There’s been a lot of criticism of the Black-Scholes model of late, on our Forum, in our blogs, in the magazine (see Haug & Taleb, Wilmott magazine, January 2008) and in other media. Most is warranted, but perhaps not all. I would now like to speak in its defence! This may seem perverse since I have been highly critical of this model for the last 15 years. But as I will explain, Black-Scholes is a remarkably robust model that copes very well even when its underlying assumptions are violated, as they inevitably are in practice. Before detailing my views on this matter, I’d like to explain how my personal relationship with the Black-Scholes model has evolved.

I was introduced to options in around 1987, well before the October crash, while I was a postdoc researching in various problems of industrial/applicable maths. For a while I researched in several areas of finance simultaneously: technical analysis; chaos theory; stochastic calculus. (Thanks to the technical analysis I was short the market coming into the crash of ’87 but sadly only on paper!). I quickly dropped the TA and chaos theory, the latter seemed like a dead end, it was too easy to construct ‘toy models’ that looked plausible but were useless in practice. And so I began to focus on classical quant finance. Being in a maths department before most maths departments had heard of quant finance I had to rely on reading the literature in order to learn the subject. There were no courses for me to attend and no one more experienced to speak to. In those days whenever I read a paper I tended to believe everything in it. If the paper referred to volatility as a constant then I would believe that it was a constant. Black-Scholes was to me a good model, which just needed a minor bit of tweaking. My research from that era was on making small improvements to Black-Scholes to allow for transaction costs, and on the pricing of exotic derivatives in a constant-volatility world. This was the first phase in my relationship with Black and Scholes.

The second phase was as a consultant working for various investment banks, hedge funds and software companies. I was still in academia but moonlighting on the side. In this new capacity I finally got access to real data and was now speaking to practitioners rather than academics. (Fischer Black himself contacted me about the possibility of working for Goldman Sachs, and at this time I got to know Emanuel Derman. For a while I was sorely tempted to join them, but ultimately such a position would not have suited my personality.) It didn’t take long for me to realise how unrealistic were the assumptions in the Black-Scholes model. For example, volatility was certainly not constant, and the errors due to discrete hedging were enormous. My research during the mid and late ’90s was on making more dramatic improvements to the models for the underlyings and this was also the era when my interest in worst-case scenarios began. I worked with some very talented students and postdocs. Some great ideas and new models came out of this period. This was the height of my anti-Black-Scholes views.

A couple of years after leaving academia I became a partner in a volatility arbitrage hedge fund, and this was the start of phase three. In this fund we had to price and risk manage many hundreds of options series in real time. As much as I would have liked to, we just weren’t able to use the ‘better’ models that I’d been working on in phase two. There just wasn’t the time. So we ended up streamlining the complex models, reducing them to their simplest and most practical form. And this meant using good ol’ constant volatility Black-Scholes, but with a few innovations since we were actively looking for arbitrage opportunities. From a pragmatic point of view I developed an approach that used Gaussian models for pricing but worst-case scenarios for risk management of tail risk. And guess what? It worked. Sometimes you really need to work with something that while not perfect is just good enough and is understandable enough that you don’t do more harm than good. And that’s Black-Scholes.

I had gone from a naïve belief in Black-Scholes with all its simplifying assumptions at the start of my quant career, via some very sophisticated modelling, full circle back to basic Black-Scholes. But by making that journey I learned a lot about the robustness of Black-Scholes, when it works and when it doesn’t, and have learned to appreciate the model despite its flaws. This is a journey that to me seems, in retrospect, an obvious one to take. However, most people I know working as quants rarely get even half way along. (As discussed elsewhere, I believe this to be because most people rather like being blinded by science.)

My research now continues to be aimed at questioning commonly held beliefs, about the nature of ‘value,’ about how to use stochastic calculus to make money rather than in a no-arbitrage world, about the validity of calibration (it’s not valid!), and how people price risk (inconsistently is how!). All the time I strive to keep things understandable and meaningful, in the maths sweet spot that I’ve mentioned before.

That’s my journey. But what about the criticisms of Black-Scholes? There are several main ones: Black-Scholes was known well before Black, Scholes and Merton; traders don’t actually use Black-Scholes; Black-Scholes doesn’t work.

I will happily accept that the Black-Scholes formulae were around well before 1973. Espen Haug (“Collector”) has done an excellent job hunting down the real history of derivatives theory (see his Models on Models). Ed Thorp plays a large role in that history. In the first issue of our magazine (Wilmott magazine, September 2002) the cover story was about Ed Thorp and his discovery of the formulae and their use for making money (rather than for publication and a Nobel Prize!). Ed wrote a series of articles “What I Knew and When I Knew it” to clarify his role in the discovery, including his argument for what is now called risk-neutral pricing. I particularly like the story of how Fischer Black asked Ed out to dinner to ask him how to value American options. By the side of his chair Ed had his briefcase in which there was an algorithm for valuation and optimal exercise but he decided not to share the information with Black since it was not in the interests of Ed’s investors! Incorrect accreditation of discoveries is nothing new in mathematics, but usually there’s a quid pro quo that if you don’t get your name attached to your discovery then at some stage you’ll get your name attached to someone else’s!

They say traders don’t use Black-Scholes because traders use an implied volatility skew and smile that is inconsistent with the model. (Do these same people complain about the illegitimate use of the ‘bastard greek’ vega? This is a far worse sin.) I think this is a red herring. Yes, sometimes traders use the model in ways not originally intended but they are still using a model that is far simpler than modern-day ‘improvements.’ One of the most fascinating things about the Black-Scholes model is how well it performs compared with many of these improvements. For example, the deterministic volatility model is an attempt by quants to make Black-Scholes consistent with the volatility smile. But the complexity of the calibration of this model, its sensitivity to initial data and ultimately its lack of stability make this far more dangerous in practice than the inconsistent ‘trader approach’ it tries to ‘correct’!

The Black-Scholes assumptions are famously poor. Nevertheless my practical experience of seeking arbitrage opportunities, and my research on costs, hedging errors, volatility modelling and fat tails, for example, suggest that you won’t go far wrong using basic Black-Scholes, perhaps with the smallest of adjustments, either for pricing new instruments or for exploiting mispriced options. Let’s look at some of these model errors.

Transaction costs may be large or small, depending on which market you are in and who you are, but Black-Scholes doesn’t need much modification to accommodate them. The Black-Scholes equation can often be treated as the foundation to which you add new terms to incorporate corrections to allow for dropped assumptions. (See anything by Whalley & Wilmott from the 1990s.)

Discrete hedging is a good example of robustness. It’s easy to show that hedging errors can be very large. But even with hedging errors Black-Scholes is correct on average. (See PWOQF2.) If you only trade one option per year then, yes, worry about this. But if you are trading thousands then don’t. It also turns out that you can get many of the benefits of (impossible) continuous dynamic hedging by using static hedging with other options. (See Ahn & Wilmott, Wilmott magazine, May 2007 and January 2008.) Even continuous hedging is not as necessary as people think.

As for volatility modelling, the average profit you make from an option is very insensitive to what volatility you actually use for hedging (see Ahmad & Wilmott, Wilmott magazine, November 2005). That alone is enough of a reason to stick with the uncomplicated Black-Scholes model, it shows just how robust the model is to changes in volatility! You cannot say that a calibrated stochastic volatility model is similarly robust.

And when it comes to fat tails, sure it would be nice to have a theory to accommodate them but why use a far more complicated model that is harder to understand and that takes much longer to compute just to accommodate an event that probably won’t happen during the life of the option, or even during your trading career? No, keep it simple and price quickly and often, use a simpler model and focus more on diversification and risk management. I personally like worst-case scenarios for analyzing hedge-fund-destroying risks. (See anything from the 1990s by Hua & Wilmott.)

The many improvements on Black-Scholes are rarely improvements, the best that can be said for many of them is that they are just better at hiding their faults. Black-Scholes also has its faults, but at least you can see them.

As a financial model Black-Scholes is perfect in having just the right number of ‘free’ parameters. Had the model had many unobservable parameters it would have been useless, totally impractical. Had all its parameters been observable then it would have been equally useless since there would be no room for disagreement over value. No, having one unobservable parameter that sort of has meaning makes this model ideal for traders. I speak as a scientist who still seeks to improve Black-Scholes, yes it can be done and there are better models out there. It’s simply that more complexity is not the same as better, and the majority of models that people use in preference to Black-Scholes are not the great leaps forward that they claim, more often than not they are giant leaps backward.

P

Maths is fun. Many people reading this blog and the Forum get a real kick out of maths and problem solving. I’ve had many jobs and careers in the last three decades, and started various businesses, but the one thing that I keep coming back to is mathematics. There’s something peaceful and relaxing about an interesting maths problem that means you can forget all your troubles, just get totally absorbed in either the detail of a formulation, calculation or solution, or lie back and think of deep concepts.

I wonder if that’s one of the reasons quantitative finance is in such a mess.

I’m going to let you in on the big secret of quantitative finance, and you must keep this secret because if word got out then that would be the end of all masters in financial engineering programs. And universities make a lot of money from those.

Ok, the big secret...Quantitative finance is one of the easiest branches of mathematics.

Sure you can make it as complicated as you like, and plenty of authors and universities have a vested interest in so doing. But, approached correctly and responsibly, quant finance is easy.

Let’s talk about the different levels of maths you see in quant finance.

Some people try to dumb the subject down. There are plenty of textbooks that kid you into thinking that there is almost no mathematics in the subject at all. These books may dabble in the binomial model but go no deeper. Now anyone with a second-year undergraduate knowledge of numerical methods will recognise the binomial model for the inadequate and cumbersome dinosaur that it is. I like the binomial method as a teaching tool to explain delta hedging, no arbitrage and risk neutrality. But as a way of pricing derivatives for real? No way! Watching the contortions people go through on the Forums in order to make their binomial code work is an illuminating experience. Dumbing the subject down is not good. You cannot price sophisticated contracts unless you have a decent mathematical toolbox, and the understanding of how to use those tools. Now let’s look at the opposite extreme.

Some people try to make the subject as complicated as they can. It may be an academic author who, far from wanting to pass on knowledge to younger generations, instead wants to impress the professor down the corridor. He hopes that one day he will get to be the professor down the corridor who everyone is trying to impress. Or maybe it’s a university seeing the lucrative QF bandwagon. Perhaps they don’t have any faculty with knowledge of finance, certainly no practical knowledge, but they sure do have plenty of people with a deep knowledge of measure theory. Hey presto, they’ve just launched a masters in financial engineering! Making this subject too complicated is worse than dumbing it down. At least if you only work with the binomial method you can’t do much harm, simply because you can’t do much of anything. But with all those abstract math tools at your command you can kid yourself into believing you are a derivatives genius. Never mind that you don’t understand the markets, never mind that the people using your models haven’t a clue what they are doing. I believe that the obscenely over-complicated models and mathematics that some people use are a great danger. This sort of maths is wonderful, if you want to do it on your own time, fine. Or become a finance professor. Or move into a field where the maths is hard and the models are good, such as aeronautics. But please don’t bring this nonsense into an important subject like finance and where even the best models are rubbish. Every chain has its weakest link. In QF the weakest links are the models, not the maths, and not the numerical methods. So spend more time thinking about your models and their robustness and less on numerical inversion of a transform in the complex plane.

Here’s a true story that illustrates my point quite nicely. Not long ago I was approached by someone wanting to show me a paper they hoped to get published. The paper was about 30 pages long, all maths, quite abstractly presented, no graphs. When I’d read the paper I said to the author that I thought this was a good piece of work. And I told him that the reason I thought it was good was because, unfortunately for him, I’d done exactly the same piece of research myself with Hyungsok Ahn a few years earlier. What I didn’t tell him was that Hyungsok and I only took four pages to do what he’d done in 30. The reason for the huge difference in derivations was simply that we’d used the right kind of maths for the job in hand, we didn’t need to couch everything in the most complicated framework. We used straightforward maths to present a straightforward problem. Actually, what he had done was worse than just unnecessarily obscure the workings of the model. There was a point in the paper where he trotted out the old replacement-of-drift-with-the-risk-free-rate business. He did this because he’d seen it done a thousand times before in similarly abstract papers. Furthermore, because the paper was about incomplete markets, the whole point of the model was that you were not allowed to make this substitution! He didn’t understand the subtle arguments behind risk-neutral valuation. That was the place where his paper and ours diverged, ours started to get interesting, his then followed a well-worn, and in this case incorrect, path.

If you look through the various Forums on wilmott.com you will see that we have some areas for people to talk about mathematics, research papers, etc., and then there are areas to talk about trading, general finance, etc. You will notice that the majority of people are comfortable in only either the maths areas or the trading areas. Not so many people are comfortable in both. That should tell you something, the overlap of skills is far less than one would expect or hope. Who would you trust your money to? A mathematician who doesn’t know the markets or a trader who doesn’t know maths? Ideally, find someone who is capable in both areas.

And so to the middle ground, not too dumb, not too clever for its own good. Let’s start with the diffusion equation. As every mathematician knows there are three important classes of partial differential equation: Elliptic; Hyperbolic; Parabolic. There are various standard techniques for solving these equations, some of them numerical. The diffusion equations that we see so often in QF are of parabolic type. Rather conveniently for us working in QF, parabolic equations are by far the simplest of the different types to solve numerically. By far the simplest. And our equations are almost always linear. Boy, are we spoiled! (I’ve thought of publishing the “Wilmott Ratio” of salary to mathematical complexity for various industries. Finance would blow all others out of the water!)

Or take the example of some fancy exotic/OTC contract. You start with a set of model assumptions, then you do the maths, and then the numerics. Most of the time the maths can be 100% correct, i.e. no approximations, etc. Given the assumptions, the pricing model will follow as night follows day. Then you have to crunch the numbers. Now the numerics can be as accurate as you like. Let’s say you want the value and greeks to be 99% accurate. That’s easy! It may take a few seconds, but it can usually be done. So where’s the problem? Not the maths, not the numerics. The problem is in the model, the assumptions. Maybe you get 70% accuracy if you are lucky. It seems odd therefore that so many people worry about the maths and the numerics, when it is very obvious where the main errors lie!

There is a maths sweet spot, not too dumb, not too smart, where quants should focus. In this sweet spot we have basic tools of probability theory, a decent grasp of calculus, and the important tools of numerical analysis. The models are advanced enough to be able to be creative with new instruments, and robust enough not to fall over all the time. They are transparent so that the quant and the trader and the salesperson can understand them, at least in their assumptions and use.

Because the models are necessarily far, far from perfect, one must be suspicious of any analytical technique or numerical method that is too fiddly or detailed. As I said above, the weakest link in the chain is not the maths or the numerics but the model assumptions. Being blinded by mathematical science and consequently believing your models is all too common in quantitative finance.

This is to me the reason why QF is interesting and challenging, not because the mathematics is complicated, it isn’t, but because putting maths and trading and market imperfections and human nature together and trying to model all this, knowing all the while that it is probably futile, now that’s fun!

P

Another day, another financial institution collapses. Bear Stearns, fifth largest US investment bank, has gone. I’ve worked closely with Bear brokerage in the past and quite enjoyed the experience. It’s the prime brokerage that JP Morgan is presumably after. I’m a quant not an accountant so was surprised to see that Bear’s assets were just 2-3% higher than their liabilities. If this is standard practice in this sector then crikey, we really are doomed! Who in their right mind would run a business that way? Sorry if I seem awfully naïve, but as someone who has himself run a few businesses in his time, albeit on a somewhat smaller scale, to me this does seem highly irresponsible.

Investments (although that hardly feels like the right word) in mortgage-backed products and over-zealous lending combined with one particular scenario are at the bottom of this. This scenario is that of falling house prices. But isn’t scenario analysis supposed to spot this sort of exposure? It’s not as if falling property prices are totally unheard of. As those of you who have heard me lecture will know, I always like to boil things down to everyday experiences. And according to my experience there is a one in three chance of losing money in property! (Like most people with similar experiences it was the early 90s to ‘blame’ in my case.) And a one in three chance is not exactly the 10 standard deviation excuse du jour! It has been suggested that many bank employees are too young to have experienced negative equity and therefore it is off their radar, but if that is the case then what is the point of risk management at all? What is the point of all those risk management qualifications that are springing up like mushrooms? It has also been suggested that senior people don’t have a clue about the instruments that their bank is trading. So I really would like to know how they fill their days, whatever they are doing it is clearly not productive.

Are those in positions of responsibility at Bear Stearns blameless? Did senior management really think that their downside was tolerable, that Value at Risk and stress testing were giving an accurate picture of potential losses and their probabilities? Again we come back to that old problem, if there’s no downside then irresponsible people will prosper at the expense of the rest of us. And it seems that only the irresponsible rise to positions of responsibility in this business. Ironic.

Or maybe they are so lawyered up as to feel invincible. I am sure there will be civil suits in some of these cases because you can guarantee that the lawyers will, as always, be the big winners. They are paid to ensure that your back is covered no matter how unethical your behaviour, and then they are paid again when you are inevitably sued.

On the subject of ethical behaviour, don’t some of these risk management courses teach about ethics? Or does understanding ethics these days amount to knowing who are the best lawyers? Personally, if I know someone had to go on a course to learn business ethics then I would ask myself whether that’s a person I can trust. A risk management qualification is just another preventative measure against being sued, just like the “Mind the Step” signs in restaurants? What, you broke your leg? Not our fault, mate, didn’t you see the sign? (I was disappointed to discover recently, but not surprised, that they’ve taken peanuts out of Revels chocolates. Some people suffer from allergies, and presumably can’t read, so we all have to do without.)

Risk management must be consistent with protecting the wider interests of the institution rather than being easy to manipulate towards the narrow interests of some employees. At present the concept of risk management only exists to make it easier for people to take risks that common sense would suggest are stupid, but risks that people still want to take because of the huge upside for these same people in terms of bonus. Let’s face it, that’s what rules and regulations are for. As Madness said in Baggy Trousers “All I learnt at school was how to bend not break the rules.”

I’d now like to explain how I think risk management should work. It’s a simple combination of standard practices that I have used very successfully in the past. It’s not exactly earth shattering, but it shows how to focus your attention on what matters. I will also finish with a small proposal for how to approach scenario analysis.

Roughly speaking, I tend to think in terms of three different levels or classes of risk management. These are

Level 1: Probabilities and VaR

Level 2: Worst-case scenarios

Level 3: Invasion by aliens, “It’s the end of the world as we know it, and I feel fine” (REM this time!)

Level 1: Typical day-to-day markets for which it is acceptable to work with probabilities and even possibly normal distributions. Correlations, while never exactly trustworthy, will not be a deciding factor in survival or collapse. Use probabilities and talk about Value at Risk by all means. This is really just classical mid 1990’s risk management, with not too much worrying about fat tails. To some extent trust in a decent amount of diversification. The rationale behind this is simply that you never know what your parameters or distributions really are and so you are better off with simple calculations, more instruments and plenty of diversification. You may not make a profit but at least you won’t be killed during a quiet day in the market.

Level 2: Situations which will cause your bank or hedge fund to collapse. Test your portfolio against a wide range of scenarios and see the results. But since these are situations resulting in the collapse of your institution you must never, ever talk about probabilities, except in terms of how many centuries before such events may happen. I would much prefer you work with worst-case scenarios (as in the very simple concept of CrashMetrics). I sometimes use the example of crossing the road. Imagine it’s late, it’s dark, and it’s raining. If you cross the road there is a 5% chance of being hit by a bus and killed. That does not mean that tomorrow 95% of you goes to work! No, you assume the worst, because it is so bad, and cross the road elsewhere.

Certainly there is little role for Extreme Value Theory (EVT) in its fiddly, detailed sense. Consider these two statements about the same portfolio: “According to Gaussian distributions the expected time to bank collapse is 10^25 years” and “According to EVT the expected time to bank collapse is 50 years.” The difference between these statements should only be of academic interest. Such a portfolio must be protected asap. Of course, many people would be happy with such a portfolio because 50 years is still longer than a trading career. Such people should not be in positions of responsibility. As I said above, “risk management must be consistent with protecting the wider interests of the institution rather than being easy to manipulate towards the narrow interests of some employees.” To recap, if it’s bad enough to cause bank/fund collapse you don’t look at probabilities. Handle extreme events with worst-case scenario analysis.

Level 3: Scenarios which are so dire as to affect the world directly. I always use the example of invasion by aliens as an example, since there are whole bodies of literature and movies that have explored the effects of such an event, but we have little idea of the probability! If your hedge fund will collapse in the event of invasion by aliens, or drying up of oil supplies, or decimation of the world’s population by bird flu, then I wouldn’t necessarily change your portfolio! You’ll have other things to worry about!

Finally, a small proposal. I would like to see risk management forced to engage in the following task, the reverse engineering of a bank collapse. Start with your current portfolio and imagine being called into the big boss’s office to be told that the bank has lost $50billion. Having put yourself in the frame of mind of having already lost this amount, now ask yourself what could have caused this to happen. As Einstein said “Imagination is more important than knowledge.” This should be the mantra for those in risk management. There is always going to be something that will come as a surprise at the time but with hindsight you realise could have been expected (if not necessarily predicted). Once you have figured out what could have caused this loss then you ask about the likelihood of this happening. The result of that analysis then determines what you should do with the portfolio. If, for example, the answer is simply that a fall in property prices caused the loss then you must get out this very instant, before it actually happens. You see the idea, work backwards from the result, the loss, rather than pick (possibly convenient) scenarios and look at the effects. Then estimate the likelihood of the chain of events happening, and act accordingly. Going the other way is more open to abuse. Scenario testing is a beautiful concept, if one gets to choose the scenarios to test. And those of weak character will not, of course, test any scenario that might jeopardize a juicy trade.

P

I’ve been critical of much of quant modelling for many years. I don’t like the assumptions, the models, the implementations. I’ve backed this up with sound reasons and wherever possible tried to find alternative approaches that I think are better. I don’t honestly expect to change the world, much, but, hey, I do what I can. Human nature is such that very often things have to go from bad to worse to bloody awful before the necessary paradigm shift happens. I hope we are close to that point now.

Who am I kidding? As another hedge fund disappears thanks to mishandling of complex derivatives, I predict that things are going to get even worse.

When it was just a few hundred million dollars here and there that banks were losing we could all have a good laugh at the those who had forgotten about convexity or whatever. But now the man in the street has been affected by these fancy financial instruments. It’s no longer a laughing matter.

Part of the problem is that many of the people who produce mathematical models and write books know nothing about finance. You can see this in the abstractness of their writing, you can hear it in their voices when they lecture. Sometimes they are incapable of understanding the markets, mathematicians are not exactly famous for their interpersonal skills. And understanding human nature is very important in this business. It’s not enough to say “all these interacting humans lead to Brownian Motion and efficient markets.” Baloney. Sometimes they don’t want to understand the markets, somehow they believe that pure mathematics for its own sake is better than mathematics that can actually be used. Sometimes they don’t know they don’t understand.

Banks and hedge funds employ mathematicians with no financial-market experience to build models that no one is testing scientifically for use in situations where they were not intended by traders who don’t understand them. And people are surprised by the losses!

I realized recently that I’ve been making a big mistake. I’ve been too subtle. Whenever I lecture I will talk calmly about where models go wrong and where they can be dangerous. I’ve said CDO models are bad because of assumptions about correlation. I’ve pointed out what you can do to improve the models. I’ve talked about hidden risks in all sorts of instruments and how sensible use of mathematics will unveil them. I’ve explained why some numerical methods are bad, and what the good methods are. But, yes, I’ve been too subtle. I now realise that one has to shout to be heard above the noise of finance professors and their theorems. Pointing people in the right direction is not enough. Screaming and shouting is needed.

So here, big and bold, gloves off, in capital letters (for this seems to help), are some fears and predictions for the future.

THERE WILL BE MORE ROGUE TRADERS: While people are compensated as they are, while management look the other way to let the ‘talent’ do whatever they like, while people mistake luck for ability, there will be people of weak character who take advantage of the system. The bar is currently at €5billion. There will be many happy to stay under that bar, it gives them some degree of anonymity when things go wrong. But that record will be broken.

GOOD SALESMEN WILL HOODWINK SMART PEOPLE: No matter what you or regulatory bodies or governments do we are all a pushover for the slick salesman.

CONVEXITY WILL BE MISSED: One of the more common reasons for losing money is assuming something to be known when it isn’t. Option theory tells us that convexity plus randomness equals value.

CORRELATION PRODUCTS WILL BLOW UP DRAMATICALLY: This means anything with more than one underlying, including CDOs. Stop trading these contracts in quantity this very minute. These contracts are lethal. If you must trade correlation then do it small and with a big margin for error. If you ignore this then I hope you don’t hurt anyone but yourself. (I am sometimes asked to do expert-witness work. If you blow up and hurt others, I am very happy to be against you in court.)

RISK MANAGEMENT WILL FAIL: Risk managers have no incentive to limit risk. If the traders don’t take risks and make money, the risk managers won’t make money.

VOLATILITY WILL INCREASE ENORMOUSLY AT TIMES FOR NO ECONOMIC REASON: Banks and hedge funds are in control of a ridiculous amount of the world’s wealth. They also trade irresponsibly large quantities of complex derivatives. They slavishly and unimaginatively copy each other, all holding similar positions. These contracts are then dynamically hedged by buying and selling shares according to mathematical formulae. This can and does exacerbate the volatility of the underlying. So from time to time expect to see wild market fluctuations for no economic reason other than people are blindly obeying some formula.

TOO MUCH MONEY WILL GO INTO TOO FEW PRODUCTS: If you want the biggest house in the neighbourhood, and today not tomorrow, you can only do it by betting OPM (other people’s money) big and undiversified. There are no incentives for spreading the money around responsibly.

MORE HEDGE FUNDS WILL COLLAPSE: You can always start a new one. Hell, start two at the same time, one buys, the other sells!

POLITICIANS AND GOVERNMENTS WILL REMAIN COMPLETELY IN THE DARK: Do you want to earn £50k p.a. working for the public sector, or £500k p.a. working for Goldman Sachs? Governments, who are supposed to set the rules, do not even know what the game is. They do not have the slightest clue about what happens in banks and hedge funds. Possibly, for the same reason, London will lose out to New York as a world financial centre.

P

One of the more quantie aspects of recent financial crises has been the valuation of CDOs, the highly complex credit instruments depending upon the behaviour of many, many underlyings.

Now your typical quant favours just one tool to capture the interaction of two assets, and that tool is correlation. Of course, this is a very unsubtle tool which is being used to capture the extremely subtle interactions between companies. And when you have 100 underlyings the number of correlations will be 100x99/2=4,950. All of them unstable, all of them meaningless. Yet you will often find complex derivatives being priced using such wildly nonsensical data. Sometimes, in the interests of simplicity, some instruments are priced assuming all correlations are the same. The rationale behind this might be to return some robustness, in the sense that you might be more sure of one parameter than of 4,950 of them. If only it were that simple!

Much more on correlation in a later blog.

Returning to the subject of CDOs. I conducted a simple experiment on a CDO with just three underlyings. Really just a toy model to illustrate some important issues. I started by assuming a single correlation (instead of three) to capture the relationship between the underlyings, and a ‘structural model.’ I then looked at the pricing of three CDO tranches, and in particular their dependence on the correlation. Look at the figure above, but ignore the exact numbers. First observe that the Senior Tranche decreases monotonically with correlation, the Equity Tranche is monotonically increasing, with the Mezzanine Tranche apparently being very insensitive to correlation.

Traditionally one would conduct such sensitivity experiments to test the robustness of ones prices or to assist in some form of parameter hedging. Here, for example, one might conclude that the value of the Mezzanine Tranche was very accurate since it is insensitive to the correlation parameter. For a single correlation ranging from -0.25 to +0.5 the Senior Tranche value ranges from 0.643 to 0.797, the Equity Tranche from 0.048 to 0.211, and the Mezzanine Tranche from 0.406 to just 0.415. (Remember, don’t worry about the numbers in this toy model, just look at the structure.) If you are confident in your valuation of the Mezzanine Tranche, then so will the next bank, and with competition being what it is, bid-offer prices will converge.

Such an analysis could not possibly be more misleading, such a conclusion could not possibly be more incorrect and such a response could not possibly be more financially dangerous.

Consider a more interesting, and more realistic, world in which correlation is state-dependent. Now allowing correlation to vary from -0.25 to +0.5, but not constant, and depending on ‘state,’ you will find that the Senior Tranche still varies from 0.643 to 0.797, the Equity Tranche still varies from 0.0408 to 0.211, but now the Mezzanine Tranche varies from 0.330 to 0.495, a factor of 18 in the sensitivity compared with the traditional naïve analysis. The reason is simple, inside the Mezzanine Tranche structure there is a non-monotonic sensitivity to correlation which is masked when calculating the value; sometimes more correlation is good, sometimes more correlation is bad. (For the Senior Tranche correlation is always bad, for the Equity Tranche correlation is always good.)

Why on earth people thought it a good idea to measure sensitivity to a parameter that has been assumed to be constant escapes me still.

The moral of this example is simple, there is far more risk inside some of these instruments than you could ever hope to find with classical analyses. Stop using such convoluted models, use more straightforward models and start thinking about where your models’ sensitivities really lie. Your models can fool some of the people all of the time, and all of the people some of the time, but your models cannot fool all of the people all of the time.

P

One of the first lessons in any course on quantitative finance will be about portfolio construction and the benefits of diversification, how to maximize expected return for a given level of risk. If assets are not correlated then as you add more and more of them to your portfolio you can maintain a decent expected return and reduce your risk asymptotically to zero. (Risk falls off like the inverse square root of the number of different uncorrelated assets.) Colloquially, we say don’t put all your eggs into one basket.

Of course, that’s only theory. In practice there are many reasons why things don’t work out so nicely. Correlations never behave as they should, the relationship between two assets can never be captured by a single scalar quantity. We’ll save discussion of correlation for another time! For the moment I’m more worried about people or banks not even attempting to diversify.

Part of the problem with current mechanisms for compensation is that people are encouraged to not diversify. I don’t mean “not encouraged to,” I do mean “encouraged to not.” Imagine you have just graduated from a respectable Ivy League university with a thorough academic understanding of risk and return but without a similar level of appreciation of politics in an employment environment. You start your job as a junior trader determined to help your bank, and yourself, make money. You thus look for opportunities that diversify some of the bank’s risk while maintaining a good, solid return. Now if, as a result of the diversification you find yourself either losing money when the others make it, or making it when they lose it, then you are stuffed. Lose money when all around are making it, you’re fired. Make money when all around are losing it? Expect a big bonus? No way! Your profits will help to bail everyone else out and no one gets a bonus, even you. No, you should do the same as everyone else. As Keynes said, “It is better to fail conventionally than to succeed unconventionally.”

There are many ways to diversify, across contracts, asset classes, time horizons, what letter of the alphabet the contract starts with, etc. Even across models. I am in two minds about diversifying using different models possibly for the same contract. The scientist in me obviously wants to see each bank trying to find the best model, but I can appreciate that less harm might be done if people pick prices and greeks at random (my slightly cynical view of using multiple models!).

My scientist within would prefer each bank/hedge fund to have ‘one’ model, with each bank/hedge fund having a different model from its neighbour. Gives Darwin a fighting chance! I see so many banks using the same model as each other, and rarely are they properly tested, the models are just taken on trust. (And as we know from everyone's problems with calibration, when they are tested they are usually shown not to work but the banks still keep using them. Again, to be discussed later.)

There are fashions within investing. New contracts become popular, profits margins are big, everyone piles in. Not wanting to miss out when all around are reaping huge rewards, it is human nature to jump on any passing bandwagon. Again this is the exact opposite of diversification, often made even worse because many of those jumping on the bandwagon (especially after it’s been rolling along for a while) don’t really have a clue what they are doing. To mix metaphors, many of those on the bandwagon are in over their heads.

The key point to remember is something that every successful gambler knows (a phrase I use often, but shouldn’t have to), no single trade should be allowed to make or break you. If you trade like it is then you are doomed.

We all know of behavioural finance experiments such as the following two questions. First question, people are asked to choose which world they would like to be in, all other things being equal, World A or World B where

A. You have 2 weeks’ vacation, everyone else has 1 week

B. You have 4 weeks’ vacation, everyone else has 8 weeks

The large majority of people choose to inhabit World B. They prefer more holiday to less in an absolute sense, they do not suffer from vacation envy.

But then the second question is to choose between World A and World B in which

A. You earn $50,000 per year, others earn $25,000 on average

B. You earn $100,000 per year, others earn $200,000 on average

Goods have the same values in the two worlds. Now most people choose World A, even though you won’t be able to buy as much ‘stuff’ as in World B. But at least you’ll have more ‘stuff’ than your neighbours. People suffer a great deal from financial envy.

In banking the consequences are that people feel the need to do the same as everyone else, for fear of being left behind. Again, diversification is just not in human nature. Now none of this matters as long as there is no impact on the man in the street or the economy. (Although the meaning of ‘growth’ and its ‘benefits’ are long due a critical analysis.) And this has to be a high priority for the regulators, banks clearly need more regulatory encouragement to diversify.

Meanwhile, some final quick lessons. Trade small and trade often. Don’t try to make your retirement money from one deal. And work on that envy!

P

For every buyer there is a seller and vice versa. So at a first glance derivatives is a zero-sum game, someone wins and someone loses, and the amounts are identical. Therefore there can be no impact on the rest of us or on the economy if two adults want to bet large sums on money on the outcome of what may just be the roll of a dice. Well, it isn’t that simple for at least two reasons.

First, many of those trading derivatives are hedging with the underlying and this can affect the behaviour of the underlying: hedging positive gamma can decrease volatility and hedging negative gamma can increase volatility. When hedging positive gamma (i.e. replicating negative gamma) as the price rises you have to sell more of the underlying, and when the price falls you buy back, thus reducing volatility if your trades are in sufficient size to impact on the market. But hedging negative gamma is not so nice, you buy when the price rises and sell when it falls, exacerbating the moves and increasing volatility. The behaviour of stocks on which there are convertible bonds is often cited as a benign example, with the rather more dramatic '87 crash, replicating a put i.e. hedging negative gamma, as the evil version. (See Wilmott, P. and Schonbucher, P 2000 The feedback effect of hedging in illiquid markets. SIAM J. Appl. Math. 61 232—272, also PS’s dissertation.) You will probably find some reluctance for people to sell certain derivatives if they are not permitted to dynamically hedge. (Not that it works particularly well anyway, but that is what people do, and that is what most pricing theory is based on. Static hedging with other derivatives is better, and does not cause such (in)stability problems.)

(We’ve had newspaper headlines about damage done by excessive risk taking, whether by single, roguish, individuals or by larger institutions such as hedge funds, or banks and corporates investing in products they don’t fully understand. I expect it won’t be long before the attempt to reduce risk is the cause of similar headlines!)

Second, with the leverage available with derivatives it is possible, and actually rather simple, for people to trade so much as to get themselves into a pickle when things go wrong. This has many consequences. For example a trader loses his bank so much money that the bank collapses or is taken over, job losses ensue and possibly the man in the street loses his savings. Is wealth conserved during this process, as would be the case in a zero-sum game? I think not.

Of course, we don’t know what proportion of derivatives trades are being used for hedging, speculation with leverage, etc. and how many are being dynamically hedged. But while derivatives trading is such a large business and while pricing theory is underpinned by dynamic hedging then we can say that the game of derivatives is not zero sum. Of course, this should spur on the implementation of mathematical models for feedback…which may in turn help banks and regulators to ensure that the press that derivatives are currently getting is not as bad as it could be.

P

Some models are better than others. Sometimes even working with not-so-good models is not too bad. To a large extent what determines the success of models is the type of market. Let me give some examples.

Equity, FX and commodity markets: Here the models are only so-so. There has been a great deal of research on improving these models, although not necessarily productive work. Combine less-than-brilliant models with potentially very volatile markets and exotic, non-transparent, products and the result can be dangerous. On the positive side as long as you diversify across instruments and don't put all your money into one basket then you should be ok...at least as long as the market overall is going up!

Fixed-income markets: These models are pretty dire. So you might expect to lose (or make) lots of money. Well, it's not as simple as that. There are two features of these markets which make the dire modelling less important, these are a) the underlying rates are not very volatile and b) there are plenty of highly liquid vanilla instruments with which to try to hedge model risk. (I say "try to" because most model-risk hedging is really a fudge, inconsistent with the framework in which it is being used.)

Correlation markets: Oh, Lord! Instruments whose pricing requires input of correlation (FI excepted, see above) are accidents waiting to happen. The dynamic relationship between just two equities can be beautifully complex, and certainly never to be captured by a single number, correlation. Fortunately these instruments tend not to be bought or sold in non-diversified, bank-destroying quantities. (Except for CDOs, of course, see below.)

Credit markets: Single name instruments are not too bad. Again problems arise with any instrument that has multiple 'underlyings,' so the credit derivatives based on baskets...you know who you are. But as always, as long as the trades aren't too big then it's not the end of the world.

Where's the 'science' in this? The science comes in accepting right from the start that the modelling is going to be less than perfect. It is not true that one makes money from every instrument because of the accuracy of the model. Rather one makes money on average across all instruments despite the model. These observations suggest to me that less time should be spent on dodgy models, meaninglessly calibrated, but more time on models that are accurate enough and that build in the benefits of portfolios.

P

I attended some of the recent Savoy auction by Bonhams this week and I couldn't resist observing the events from a quant perspective! In particular, I was drawn back again to the question of valuation versus supply and demand. We are taught that value comes from some complicated mathematical analysis involving lognormal random walks and stochastic calculus. However, we all ought to know that value comes about by a more obscure and more interesting and usually ad hoc procedure, often involving little logic and certainly no maths, and sometimes quite a lot of emotion. (Think women and shoes.) All of this was seen at the Savoy auction.

Yes, there were people tut-tutting at the amount some were willing to pay for an ashtray, but they weren't those doing the buying. Those who bought the ashtrays probably had some doubts at the time, and also shortly afterwards, even this morning and maybe when they collect the ashtrays, but in the long run they'll at least have a funny story about themselves. (The latter not so easy to assign a value to, and certainly not risk neutral!)

Near the end of the three days there was a boring patch with 50 Savoy double beds going under the hammer, one after another. To amuse myself and in the spirit of scientific curiosity, I wrote down the 'time series' of prices for these identical items. Now here was a room full of the same people, bidding for identical items with a known and limited supply, but even in this rather dull scenario the results were interesting, the plot of the times series is shown. Observations: the price did settle down to a value around £50, but that wasn't exactly stable; absentee bids mostly caused dramatic increases in the price (I'm sure economists will get excited about 'information' at this point, but this was the least interesting observation); later absentee bids were very low, people perhaps hoping for drying up of demand(?); a few lucky or clever people even got the price down below the £50; individual bidders did not seem to show consistency in their bidding; losing bidders often took a break for a few lots before coming back in. None of this is other than perfectly obvious (and much already covered in auction theory, I hope) but in my experience you really have to keep reminding quants that they are human beings as well, and that they should draw inspiration from the mundane.

I bumped into Uri Geller at the auction. He had just successfully bid for...you guessed it, spoons! He's a very nice gentleman, and very kindly gave a few of us a private display of spoon bending!

P

A century or two ago, finance was the career for the less talented members of the family. Sons of the aristocracy would eventually go to sit in the House of Lords, while overseeing their property. One son would join the military, Catholic families would send a son off to the church. Perhaps if they were of an enquiring mind one son might become a scientist. But if a son turned out to be intellectually challenged he would be sent off to be 'something in the City.' This didn’t require any more brains than that required for an arts degree. This was the finance-is-for-artists (and long lunches) period, now long gone.

More often one now finds proper scientists working in finance. They have the analytical skills needed by investment banks and hedge funds. I imagine some must start out being frustrated by the lack of an established rigorous foundation for the subject. Where are the conservation laws? Where are the experimental results and the hypotheses? Quantitative finance has a well-used set of tools, but the popular models are essentially ad hoc.

Those in trading are undoubtedly pragmatists who really don’t care for the port-and-cheese side of finance, nor for compact theories. Can it be put in a spreadsheet and does it make money? That’s all that matters.

Unfortunately, most of the theory is built by axiomatists who really seem to believe in their models. These are the ones to be really frightened of. Speaking to them is like speaking to a god botherer, "there is but one stochastic volatility model and its name is Heston." (News flash: God and complete markets are simplifying assumptions that make life easier for the unimaginative, you aren't meant to believe in them once you've grown up!)

My feeling is that the best type of 'ist' working in finance is a pragmatic scientist, combining the curiosity and the scepticism of the scientist with the get-the-job-done attitude of the pragmatist.

P

That's it in a nutshell, supply and demand. Everything is driven by supply and demand. And if you want any financial principle as the foundation for a (scientific) theory then this is it, just like we have conservation laws in the physical sciences.

Of course, quantifying this may not be that easy. Attempts to explain asset prices by modelling interacting agents have not been fantastically successful, whereas simply saying that the end result of all those interactions is a stochastic differential equation, has been.

I do sometimes wonder if the typical etiolated quant has ever been into a shop and experienced supply and demand first hand by buying a pint of milk. Whenever a quant calibrates a model to the prices of options in the market he is saying something about the information content of those prices, often interpreted as a volatility, implied volatility. But really just like the price of a pint of milk is about far more than the cost of production, the price of an option is about much more than simple replication. The price of milk is a scalar quantity that has to capture in a single number all the behind-the-scenes effects of, yes, production, but also supply and demand, salesmanship, etc. Perhaps the pint of milk is even a 'loss leader.' A vector of inputs produces a scalar price. So, no, you cannot back out the cost of production from a single price. Similarly you cannot back out a precise volatility from the price of an option when that price is also governed by supply and demand, fear and greed, not to mention all the imperfections that mess up your nice model (hedging errors, transaction costs, feedback effects, etc.).

Supply and demand dictates everything. The role of assumptions (such as no arbitrage) and models (such as the continuous lognormal random walk) are to simply put bounds on the relative prices among all the instruments. For example, you cannot have an equity price being 10 and an at-the-money call option being 20 without violating a simple arbitrage. The more realistic the assumption/model and the harder it is to violate in practice the more seriously you should treat it. The arbitrage in that example is trivial to exploit and so should be believed. However, in contrast the theoretical profit you might think could be achieved via dynamic hedging is harder to realize in practice because delta hedging is not the exact science that one is usually taught. Therefore results based on delta hedging should be treated less seriously.

To summarize: Supply and demand dictate prices, assumptions and models impose constraints on the relative prices among instruments. Those constraints can be strong or weak depending on the strength or weakness of the assumptions and models.

P

Having for most of my quant career attacked the majority of mathematical modelling in finance for being 'unscientific' (in the sense that the theories are rarely tested before being used, and when tested usually fail miserably) I feel somewhat heartened by the recent anti-Black-Scholes movement.

Unfortunately this countermovement, although healthy in provoking debate, also does not quite match my (presumably rather high!) standards of rigour. With the aim of putting some science back into the quant debate I'm going to spend a few blogs highlighting what I think are the weaknesses of financial modelling, and its strengths. I will even be defending Black-Scholes at times! Being scientific does not mean being without emotion, so although my reasoning will be logical my language will almost certainly, and as always, get quite demonstrative.

Topics to look out for, in no particular order: supply and demand; accuracy in different markets; distributions and fat tails; volatility and robustness; hedging errors; diversification; correlation; etc.

In the meantime, listen to the recording of BBC Radio 4's recent More or Less programmme which discusses what quants do in the context of recent market upsets. The podcast of this programme may be found here. (The quant section starts after 9mins 25 secs.)

P

I couldn't resist this rather trivial blog, just a comment really on happenings at a recent quantie dinner. In attendance, going clockwise around the table at Union Square Cafe, PW, Bruno Dupire, Salih Neftci, Peter Carr, Jim Gatheral and Emanuel Derman.

Discussing the validity of all this Black-Scholes stuff that has got so much bad press recently, JG says "I have a nice apartment in **** thanks to Black-Scholes being correct" to which yours truly responded "Well, I have a nice flat in **** thanks to it being wrong!" Now, you can sort of see how that can be! It depends upon a) to what use you put BS and, crucially, b) how much profit margin you can add to any deal!

Not naming any more names, it was clear from the rest of the conversation (which concerned numerical integration in infinite-dimensional spaces!) that, if this sample is to be trusted and extrapolated, a) half of all quants actually believe all this math finance modelling nonsense, b) one third of all quants don't, and are rather concerned for the mental health of the first half, and c) one sixth of all quants either don't care or have maybe been enjoying the excellent wine list at Union Square Cafe too much!

P

Nassim Taleb and I were lecturing in Mexico City at the weekend (thanks, RiskMathics!). In our free time we visited Teotichucan with its two impressive pyramids. The larger (Pyramide del Sol) is the third largest pyramid in the world by volume. The two pyramids and the surrounding temples, buildings, arenas, etc. were constructed about two thousand years ago and took two hundred years to complete.

Pyramide del Sol

The people of Teotichucan were divided into three castes. The lowest being the farmers, the middle being the builders (15,000 involved in the construction of pyramids and temples), and the highest being the priests and the applied mathematicians. Yes, our guide really did specify that they were applied mathematicians!

A priest and an applied mathematician.

Watching Gordon Brown speaking at the Labour Party conference I was reminded of that quotation by the insightful Georges Clemenceau, "Not to be a socialist at 20 is proof of want of heart, to be one at 30 is proof of want of head." So perhaps Labour's transformation over the last ten years has not been a cynical ploy to get power but instead a perfectly natural maturing that comes with age.

Entirely unoriginal, but I couldn't resist taking this photo from the dais before my recent lecture at the HSBC Global Markets Conference, Latin America, held at Los Cabos, Mexico. (One of the audience members suggested it should be entitled "Hands up if you thought Wilmott's talk was the best of the conference?" But he suggested that before I spoke.)

Los Cabos is a desert-like resort near the tip of Baja Sur. Multi-million dollar properties are being snapped up by retiring Americans. Call me old fashioned, but with global warming isn't a desert the last place you would want to buy a retirement home?

A very warm climate, and an even warmer audience. Thank you, HSBC!

And so my travels take me to the OMX in Stockholm where I had been invited to give a few lectures (fear and greed, blackjack and vol arb). The other two speakers were Joe Corona and Marc Faber, both of whom were foretelling the imminent collapse of global stockmarkets. It seems that at the moment we all want to hear that the end is nigh. Certainly, it makes for an entertaining speech, and nothing brings people together better than knowing that everyone is going to suffer, and not just you. 'Hope' is the trouble, as John Cleese says in Clockwise "I can take the despair. It's the hope I can't stand." My own view is equally pessimistic, but rather than blaming house prices and climate change, I tend to favour the 'global village' villain, i.e. thanks to easy long-distance travel and technology/telecoms we can no longer rely on Darwin to get us out of a mess. Survival of the fittest is not going to work when we all live or die as a single organism. (Mind you, this won't be an argument that will worry Americans, since only 50% of them believe in evolution. Praise the Lord, and pass the ammunition.)

The moderator for some of the lectures was Jon Briggs, famous as the voice of The Weakest Link. A true professional, he was totally unperturbed by the audience's refusal to laugh at his jokes. Don't get me wrong, they were a keen and bright audience (for example, no snoring even after a particularly heavy session the night before), but they were very shy and would not ask questions about the lecture nor would they laugh unless a joke contained a reference to suicide or sex.

P

The Cannes Film Festival, of sorts, began in 1939 as a palatable alternative to the Venice Film Festival which had by then developed a nasty habit of giving all its awards to chums of Hitler and Mussolini. The Festival as we now know it began in 1946, and this year is the 60th Cannes (lack of money meant that there was no festival in 1948 and 1950). Last week I was lucky enough to attend the opening days as a guest of one of the few British films in any of the main competitions.

To draw similarities with more familiar terrain, the festival is not unlike a larger version of a major quant conference, but with more sun, sand and sex. There are the big premieres and innumerable smaller movies. The celebs attend only the biggest screenings, but will often sneak off as soon as they can. You can see the parallels. (I personally am more tolerant of boring, unimaginative films than I am of boring, unimaginative quant research!)

Being one of the plebs, I did actually see a few movies (as well as doing a decent amount of partying). Michael Moore's Sicko was excellent. I am sure it will be easy to pick holes in his data, but the big picture is spot on. I was at the premiere of Leonardo di Caprio's Al-Gore-without-the-science-but-with-a-real-Native-American-chief-instead 11th Hour. Daryl Hannah sat in front of me, you can see the back of her head in the photo below. (I know what you are thinking!)

Another film I must mention is the French Heros. Thirty minutes too long, it really drags, but then the main protagonist puts on the clown makeup, a Gene Kelly song, and some drum'n'bass and suddenly it's a whole new film. Worth sitting though the first 90 minutes for some memorable images towards the end.

Paul's Tips for Cannes:

1. Spend a few hours early on familiarizing yourself with all the venues and the system generally. Don't believe anything that anyone tells you, no matter how official they are. Printed info is usually correct however.

2. To get into the more exclusive screenings you must be a producer, director or someone on the buy side. (You need a pass with an 'R' on it.) Actors, editors, DOPs, etc. can get into many events, but PR people can't get into anything!

3. For the ladies, wear flat shoes whenever possible. You will spend many hours doing "the walk", up and down La Croisette, from screenings, to parties, etc. and back.

4. Cannes is surprisingly inexpensive. Except for taxis, that is. There is neither rhyme nor reason to the pricing, all that is certain is that they cost an arm and a leg.

5. In the screenings the celebs have reserved seating about one quarter of the way back from the screen. That's except for DH, who always sits in front of me. She must like younger men!

P

At Espen Haug's recent book launch I gave a lecture on how to win at Blackjack, and the Kelly criterion for money management. The MP3 of that lecture, introduced by Einar Bonnevie, is attached (see 'Download' link below). In a few weeks' time we will probably put the Powerpoint lecture on wilmott.com as well.

P

Espen was kind enough to invite me to speak at the launch of his Derivatives: Models on Models at the Oslo Stock Exchange last week.

My talk was on Blackjack (I hope to have it on the site soon) and Espen spoke about the missing history of derivatives (his lecture also to be posted shortly).

This was followed by lots of very nice and free (NNT, are you reading this?!) food and drink, and an exhibition of some derivatives art.

The evening ended for some of us at around 2am, shortly after a visit to "Zinatra's" for karaoke. (Fortunately Mrs W was there to sing Smells Like Teen Spirit, I swear I was only miming!)

Had dinner with Nassim Taleb last night, at Carluccio's off Russell Square. I realise now how he got into this Black Swan thing...his entire life is lived in the tail, every time we meet there is some major event happening. Last night it was his entry into the New York Times bestseller list at an impressive (even to him!) number 5. The photo was taken a couple of minutes after him getting the news from his publisher.

I only see NNT a few times a year, but always there is excitement. Unfortunately, not always of the good kind. We were walking along Bishopsgate a few years ago when a couple of planes flew into the WTC. Our discussions had to be cut short since he was managing a hedge fund which had a lot of tail exposure...long, of course. Another time we were giving one of our famous training courses together when we almost had to evacuate the building...it was the 21 July failed bombings in London.

I thought my life was quite unusual but I can't compete! And he seems to get younger as he gets older, clearly this life is good for him. (And, yes, Nassim, you are even getting slimmer!)

Nassim Nicholas Taleb's The Black Swan is available from all good online bookshops, such as here.

P

Paul Shaw (CEO of 7city Learning) and I have just returned from a week in China, starting in Shanghai and then going on to Beijing. We were invited there to explain about the Certificate in Quantitative Finance. I spoke at various exchanges, at universities and with government officials. We met with Mr Xia Bin, Director General and Research Fellow of the Financial Research Institute, which is the finance arm of the Development Research Center of the State Council of the P.R.C. The FRI will be endorsing the CQF as the best-practice financial engineering qualification for those working in or wanting to work in the emerging derivatives market in China.

As well as being a very successful business trip it was also an impressive social whirl. I have never eaten so much so consistently in all my life. Every day we had at least one 'banquet' (usually two) of typically a dozen courses. These banquets would sometimes take place in what I can only describe as Chinese versions of country estates, just a few of us and the staff in the middle of acres of lakes and lawns. What surprised me was that in eight days we only had two, small, bowls of rice. Given the vast amount we ate, with so few carbs, I think I can now vouch for the Atkins diet! The photo shows me having a late-night snack of scorpion, bought at a street food stall in Beijing.

If all goes well I hope to return to China later this year.

The Market Price of Risk is a much-neglected quantity. It is a concept that you'll find in models of incomplete markets. In a nutshell, if a market is incomplete and you can't hedge away some risk then you have to say how that risk is valued. The Market Price of Risk (MPR) quantifies this, and allows you to price all derivatives on the same underlying(s) consistently with each other. (If they have the same source(s) of risk then those risks ought to be treated alike.)

You don't see it discussed much because we tend to talk about the risk-neutral world alone, whereas the MPR defines the difference between the real and risk-neutral worlds. (You'll find the MPR in the drift of the stochastic variables.) And once you calibrate to the market prices of derivatives you won't see it anymore. (That's why it's hard to spot in the HJM and BGM models which calibrate right from the start.)

Nevertheless this quantity is very important since it levels the playing field for all investments, no matter how complex. You see it in Markowitz's Modern Portfolio Theory, and the MPR for each source of randomness and the correlations between them can be used to choose optimal portfolios.

But what does the MPR look like? Is it a time-stable constant, representing the compensation for taking risk of rational investors. Is it slowly varying representing the changing attitude towards risk of different generations? No, neither of these. The figure above shows what the MPR looks like for US interest rates. (Technically, this is the Market Price of Spot Interest Rate Risk.) It appears random. The spikes can be interpreted as over or undercompensation for taking risk, fear versus greed.

Details of how to find this MPR and how to use it for modelling as a second stochastic factor (incidentally, introducing the Market Price of Market Price of Risk Risk!) and also for trading, as in stat arb, can be found in Ahmad, R & Wilmott, P 2007 The Market Price of Interest-rate Risk: Measuring and Modelling Fear and Greed in the Fixed-income Markets. Wilmott magazine, January 64-70

The book FAQs in QF (FAQQF, pronounce it how you will!) has just been released. You can hear and 'see' me read the preface in the attached MP4 video. (If I sound a little tired in the video it's because I was just recovering from a head cold.)

The book can be bought here.

P

Following on from Emanuel's Continental Airlines spelling-error blog...

Virgin Atlantic is the worst airline for such errors, The Independent newspaper now matches The Grauniad typo for typo, even the BBC's standards have dropped so that on news bulletins you will, fortunately still only occasionally, find mistakes on their 'tickertapes.'

Now here's my favourite.

I was at a music festival this summer. There I saw a young gentleman sporting a fluorescent green Mohican. His chest was bare, and on it, in a very large gothic font, were the words:

"Life Wont Wait"

Indeed, nor will apostrophes. At least, you can't accuse him of not practising what he preaches.

P

Earlier this year, I gave the keynote speech at the Palisade Conference in exotic Heathrow (Monte Carlo itself has gone terribly downmarket of late, dahlings). This speech is now available as an audio podcast. Palisade incorporate Monte Carlo simulation tools as part of their powerful risk analysis software.

Download the attached.

The Connaught Square Squirrel Hunt is the world’s only urban hunt and the first hunt club founded after the 2004 hunting ban. Their recent meet was on Sunday, 17th September, starting, as they usually do, with a glass of port at the Duke of Kendal pub, Connaught Street.

The CSSH drag-hunt squirrels across Hyde Park, meaning that one member runs through the park with an old sock on a string, and the dog chases after this pretend 'squirrel.' Close on the paws of the dog are the hundred or so sweating followers, trying to keep up. After no more than thirty seconds the dog catches the sock, has a good chew, and after a short break for everyone to catch their breath, it all starts again.

Here's Dillon and master, on 'horseback.' On the way to Hyde Park we all stop outside 29 Connaught Square, the retirement home for Tony Blair, for a photo opp for the horde of media. The hunt say they named themselves after this square to remind TB about what they call his worst piece of legislation.

The 'squirrel' is just an old sock, but Dillon still seems to enjoy the chase.

Your fearless reporter gets trampled by the mob. And Dillon gets his breath back for a repeat performance.

The purpose of the hunt is explained on their website as follows.

It is absurd that the Hunting Act prohibits you from encouraging a dog to chase a squirrel. More than that, it is frightening that your dog could be put down and you could be fined £5,000 for saying “Go on Rover, get after it!”.

What is really sinister though, is that if a policeman thought you might want your dog to chase a squirrel in the future, he could raid your home and confiscate evidence.. All this without a warrant from a Magistrate: even a suspected burglar has more rights than a suspected Squirrel-Hunter!

When we realised that the innocent habits of most dog-owners were illegal, it was clear that this should be publicised as much as possible. People should be made aware of the law and how to act within it. And the more people that think the Hunting Act is an ass, the more people will support its repeal.

Not wanting to compete with the Collector's 'The World is My Office,' of course, but even on top of a volcano one can check one's emails.

This is Haleakala, the volcano on the East of Maui. Haleakala hasn't erupted since 1790. (Kilauea on neighbouring Big Island has been in a state of eruption for twenty years, and you can see the lava running into the ocean.) The summit of Haleakala is just over 10,000 feet above sea level, but if it weren't for the other 20,000 feet being submerged this volcano would be taller than Mount Everest.

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As promised, here is some of the mathematics behind hedging options when you think that there is an arbitrage opportunity.

Let's keep the problem simple. You are in a Black-Scholes world. Volatility is constant. But the market is pricing an option using the wrong volatility, implied volatility is lower than actual volatility, so the option is cheap. You buy the option, but then to make money you must hedge away market risk using the underlying. Do you use a delta based on the actual volatility or on implied volatility?

Case 1: Hedge using actual volatility

If you use actual volatility then you make a guaranteed profit, whose present value is the difference between Black-Scholes using actual volatility and Black-Scholes using implied:

where is actual volatility and is implied. This is the guaranteed profit, regardless of how the stock behaves, it is path independent. Unfortunately, you get wild P&L swings hedging this way.

Case 2: Hedge using implied volatility

It is more common to hedge using implied volatility. Now the profit is path dependent, but at least there aren't the wild swings from day to day. Each time step you make a profit of

This is always positive (as long as actual volatility is greater than implied) but the total by expiration depends on how the stock behaves.

The expected profit is then

This depends on the growth rate of the stock . Note that this is the expected profit. Although you won't lose money hedging this way you will not know a priori how much profit you will make.

So would you rather have a known profit but with daily P&L swings, or a positive daily P&L but random final profit? Perhaps the deciding factor is that when you hedge with implied volatility you will make a profit whenever you are on the right side of the trade (buy when implied vol is lower than actual and sell when it is higher). You can see this from the above formula for the profit each time step.

All of the details are contained in Ahmad & Wilmott 2005 "Which free lunch would you like today, Sir?" Wilmott magazine, November issue.

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I continue to be staggered by the depth and detail of some people's understanding of complicated quant models while these same people have absolutely no appreciation of the bigger picture. A case in point is that of volatility modelling.

If you really get into the Heston stochastic volatility model you will find yourself having to do some numerical integration in the complex plane (thanks to the transform methods used to solve the governing equation). This can be quite tricky to do in practice. Is all that effort worth it? Well, in part this depends on how good the model is. So you might think people would test the accuracy of the model against the data. Do they do this? Rarely. It is deemed sufficient to calibrate to a static dataset of option values regardless of the accuracy of the dynamics of that dataset. Yes, I know you then hedge with vanillas to reduce model risk, but this is a fudge that is completely inconsistent with the initial modelling. The cynic in me says that the benefit of modelling in such oblivion is truly tested by the state of your bank balance at the end of the year. If you get a bonus, does it matter? I don't have too much of a problem with that, depending on where you are in the management structure. However, I suspect that this is not most people's justification for their inaccurate modelling. I suspect that people really do believe that they are doing good work, and the more complicated the mathematics the better.

So, many know all the ins and outs of the most advanced volatility models based in the classical no-arbitrage world. Well, what if your job is to find volatility arbitrage opportunities? "There's no such thing as a free lunch" is drummed into most quants, thanks to academics and authors who take an almost axiomatic approach to our subject (see Derman’s recent blog). Those who know the details of volatility arbitrage are few and far between. Take the example of how to hedge when you think that options are mispriced.

You forecast volatility to be much higher or lower than current implied volatility. Clearly this is an arbitrage opportunity. But to get at that profit you must hedge stock risk. Now, working within an otherwise very simple Black-Scholes world but with two volatilities, implied and forecast, how should you hedge and how much profit will you make?

I have attached an audio recording (MP3) of the lecture I gave on this topic at a recent conference in Amsterdam.

In my next blog I will give some of the details of this problem.

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Or maybe it really happened. Well, here are the photos to prove it. Hedgestock 2006.

Knebworth House, and some flash motors.

David Harding of Winton Capital. Winton Capital had one of the more laid-back tents.

The Credit Suisse tent. Thanks to them for inviting me to Hedgestock to give a little speech.

Jimi Hendrix is doing somersaults in his grave.

All together now, "One, two, three, what are we fighting for?"

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"Celebrating Derivatives" was name of the conference I attended in Amsterdam on Thursday. (I will upload an audio file of my lecture shortly, assuming the recording is acceptable.) I arrived towards the end of the conference so missed almost all of the talks except for Jim Gatheral speaking on volatility forecasting. This seems to be all the rage at the moment. I don't mean being late or Jim Gatheral are all the rage, anyway no more or less than usual, rather vol forecasting is.

The panel discussion (John Hull, Claudio Albanese, JG, Antoon Pelsser and me, overseen by Ton Vorst) was great fun. My favourite bit was John Hull trying to persuade the audience of the value of an MBA above more mathematical programmes. Now JH is a lovely man, but even he couldn't convince this audience! (Do you really get more interpersonal and business skills on an MBA than you should have picked up naturally by the age of six? I don't think so.)

JH had been talking about CDOs and CDO^2s, mentioning that he knew of some CDO^3s that had been done. Exp(CDO) anyone?

The photo above is of the gift I was given by the organizers. All of the speakers were given their own personalized cartoon...as JG said "of a lot of bald guys."

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I have attached an audio recording of a recent open evening for the CQF in New York. This may be of interest to anyone who would like to but can't make the final open evening for the June programme, in London on Thursday 25th May.

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In April 2006 I spoke at an IQPC conference in London on Correlation Trading. My contribution concerned three topics, all of which I have used successfully in practice:

- The first is a useful technique for backing out reliable stochastic models for random financial variables. It uses the increments in the variable to estimate the volatility structure (in this case, the volatility of volatility) and the steady-state distribution to back out the drift structure. It's a technique I've used for modelling volatility, interest rates and commodity prices.

- Second is the profit you make from buying/selling and hedging incorrectly priced options, simple volatility arbitrage.

- Third, CrashMetrics, the ever so simple stress test for extreme markets when assets become very highly correlated.

I'll be speaking on the second of these subjects in detail at a conference in Amsterdam on June 1st.

The attached (rather large, 17Mb) file is the audio recording of my London lecture.

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