioning at this point that we've now seen the three main ways that quants eliminate, or at least reduce, risks. To recap, we have the following techniques:

• Diversify. Used in MPT – exploiting correlations between assets to reduce risk.
• Delta hedge. An extreme form of correlation exploitation, if you like. Assuming that the option pricing model perfectly captures the option's dependence on the price of its underlying asset (it won't), then the correlation between an option and the underlying is also perfect, and thus you can theoretically eliminate risk entirely.
• Worst case. Some things are out of your control, but could be controlled and exploited by others, and they aren't simply random. Here you assume that the worst happens to give the worst possible outcome. The good news is that the worst is unlikely to happen, and so there is extra profit to be made. Which is one reason why exciting new financial products are often designed to be as complicated as possible.

Decision Cost

Early exercise is a simple example of a feature that you see in lots of sophisticated financial instruments. It's called a “decision” feature for obvious reasons. It's not always the option holder who gets to make the decision though, it can also be the writer, as in the case of a callable bond in which the issuer can call back the bond for a pre-specified amount. Or it could be a third party who makes the decision affecting the contract's value. Whoever makes the decision, the same valuation principle applies – value the contract by assuming that the worst happens (if it's not you making the decision) or the best (if you can make it).

As financial derivatives got more and more complex, so you would also see decision features cropping up in some of them. And they added a lot of mathematical interest to quant finance. Classical derivatives theory, as presented by Black and Scholes in 1973, was very similar to the physical problems of heat transfer and diffusion. Heat transfer is about how temperature changes through a medium, how the radiators in your living room will warm up the surroundings, for example. Diffusion is about how particles move through a medium, such as contaminants in a river. With derivatives, the probability distribution of asset prices is diffusing in time, and becoming less concentrated or certain as we go further into the future. But in all these cases, the geometry of the problem remains stable. A radiator heats your room by distributing heat, but – unless you have some really serious plumbing issues – doesn't itself move or change shape. Similarly, the buyer of a European option only cares about the price at the exercise time, so that restriction imposes a kind of fixed boundary on the problem.

In contrast, the mathematics one finds in American options is much trickier, because now the exercise time can move around. An analogous problem from the physical sciences would be something like the melting of an iceberg. Heat flows through the ice and the surrounding water. As ice melts, or as water freezes, the boundary between the ice and water moves. This boundary can be represented by physical coordinates as a function of time. Think of the ice/water boundary being like the exercise/hold boundary in the American option. It's not immediately obvious how that boundary changes, and in fact finding that boundary is part of the problem. In mathematical language it's called a “free boundary.” In the American option the boundary to be found is the line between where it is optimal to exercise the option, and where it is optimal not to exercise.

This simple change to the contract specification, from European to American exercise, adds a great deal of interest for mathematicians. In most situations the equations cannot be solved directly, so approximate solutions must be obtained using computer simulations. Many PhDs have been written just on this one topic. And it's typical of how mathematically interesting quantitative finance was becoming, even in the early years after the Black–Scholes model was published.

New Flavors

In its early years, quantitative finance was mainly practiced in the halls of academe, or through the occasional consulting gig. In 1983, Fischer Black became one of the first full-time quants, when he left the University of Chicago to set up a Quantitative Strategies Group at Goldman Sachs. Recruiting newly minted math and physics PhDs to join the group was easy. The Cold War had led to a bubble in science education, as Americans tried to out-science their Soviet rivals; but there weren't enough actual jobs developing space laser systems or whatever to employ the graduates, with the result that many were happy to make the switch to finance – especially since the pay was much better.

European and American options are called in the jargon “vanilla” options, because of their simplicity and ubiquity, and they are usually traded on an exchange such as the CBOE. As quantitative finance developed into a profession, quants turned their collective genius to inventing contracts with more and more complex behavior. These contracts are typically not traded on an exchange, but might be designed for a particular client. They are called “over the counter” (OTC).⁵ We'll describe a few to help you understand how traders and quants think:

• Barrier option. Suppose you think that a stock is going to rise, but only a little bit. You'd consider buying a call option for its leverage. However, the price you pay for the call represents the potential for the stock to rise enormously, and you don't think this will happen. So you could buy an up-and-out call option. This contract pays off like a call option if the stock rises, but if the stock rises so far as to hit some pre-set trigger level any time before expiration, then it “knocks out” and becomes worthless. This is perfect for you, you just choose a trigger level above where you think the stock might rise. And this contract can cost far less than a vanilla call.

• Lookback option. Imagine you were the world's greatest investor. Your timing is perfect, such that you always buy at the lowest stock price and sell at the highest, during some timeframe. Dream on? No, there exists a contract that pays off exactly that amount, the difference between the highest and lowest price over some period. It's called a lookback option. It makes you the perfect trader. Oh, but it's very expensive.

As mentioned above, options don't have to be based on shares. They can have anything as an underlying. Commodity prices, exchange rates, etc. Let's look at an option that might be perfect for you if you run a business that sells stuff to a foreign country:

• Asian option. You manufacture widgets. You sell them abroad at a fixed price in the foreign currency. Your sales are fairly regular. Your skills are in manufacturing, not in forecasting exchange rates, so you really don't want to be exposed to exchange-rate risk. Quite frankly you'd like to focus on manufacturing and outsource any currency hedging. Well, you can. All you need is an Asian option. (Again, the name has nothing to do with Asia, except that in 1987 its American inventors happened to be working in Tokyo.) This contract has a payoff that depends on the average exchange rate over some period. And since your sales are regular, it's the average exchange rate that you are exposed to.

All of the above contracts require only fairly minor extensions to the Black–Scholes model, and all are still based on the idea of delta hedging to construct a risk-free portfolio. Such contracts are called “exotics” or “structured products” (the latter tends to be when the instruments have interest-rate exposure, of which more shortly). There are countless new products. Here's one that introduces us to new modeling challenges:⁶

• Multi-asset options. So far the derivatives we've seen have had a single underlying. This means that the payoff only depends on the behavior of a single financial quantity, such as a share or an exchange rate. It's not difficult to imagine a contract that pays off something to do with several assets. For example, the payoff is the best performing out of ten shares. Or with underlyings in several asset classes, such as a contract that pays off in dollars on a share that is quoted in sterling. When there are multiple underlyings, you have the same problem that you have with MPT, how to model the relationships between assets. Do you rely on correlation, given that it is so unstable?

You can't Always Delta Hedge

And then there is the bond market. Bonds represent loans, and so their value as investments fluctuates depending on interest rates. A bond which pays 8% might seem like a reasonable investment if the base interest rate is only 6%, but it loses some of its attractiveness if base rates suddenly go to 12%. Bond prices therefore tend to be inversely correlated with base rates. In the 1980s the Federal Reserve's benchmark rate was bouncing in the range of 6–19% as its central bankers struggled with the effects of stagflation. This made bonds an exciting area to work in, and it was only a matter of time before option theory was being adapted to model bonds as a kind of derivative, with interest rates as the underlying. The aim of the modeling was actually not so much to price the bonds themselves, whose prices are what they are in the market, but to value the more complex, non-traded exotics, the structured products.

Now, while option theory could be applied to bonds by assuming that interest rates follow a random walk, there is an important difference between things like stocks, currencies, and commodities on the one hand, and things like interest rates and credit on the other. You can find derivatives with all of these, and more, as underlyings, but the first group are easier to model than the second. This is because the basic quantity that one models in the first group is traded.

Stocks and shares are traded, and so it's easy to hedge options based on them. The same is true for options on currencies and commodities.

However sometimes, in fact extremely often, we have options on things that aren't traded. And that presents valuation problems. If you can't construct a risk-free portfolio with options and its underlying then you're back in Markowitz's world, no longer at that single risk-free dot but out in the wide-open spaces of non-zero risk, hard-to-measure expected returns, and market prices of risk.

The most important such options are surprisingly those based on interest rates. And that's because interest rates aren't traded. No, seriously. Bonds are traded, so are swaps, but these aren't the same as the rate that you are getting from your bank right now. That 0.5% isn't traded. If it helps, think of bonds as like the above Asian options, the bond's value depends on the average of a fluctuating instantaneous interest rate until maturity.

This presents a modeling problem, because even the simplest zero-coupon government bond becomes like a derivative of an interest rate. We can go through the whole Black–Scholes dynamic hedging argument, but to eliminate interest-rate risk we have to do something clever like hedging a one-year bond with, say, a one-month bond.

Delta hedging, that most fundamental and crucial idea from the early days of quantitative finance, is much harder in some markets than in others. But this wasn't going to stop the quants pretending that it worked.

Market Price of Risk Again

This might all seem a bit esoteric, but it's of great importance in the story of quantitative finance. For this is possibly the first, or most important, time that quants started cheating. Or maybe let's just call it brushing things under the carpet.

The difference between the traded and the untraded is one of the key distinctions between good models and poor. When the underlying is easy to hedge with, then the Black–Scholes derivation leads to an equation for the value of an option. That's one equation for one unknown. But if we go through the same derivation line by line for an interest-rate product, we now find that we still end up with one equation but two unknowns. In the above example the unknowns would be the one-year bond and the one-month bond. This means that we don't have a unique value for either of them. Instead, we can only value them relative to each other. It turns out that we can value all interest-rate derivatives if we bring in one unifying function – the market price of risk for the interest rate. The hedging argument tells us that all fixed-income instruments should receive the same compensation for taking risk, that's the same market price of risk, since they all have the same risk exposure – interest rates. We are back with Harry Markowitz, just not at that left-hand risk-free dot where we'd like to be, feeling safe and comfortable.

So hopefully – and this is where things get mushy – we can treat this market price of risk as some kind of fixed parameter of the system, which will give the extra piece of information needed to solve our equations.

It looks like we are back on track again. And superficially it does appear that way. Sadly though, this hedging isn't quite so trouble free.

When you read the economics or finance textbooks you get the impression that the market price of risk is something nice. Like three. After all, it's the measure of how much compensation above the risk-free rate one requires for taking a unit of risk. How rational is that? But in practice the market price of risk is unstable. It's also different for each source of risk; each stock has one, so do rates, currencies, etc. And it's not easy to measure.

Figure 5.1 shows a plot⁷ of the market price of risk for US short-term rates. It's definitely not a simple three.

This is one of Paul's favorite financial graphs, because it shows what should be blindingly obvious.⁸ The market price of risk is not that nice, stable quantity implied by the textbooks. It's all over the shop. One day it's high, one day it's low. Some days it even has the wrong sign – people are paying to take risk. Surely everyone, bar economists, knows that's just kinda how people are. In the figure are labels “fear” and “greed.” These are totally unscientific, but we can say that where the spikes are particularly large represents a fearful market, where a greater compensation than usual is required for risk taking. Greed is when the compensation has the wrong sign. That's like buying a lottery ticket, with its negative expectations.

Black–Scholes relies on knowing the volatility, which is already an unstable parameter, but now it has been joined by something even worse. And second, the assumed correlation between the one-year bond and the one-month bond isn't seen in practice; the correlation is far from perfect. The perfect correlation was a by-product of assuming that the instantaneous forward rate was the sole driver for all values. Typically there will be more than one random factor governing a valuation.

It was at this point – as quants issued more and more of these complex, unhedgeable instruments – that they made a sort of collective decision to not worry. At precisely the point where they should have. The Black–Scholes model was looking pretty good for traded underlyings, and so they wanted to use it when the underlying was not traded, even though it came with some major drawbacks. It was tempting to look the other way. Let's not frighten the horses when there are so many dollars at stake. Slippery slopes, slippery slopes. The quants had to become masters of cognitive dissonance. Not worrying became part of the job – along with those ill-fitting suits.

Getting Carried Away

If you don't worry too much about the practicality of valuation when the underlying is not traded, then you won't have a problem with derivatives based on the following:

• Credit. Businesses are risky. They issue contracts, such as bonds, that are exposed to risk of default (as well as interest rates). Risk of default is not traded, and is hard to model, but has historically been one of the largest concerns of banks. Traditionally it was handled by careful screening of loan candidates, and diversification. But that is a lot of work. Another approach, discussed further below, is tools such as credit default swaps (CDSs). These can be considered a form of insurance, which pay out in the case of “credit events” such as default. To price them, some people build models that take ideas from fundamental analysis, others just treat bankruptcy as a random event triggered by something like a coin toss. Credit derivatives models are therefore dubious.
• Macro. A portion of that credit risk is due to the overall state of the economy – bankruptcies go up during a recession. To protect against this, some derivatives use economic variables such as a manufacturing or payroll index as the underlying. Hard to model, and nigh on impossible to hedge.
• Inflation. You can have instruments with inflation as the underlying. Again this is not traded. And it's also something that governments try to control, albeit without perfect success.
• Property. There are also property derivatives, based on real-estate indices such as the S&P/Case-Shiller Home Price Indices in the USA. These can be used to hedge property risk or speculate on the property market. Although property is traded, it is so illiquid that tradability is virtually irrelevant.
• Energy. Energy derivatives have wonderful potential for modeling. Energy is difficult to store, its value changes erratically and to an extreme degree. Hedging is difficult. See the discussion of Enron below.

• Weather. Weather has a huge impact on many individual businesses and the economy as a whole. So if you want insurance, you might consider weather derivatives. But you can't hedge weather by buying or selling rain or snow. At least not directly. You can think of fun examples like buying shares in an umbrella manufacturer to hedge rainfall, or using the commodity orange juice as a hedge against the sun. Both are a long way from perfection.

We are starting to see the appeal of quant finance to the mathematician. (And maybe the salary has a slight influence too.) We have mathematical modeling, financial concepts to turn into mathematical principles. We have differential equations and free boundaries. Sometimes we have nice formulas. If we can't find formulas then we have to do some complicated numerical analysis. And complicated can be fun.

It can also be expensive if you get it wrong. Remember that dynamic hedging plays two roles in quant finance. It is used in a mathematical sense to determine the price of an option, as in the Black–Scholes formula. A trader can use the formula without actually buying or selling any stocks. However, hedging can also be used by the option writer as a method to (theoretically) eliminate risk on the option. A bank can issue options while delta hedging at the same time, and make money on the commission. So if instead you are selling an option where you can't trade the underlying, you might be able to come up with a theoretical price using the model, but it is impossible in practice to eliminate risk.

A separate issue is that, while derivatives can be used to reduce risk, just as often they are used as a way to make highly leveraged bets – so models are critical for risk assessment. In the 1990s a number of large companies, such as Procter & Gamble and Metallgesellschaft, experienced huge losses from derivatives trading. The wealthy Orange County in California was driven to bankruptcy by using derivatives to bet on interest rates. In energy derivatives, the undisputed leader was the Houston-based firm Enron. In 2000 it reported revenues of over $100 billion, which worked out at over $5 million per employee.⁹ The next year it went bankrupt. But these were just warm-up acts for what was to follow.

From the Sublime to the Ridiculous

Derivatives are obviously not the most stable of financial instruments, and should be handled with extreme care. We'll talk about the importance of good models in detail later, but here we just make a few brief comments.

First of all it's nice if the models are robust and internally consistent. In this category we'd put derivatives with shares, indices, exchange rates, and commodities as the underlying. As long as they aren't multi-asset contracts. At least these underlyings are traded and so the model fudging is minimal. And quants tend to all use similar models here.

Interest-rate models are not great. There are many, many of them. Different people use different models for the same instrument. And the inability to hedge consistently within the model can be a problem. On the down side, there's also the fact that the market in interest-rate derivatives is huge. The potential for a systemic disaster is therefore equally large. On the positive side, however, interest rates are dull, dull, dull. There is usually so little volatility in interest rates that perhaps none of this matters. At least this is true at the time of writing.

Credit-risk models are worse. You can't hedge. You don't know how to model default. Default isn't random, governed by the roll of a die – it's a business decision. There's no data for specific companies, since bankruptcy tends to be a one-off event. Volatility in risky businesses can be huge. And the market in credit instruments is large. Credit modeling is so bad, and credit instruments so dangerous, that it's worth having a closer look at these, in particular the infamous collateralized debt obligation (CDO) instrument.

Before explaining what quants do with CDOs, let's be clear, they are wonderful instruments. It's not that they are frightening per se, the worry is more to do with their abuse.

CDOs are a class of financial instrument in the family of asset-backed securities (ABSs). An ABS is a financial instrument whose value or cash flows are linked to a pool of typically illiquid underlying assets. These underlying assets could be things like property rental income, credit cards, student loans, car loans, mortgages, etc. Being pooled together so that you have thousands of student loans in the pot is meant to help with diversification and therefore control risk. Sometimes the ABS passes through a “special-purpose vehicle,” a specially created legal entity, designed to obscure risk or hide investments from prying eyes… including those of shareholders.

CDOs were originally invented in the late 1980s, but only really caught on when bankers hit upon the mother lode of risk: subprime mortgages. Consider, for example, the city of Detroit. Today, Detroit has a reputation as the zombie apocalypse of the real-estate world: wild dogs roaming streets of abandoned buildings; gang-run ‘hoods where the police fear to tread; brain-eating zombie politicians. Of course it's not that bad, but in 2014 someone was offering to swap an “investor special” three-bedroom house for “a new iPhone 6 or a new iPad,” which is usually a sign that the market is coming unstuck. A big contrast from ten years earlier, when home prices were booming, as lenders such as Countrywide Financial swamped the area with easy credit, pushing mortgages at anyone who could sign their name – no job or credit history required.¹⁰ Already in 2004 some 8% of houses in Detroit paid for by subprime loans had been seized by the banks, and a similar scenario was unfolding around the rest of the country, but the lenders didn't care. Why? Because they had found a way to repackage the risk, using CDOs, and sell it off to other people. The mortgage on a house in Detroit could end up being owned by a German bank wanting to diversify its holdings.

From the point of view of most observers, this was all to the good. As the IMF approvingly noted in 2006, its regulatory antennae all aquiver: “The dispersion of credit risk by banks to a broader and more diverse group of investors, rather than warehousing such risks on their balance sheets, has helped to make the banking and overall financial system more resilient.”¹¹ They were echoed by Ben Bernanke, who announced the same year that “because of the dispersion of financial risks to those more willing and able to bear them, the economy and financial system are more resilient.”¹²

So how did this “dispersion of credit risk” work? The CDO is a clever way of taking the cash flows from many underlying assets, let's say they are mortgages, pooling them all together, and then paying them out in tranches. Investors can choose which tranche to buy, some are far riskier than others. In detail, we might have something like the following:

• The payments, interest, and principals of 1000 mortgages go into one pot.
• As the payments go into the pot they pile up and are paid out to investors.
• Those investors with the senior tranche get paid first, then the mezzanine tranche, and last of all the junk tranche. (There would typically be more tranches than in this example.) The further down you are, the higher the risk that the pot won't be full enough, thanks to mortgage defaults, and you won't get paid.
• Each of these tranches comes with a credit rating. The topmost would be AAA, then AA, etc. The higher the credit rating, the greater the cost of the tranche, ceteris paribus, or equivalently the lower the expected return. As you go further down the tranches the risk gets higher, and the expected return gets greater. (That's MPT again.)

So far so good. You decide how much risk you can bear, look at whether the corresponding expected return is sufficient, and choose your tranche. As financial investments they are wonderful things.

Hold it Together

However, from a quant finance modeling perspective these instruments are horrendous. In a chapter on credit derivatives in a book published in 2006, Paul wrote “… credit derivatives with many underlyings have become very popular of late… I have to say that some of these instruments and models being used for these instruments fill me with some nervousness and concern for the future of the global financial markets.” (This didn't stop him almost losing a fortune in 2008. Idiot!)

One of the models used to value CDOs is the “copula.” This is a mathematical idea in probability theory that helps you analyze the behavior of multiple random variables, here the random variables being default. We’ll describe and critique this model here, but note that with CDOs it's not so much on any particular model that we can pin blame for the credit crisis that hit in 2008. No, it's more a problem that there's no model that is going to give you a value that you'd be able to sell at while giving you a mechanism for hedging risk.

The copula model tells you about the probabilistic behavior of multiple random variables in terms of the random behavior of the variables individually. Up until now we have concentrated on derivatives such as call options, which have a single underlying, such as a stock. The copula technique gives you a way to generalize that approach to a portfolio of assets. The connection to CDOs is clear, since there are many, many individual underlyings, the mortgages, say. But we only care about the tranches, which are complex amalgamations of the mortgages. Valuing these tranches depends critically on the degree of correlation between the securities. If they are highly correlated, then even senior tranches risk being exposed to a wave of defaults.

The copula method was originally based on an actuarial technique, used to address something known as the “broken heart syndrome.” The death of a person's partner significantly increases the probability of their own death over the next year, which affects (lowers by a few percent) the price of a joint annuity.¹³ The copula (from the Latin for “fasten together”) was a way to calculate the probability of both partners dying at or before a certain age – and therefore the value of a joint annuity – while accounting for this temporal correlation. A Gaussian copula is one that makes use of the Gaussian distribution, which is another name for the normal distribution discussed earlier. (This work surprisingly didn't win its inventors a Nobel Prize, but they did pick up the Society of Actuaries' 1998 Halmstad Prize.)

The quant idea, then, was to simulate default by assuming that individual defaults are similarly correlated. If one owner defaults on his mortgage, the chances increase that the owner down the street will shortly default too. The approach sounds plausible, but it is less clear how you generalize it to a thousand mortgages. The broken heart syndrome deals with individual correlations, but what if risk is contagious? And what happens if the result is a kind of global financial coronary?

There is a very famous, in quant circles, article in Wired magazine from 2009 by Felix Salmon called “The formula that killed Wall Street,” about the copula model.¹⁴ His focus, as ours has been, was on the correlation behavior between the underlyings. We've done the combination mathematics already in Chapter 3, but let's quickly do another example. There are 1000 mortgages in the CDO, how many correlations are there? The answer is 1000 × 999 / 2, and that's nearly half a million. Who is going to try to measure those parameters? Should one go door to door interviewing the mortgagors to see if the ability to pay is correlated among neighbors? Does a simple number like correlation even capture the relationship between Mrs Smith at number 99 and Dr Garcia at number 101? Won't it change in the event of a financial shock? More on correlation anon. But what is the quant gonna do?

Model Abuse

What the quant does is say, “Let's assume that all of the correlation parameters are 0.6.”

WHHHAAAATTT?!?!

The quant has taken a subtle and sophisticated probabilistic concept like the copula, with half a million parameters, and through laziness, naivety, apathy, or something, thrown it all away by plucking a correlation parameter out of thin air and assuming it to be the same for all pairs among the 1000 mortgages.

We have seen above that using volatility in Black–Scholes is problematic, because it is unstable. We have remarked that making up a market rate of risk for interest-rate options is a way of sweeping problems under the carpet. But assuming that the inter-relationships in the complex entity known as the housing market can be adequately described through a single number – a kind of market rate of correlation – is taking model abuse to a completely new level.

Despite the criticism of the copula model from Salmon and others, there have been moderately vigorous defenses. The defenses amount to: “We never really believed the model” and “We used far more sophisticated models.”¹⁵ That may well be true. But whatever model you use, if you have a CDO with 1000 underlyings you are going to run against the problem of relating the individual underlying assets, and that is a problem far too complex to do at all accurately.

Indeed, the model's simplicity was its main selling point. Simple models aren't just aesthetically pleasing, they also have many other advantages, including being easy to communicate. As with the Black–Scholes model, the copula model soon took on a life of its own. Traders incorporated it into their working practices, using it as a communication device. Instead of implied volatility, traders could quote on implied correlation. Again, implied correlations for related securities – say, the same mortgage pool but different tranches – tended to be inconsistent, with the senior tranches giving a higher implied correlation than the lower tranches, which again should have raised some alarms. But as one trader told sociologists Donald MacKenzie and Taylor Spears, “if everyone had the same model and they all agreed on the same model it didn't matter whether it was a good model or not.”¹⁶ In particular, the model could be used by accountants and auditors to value a contract now, even though the true value would only be known in the future, after the mortgages had defaulted or not. So if the trader sold a contract for more than the model value, the profit would count toward his bonus. Without the model, “people would be in serious trouble, all their traders would leave and go to competitors.”

Skepticism about the copula technique was further neutralized in August 2004 when the world's main two rating agencies, Moody's and Standard & Poor's, both adopted the formula as a metric for valuing CDOs. Just as Black–Scholes led to a huge expansion in option trading, so the Gaussian copula galvanized the trading of CDOs. The endorsement of the credit agencies meant that regulated institutions such as pension funds could pile in. Institutions didn't even have to make their own models, they could just download Standard & Poor's “CDO Evaluator” program.

In 2004, some $157 billion in CDOs was issued. In 2006, that figure had ballooned to $552 billion.¹⁷ The growth in CDOs was facilitated by the use of CDSs. These could be used to insure against default on the loans, which allowed banks to remove risk from their balance sheets. And since these paid off only if an investment defaulted, they could also be used as a way to go short on assets, and therefore as a hedging device. By the end of 2007, the value of the CDS market, in terms of amount insured, had reached roughly $60 trillion, which was about the same as world GDP.

Hedge funds were having a field day. One firm which did well at this was Magnetar Capital – it “sponsored” CDOs by offering to buy high-paying junk tranches which no one else wanted, and hedged its risk by using CDSs to go short on the upper tranches. This position would pay off handsomely in the event that correlations blew up during a crash, so that defaults reached all the way to the upper (supposedly safe) tranches. Which of course is exactly what happened.

Pass the Parcel

The only realistic approach to valuation is to assume fairly extreme relationships between underlyings in such a way that the prices of CDO tranches become too expensive so that no one would buy them. Although that might be the mature and responsible thing to do from a valuation perspective, it's not going to win the quant any friends among the traders. All they care about is doing the deal to get the bonus. Fingers crossed that they get the bonus before it all goes pear shaped.

Even though there have been defenses of copula and other models for valuing CDOs, we suspect that a lot of this is rewriting history. Paul was there in the audience at many pre-2008 conferences where both academics and practitioners were peddling their models and risk-management software (see Box 5.1). No one was saying don't trade these, they're too dangerous. This would not have mattered if trade in these too-complicated-to-model derivatives was small, but they were incredibly big business. The reason they became big business is the interesting and dangerous part.

As children we would play pass the parcel at birthday parties. A present would be wrapped in paper prior to the party. This parcel would then be passed around a circle of children as music was played. The music was stopped intermittently and the child holding the parcel at that time would remove a layer. The music resumed, the parcel continued around the circle, and eventually the present was unveiled and kept by the child who had unwrapped it. That's sort of what happened with CDOs. But not with a nice present inside.

A mortgage lender would lend money to people to buy their own homes. This is risky for them. But they wouldn't be holding that risk for long. They could pass that on to a bank, who now temporarily held that risk. But not for long. With a little help from their investment banking quant chums they could package this up and sell it on to investors in individual, tailor-made, tranches. And once it was in the hands of the investors, it was they who took all the risk. Unless they insured it with something like a CDS, in which case the risk passed to the insurer.

It became quite the fashionable business. A bit of a bandwagon you could say. And that's really the dangerous part… the size of the market in these instruments. Not being able to value or hedge doesn't matter too much if the trades are small, but once the size gets enormous you get systematic risk that could bring down the whole system. And the size did get enormous, and that's because there was the illusion that these contracts could be valued and hedged, resulting in the biggest false sense of security in history. And it was the dubious quant models that played a key enabling role. The fact that most of the instruments were sold OTC meant that there was no visibility about institutional exposure, until the debts started to be called in.

The largest issuer of CDSs at the height of the crisis was AIG. But even they didn't keep the parcel for long – they were technically broke and passed it on to taxpayers. If you're curious who ultimately underwrote that $1.2 quadrillion worth of outstanding derivatives, it's you.

Money Crunch

The net effect of financial instruments such as CDOs and CDSs was not a reduction in risk, but a huge expansion in money and credit. We tend to think of the money supply as being something that is controlled by the central bank, while in fact the vast majority of money is created by lending from private banks. As Adair Turner, former Chairman of the UK's Financial Services Authority, notes: “Economic textbooks and academic papers typically describe how banks take deposits from savers and lend the money on to borrowers. But as a description of what banks actually do this is severely inadequate. In fact they create credit money and purchasing power. The consequences of this are profound: the amount of private credit and money that they can create is potentially infinite.”¹⁸ When you take a mortgage out with a bank, they don't take the money from the accounts of other clients, they just make up new funds. When economic conditions are good, the price of assets such as houses goes up, and these can be used as collateral for even larger loans, in a positive feedback loop. The central bank only has indirect influence on this process, by adjusting things such as the interbank lending rate.

Of course, banks have to manage their risks, which puts a cap on their lending activities. However, the invention of CDOs and CDSs, and their wholesale misuse in the early 2000s, allowed those banks to parcel the risk up and insure it or sell it on to others. Either way they got it off their balance sheets, meaning they could issue more loans, and further inflate the money bubble. The reason the credit crunch of 2007 did so much damage was that it was in fact a money crunch, similar in spirit to the one which John Law unleashed on France in the 17th century, but on a global scale. Quants didn't set out like Law to print money, but that was the emergent effect of their endeavors. Models such as the Gaussian copula, which modeled the financial world as an intrinsically stable system, and didn't account for the effect of things like sudden crashes or contagion between institutions, created a false sense of security. Simple models have many advantages, including as communication devices, but when misused they can also be a way of enforcing a type of group denial.

As MacKenzie and Spears observe, “Perhaps the modelling of derivatives in investment banking always has an aspect of what one of our interviewees memorably called a ‘ballet,’ in which highly-paid quants are needed not just to try to capture the way the world is, but also to secure co-ordinated action. Perhaps the quant is actually a dancer, and the dance succeeds when the dancers co-ordinate.” Unfortunately, this particular ballet had a tragic ending to rival that of Swan Lake, when a black swan – known as reality – swam serenely into view.

A particular property of money, which rivals quantum physics for its weirdness, is the way that it is real, in the sense that its appearance has real effects on people and the economy – the housing boom in places like Detroit meant that more Americans could buy a home than ever before – but when conditions change it can suddenly disappear into the ether, as if it had never existed. In the next chapter we look more deeply at how the quant community attempts to use mathematics to tame this mysterious substance, and ask why the results so often end in an explosion.