Derivatives E-Book

Espen Gaarder Haug

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 778

Veröffentlichungsjahr: 2013

Beschreibung

Derivatives Models on Models takes a theoretical and practical look at some of the latest and most important ideas behind derivatives pricing models. In each chapter the author highlights the latest thinking and trends in the area. A wide range of topics are covered, including valuation methods on stocks paying discrete dividend, Asian options, American barrier options, Complex barrier options, reset options, and electricity derivatives. The book also discusses the latest ideas surrounding finance like the robustness of dynamic delta hedging, option hedging, negative probabilities and space-time finance. The accompanying CD-ROM with additional Excel sheets includes the mathematical models covered in the book. The book also includes interviews with some of the world's top names in the industry, and an insight into the history behind some of the greatest discoveries in quantitative finance. Interviewees include: * Clive Granger, Nobel Prize winner in Economics 2003, on Cointegration * Nassim Taleb on Black Swans * Stephen Ross on Arbitrage Pricing Theory * Emanuel Derman the Wall Street Quant * Edward Thorp on Gambling and Trading * Peter Carr the Wall Street Wizard of Option Symmetry and Volatility * Aaron Brown on Gambling, Poker and Trading * David Bates on Crash and Jumps * Andrei Khrennikov on Negative Probabilities * Elie Ayache on Option Trading and Modeling * Peter Jaeckel on Monte Carlo Simulation * Alan Lewis on Stochastic Volatility and Jumps * Paul Wilmott on Paul Wilmott * Knut Aase on Catastrophes and Financial Economics * Eduardo Schwartz the Yoga Master of Quantitative Finance * Bruno Dupire on Local and Stochastic Volatility Models

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Author’s “Disclaimer”

Introduction

Derivatives Models on Models

Nassim Taleb on Black Swans

Chapter 1: The Discovery of Fat-Tails in Price Data

Edward Thorp on Gambling and Trading

Chapter 2: Option Pricing and Hedging from Theory to Practice: Know Your Weapon III*

1 The Partly Ignored and Forgotten History

2 Discrete Dynamic Delta Hedging under Geometric Brownian Motion

3 Dynamic Delta Hedging Under Jump-Diffusion

4 Equilibrium Models

5 Portfolio Construction and Options Against Options

6 Conclusions

Alan Lewis on Stochastic Volatility and Jumps

Chapter 3: Back to Basics: A New Approach to the Discrete Dividend Problem*

1 Introduction

2 General Solution

3 Dividend Models

4 Applications

Appendix A

Appendix B

Emanuel Derman the Wall Street Quant

Chapter 4: Closed Form Valuation of American Barrier Options*

1 Analytical Valuation of American Barrier Options

2 Numerical Comparison

3 Conclusion

Peter Carr, The Wall Street Wizard of Option Symmetry and Volatility

Chapter 5: Valuation of Complex Barrier Options Using Barrier Symmetry*

1 Plain Vanilla Put—Call Symmetry

2 Barrier Put—Call Symmetry

3 Simple, Intuitive and Accurate Valuation of Double Barrier Options

4 Static Hedging in the Real World

5 Conclusion

Granger on Cointegration

Chapter 6: Knock-in/out Margrabe*

1 Margrabe Options

2 Knock-in/out Margrabe Options

3 Applications

Appendix

Stephen Ross on APT

Chapter 7: Resetting Strikes, Barriers and Time*

1 Introduction

2 Reset Strike Barrier Options

3 Reset Barrier Options

4 Resetting Time

5 Conclusion

Bruno Dupire the Stochastic Wall Street Quant

Chapter 8: Asian Pyramid Power

1 Celia in Derivativesland

2 Calibrating to the Term Structure of Volatility

3 From Geometric to Arithmetic

4 The Dollars

Appendix: Inside the Average Period

Eduardo Schwartz: the Yoga Master of Mathematical Finance

Chapter 9: Practical Valuation of Power Derivatives*

1 Introduction

2 Energy Swaps/Forwards

3 Power Options

4 Still, What About Fat-Tails?

Aaron Brown on Gambling, Poker and Trading

Chapter 10: A Look in the Antimatter Mirror*

1 Garbage in, Garbage Out?

2 Conclusion

Knut Aase on Catastrophes and Financial Economics

Chapter 11: Negative Volatility and the Survival of the Western Financial Markets

1 Introduction

2 Negative Volatility — A Direct Approach

3 The Value of a European Call Option for any Value — Positive or Negative — of the Volatility

4 Negative Volatility — The Haug interpretation

5 Chaotic Behavior from Deterministic Dynamics

6 Conclusions

Elie Ayache on Option Trading and Modeling

Chapter 12: Frozen Time Arbitrage*

1 Time Measure Arbitrage

2 Time Travel Arbitrage

3 Conclusion

Haug on Wilmott and Wilmott on Wilmott

Chapter 13: Space-time Finance The Relativity Theory’s Implications for Mathematical Finance*

1 Introduction

2 Time Dilation

3 Advanced Stage of Space-time Finance

4 Space-time Uncertainty

5 Is High Speed Velocity Possible?

6 Black-Scholes in Special Relativity

7 Relativity and Fat-Tailed Distributions

8 General Relativity and Space-time Finance

9 Was Einstein Right?

10 Traveling Back in Time Using Wormholes

11 Conclusion

Appendix A: Special Relativity and Time Dilation

Appendix B: Relationship Between Acceleration in Different Frames

Andrei Khrennikov on Negative Probabilities

Chapter 14: Why so Negative about Negative Probabilities?*

1 The History of Negative Probability

2 Negative Probabilities in Quantitative Finance

3 Getting the Negative Probabilities to Really Work in Your Favor

4 Hidden Variables in Finance

5 The Future of Negative Probabilities in Quantitative Finance

6 Appendix: Negative Probabilities in CRR Equivalent Trinomial Tree

David Bates on Crash and Jumps

Chapter 15: Hidden Conditions and Coin Flip Blow Up’s*

1 Blowing Up

2 Coin Flip Blow Up’s

Peter Jäckel on Monte Carlo Simulation

Index

Derivatives Models on Models

© 2007 John Wiley & Sons, Ltd

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Photographs, cartoons and paintings © 2007 Espen Gaarder Haug.

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A catalogue record for this book is available from the British Library.

ISBN 978-0-470-01322-9 (HB)

Author’s “Disclaimer”

This book contains interviews with some of the world’s top modelers, researchers, quants, quant-traders, gamblers and philosophers from Wall Street and academia. Their views stand on their own and many of them would probably not agree with each other. This reflects the great depth and width that I tried to get into this book. Also, the views in most of my chapters are my own and the many great modelers interviewed in this book will not necessarily agree with me.

This book should also be read with a sense of humor. By investing in this book you are getting a perpetual American option to read whatever you want and whenever you want, it is not your obligation to read it but your option. If you think the option was overvalued after investing in it you can always try to re-sell it, if you think it was undervalued you can simply buy more copies (options).

Introduction

“Derivatives: Models on Models” is a different book about quantitative finance. In many ways it is two books in one, first of all it contains a series of interviews with some of the world’s top modelers, researchers, quants, quant-traders, gamblers and philosophers from Wall Street and academia. On the top of this you get a series of technical chapters covering valuation methods on stocks paying discrete dividend, Asian options, American barrier options, Complex barrier options, reset options, and electricity derivatives. The book doesn’t stop there it also takes you into the tails of your imagination and discusses ideas like negative probabilities and space-time finance.

The title “Derivatives: Models on Models” deserves some explanation. It was my former co-worker Dr John Stevenson1 a great model trader in J.P. Morgan that first came up with the idea of the book title “Models on Models”, a title I later changed to “Derivatives: Models on Models”. First of all the book is about derivatives models; quantitative finance, option valuation, hedging, and some non-traditional topics in finance like negative probabilities and space-time finance.

“Models on Models” has multiple implications. First of all models are only models and derivatives models are themselves based typically on more fundamental underlying models. For example most derivatives models are based on classic probability theory, that itself is a model that we often take for granted. Many models are based on the assumption of Gaussian distribution (including many of my own formulas). Some of the biggest mistakes in trading and modeling are done because we forget that our models are typically also based on some implicit non-stated assumptions. Typically such models work well (or at least to some degree) most of the time, but then sooner or later the hidden conditions will show up and often cause unexpected problems.

“Models on Models” also points back at the many interesting interviews with many of the world’s top modelers and their views on their models: Clive Granger, Emanuel Derman, Edward Thorp, Peter Carr, Aaron Brown, David Bates, Andrei Khrennikov, Elie Ayache, Peter Jäckel, Alan Lewis, Paul Wilmott, Eduardo Schwartz, Knut Aase, Bruno Dupire, Nassim Taleb, and Stephen Ross, all share their great wisdom on quantitative models, trading, gambling and philosophy about modeling. These interesting and fascinating interviews are spread throughout the book. These are modelers with very different backgrounds and personalities; it has been a great pleasure to learn from them through their publications, presentations, and in particular through the interviews for this book. All of these great people stand on their own, and are in no way directly related to my articles if not so stated except in the few cases when some of them are co-authors, Some of them like my work, some of them disagree strongly with me, compared to many of these giants of quantitative finance I am only a footnote, but either you like it or not a footnote that is growing in size!

“Models on Models” also reflects upon early often “forgotten” research and knowledge. The current quantitative finance models are in almost every case extensions that are based on early wisdom and knowledge. Many of the techniques used in finance have their background in physics, engineering, probability theory and ancient wisdom. Many of these theories have developed over thousands of years. It is easy to forget this when working with valuing some advanced derivatives instruments.2

For making it easy to remember the importance of understanding models is based on models. This concept is illustrated in an artistic photo in the centre section of the book. After seeing beautiful quantitative finance models painted on beautiful photo models you will hopefully never forget the importance of “Models on Models”. Even if the model and the underlying model potential look extremely elegant and beautiful, this does not mean that the surface and the explicitly stated assumption of the model tells the truth about real market behavior. I have myself spent considerable time as a derivatives trader and have seen many of the differences between how the model worked in theory and practice, and I am still learning, it is a life long process. Many of us love beauty and elegance, even if we naturally know beauty and elegance is far from everything. Personally I love the beauty of closed form solutions, but they have their limitations. The beautiful surface of a model says little about how the model actually behaves in practice. Derivatives models are only models and nothing more, often beautiful and elegant on the surface but the real market is much more complex and much more interesting than the model alone, the art of quantitative trading is the understanding of the interaction between the market and the models, and in particular the shortcomings of the model.

The truth about derivatives models as well as photo models is more like this; the first time you see a derivatives model you think it is beautiful and elegant and you fall in love with it. Then when you know the model better, you learn its complexity and that the beautiful and elegant surface is only part of the reality. The more you learn about the model versus reality, the complexities, the weaknesses and strengths the more you tend to like the model despite its weaknesses. Not because it makes the model better, but you know how to get around its shortcomings. Knowing the weaknesses of a model is your best strength in applying the model to the market. Academics and researchers falling blindly in love with the mathematical beauty and elegance of their models without rigorously testing it out on market data, without listening to people with trading experience are like someone falling in love with a photo of a super-model, it has often little to do with the multidimensional reality. A great modeler or trader will test the model against market data, talk with experienced traders and always be open for discussion, and most important will see how the model works over many years in the market. Love based on outer beauty can sometimes be a good start, but only love based on a deep understanding of the weaknesses, strengths and complexities of a model and its interaction with reality can make love last.

Just like fashion models anyone that has followed mathematical finance for some time must also have noticed how fashion here changes over time. In the 1970s to the mid 1980s equity derivatives were in fashion. In the 1980s and early 1990s interest rate modeling was in fashion, every researcher and modeler tried to come up with benchmark yield curve models. In the 1990s exotic options and energy derivatives came into fashion. In late 1990s and until now modeling credit derivatives has been in fashion. It is hard to predict what fashion will be next and when it will end. As the derivatives business has grown dramatically and also the number of people in it, there now tend to be several fashions going on at the same time. The quants and the modelers have a few similarities with fashion designers, sometimes disliking each other’s style, design and product. A good modeler should however in my view be open to at least discussing his own work with modelers in the opposite fashion camp. Too often modelers fall in love with their own view of the world (I do it all the time), I guess this is why science has always evolved through paradigm changes, and I don’t think that things are any different now, the human brain is more or less unchanged over thousands of years.

Even if this book covers some “serious” ideas and valuation models I see no reason why this cannot be combined with humor and fun. The great sense of humor of the many great researchers and quants interviewed is just one example. To make the book more entertaining I have also added a section filled with comic ships and quant related artistic photos, I hope and think that some of you will like it.

First of all I would like to thank the great modelers, quants, researchers, philosophers, gamblers and traders that shared their knowledge, wisdom and their views through the interviews presented in this book.

I would like to give a special thanks to, Jørgen Haug, Alan Lewis and William Margrabe who I was lucky to have as co-authors in a few of the chapters. Even if this book (besides the interviews) mostly is personal ego trip I have also included a chapter by Professor Knut Aase that is closely related to the subject of one of my chapters.

I am also grateful for interesting discussions, comments and suggestions by Alexander Adam-chuk, Gabriel Barton, Christophe Bahadoran, Nicole Branger, Aaron Brown, Peter Carr, Peter Clark, Jin-chuan Duan, Tom Farmen, Stein Erik Fleten, Edwin Fisk, Omar Foda, Gordon Fraser, Stein Frydenberg, Ronald R. Hatch, Steen Koekebakker, John Logie, Xingmin Lu, Hicham Mouline, Svein-Arne Persson, John Ping Shu, Samuel Siren, Gunnar Stensland, Erik Stettler, Svein Stokke, Dan Tudball, James Ward, Sjur Westgaard, Lennart Widlund, Nico van der Wijst and Jiang Xiao Zhong.

I had great fun working with the very talented photographers Amber Gray and Julian Bern-stein doing some artistic “Models on Models” shots for this book. For the comic strips I am grateful to the multi-talented Sebastian Conley. The comic strips began by me drafting the story, then Sebastian drew it by hand (black and white) and improved my stories before using computer software to add colors and special effects. Also special thanks are due to my very artistic friend Wenling Wang for her painting “The Path” (oil on canvas shown on page xiv) inspired by the art of ancient wisdom and the vicissitudes of today’s financial world that she has witnessed during many years of experience in the some of the top hedge funds of our time.

At John Wiley & Sons I would like to thank Caitlin Cornish, Vivienne Wickham and Emily Pears for helping me put all this material together.

Chapter 1 describes the early and partly forgotten discovery of high-peak/fat-tailed distributions in price data. Chapter 2 describes dynamic delta hedging from theory to practice. Chapter 3 is about how to value options with discrete dividend, a problem that has caused a lot of confusion over the years. Chapters 4 to 8 cover different topics in Exotic option valuation, barrier options, reset options and Asian options. Chapter 9 covers practical valuation of power derivatives. Chapters 10 and 11 look at negative volatility and a interesting symmetry in option valuation. Chapter 12 is a bizarre story about time and frozen time arbitrage. Chapter 13 covers the Relativity Theory’s implications for mathematical finance. Chapters 14 and 15 look at probabilities in finance in a non-traditional way.

FOOTNOTES

1. Dr John Stevenson was at that time working on developing quantitative model trading using one of Wall Street’s most powerful computers. Basically he was using all the other computers in the bank, and not surprisingly he was often blamed when the network crashed.

2. For a very interesting book on the history of financial economics see Rubinstein, M. (2006) A History of The Theory of Investments. New York: John Wiley & Sons, Inc.

The author has received financial support from The Non-Fiction Literature Fund in Norway.

Oil painting by Wenling Wang

Derivatives Models on Models

It is easy to forget that many of today’s financial models are based on ancient models. For example, an important ingredient in the binomial tree model is Pascal’s triangle and the binomial coefficient. The Chinese knew Pascal’s triangle long before the Europeans. Figure 0.1 shows a drawing from Chu Shih-chien’s book published in (1303). However reading Chinese from right to left we will see that he already in 1303 called this an ancient old method. Chu Shih-chien is actually referring to Yanghui (1261), even today the Chinese mathematical literature refer to this triangle as Yanghui’s (or Yang Hui’s) triangle. In India there is a book called “Bhagabati Sutra” published 300 years before the birth of Christ that indicates that already back then there was some knowledge of the binomial coefficient. Mahavira (circa 850 A.D.), writing in “Ganita Sara Sangraha” generalized the rule found in the Bhagabati Sutra.

The illustration to your left is an oil painting by Wenling Wang, a hedge fund artist in Greenwich Connecticut. She was inspired by the ancient Yanghui triangle, and the painting is full of ancient wisdom and mystery.

Figure 0.1: Drawing from Chu Shih-chien (1303)

Here together with Nassim Taleb and Benoit Mandelbrot

Nassim Taleb on Black Swans

The first time I heard about Nassim Nicholas Taleb was when, as an option trader, I came across his very interesting book on option trading, “Dynamic Hedging”. A few years later I moved to the USA to work for a Hedge Fund in Greenwich, Connecticut. On one of my first days at work I walked over to the coffee room. In order to get up to speed in a new trading job and get rid of my jet lag I needed that caffeine.

There in the coffee room I met someone who looked familiar. It was Nassim Taleb. I recognized him from the photo in his book “Dynamic Hedging”. He also recognized me from my book “The Complete Guide to Option Pricing Formulas”. I called him Taleb, but he told me to call him Nassim. Nassim was running his own trading firm in the same building, but we shared the coffee room.

Over the next three years or so I met Nassim frequently, he was happy to share his knowledge and loved to get involved in discussions on tail events, advanced option trading and phenomena outside the traditional mainframe of finance. Nassim Taleb was a original thinker, a tail event himself specializing in tail events. He was also not afraid of sharing his knowledge, probably because he knew that human nature and the bonus system in most Wall Street firms would make most traders ignore his ideas anyway. I went skiing a few times with Nassim. Nassim was a great skier and we always did the double black diamonds, something I will come back to in the end of the interview.

Nassim Taleb has more than 20 years of trading experience specializing in option trading and convex payoff structures. In addition to his book on options “Dynamic Hedging” he has also published the best selling book “Fooled by Randomness” and is at the time of writing also coming out with a new book titled “The Black Swan”.

Haug : Where did you grow up first of all?

Taleb : This question is appropriate because I’m currently writing the Black Swan, and in it I discuss something I call the narrative-biographical fallacy. The error is as follows: You try to look for the most salient characteristics of someone and impart some link between the person’s traits and his background, along causative lines. People look at my background, they see my childhood in the war, in Lebanon, and they think that my idea of the Black Swan comes from that. So I did some empiricism, I looked for every single trader I could find, who had the same background, experienced the same war, and looked at how they trade, they are all short gamma, they are all short the wings, they all bet against the Black Swan directly or indirectly. So there’s no meaningful relevance to the background, I can pretty much say now. This has been studied by a lot of researchers across fields.

Haug : What is your education and trading background?

Taleb : My trading background is more relevant to a description of my personality. I started trading very early on. My education was quite technical but initially I did not have much respect for technical careers. Math was very easy to me and convenient because the books were short and it was not time consuming. I liked its elegance and purity but I feared committing to it career-wise by becoming an engineer — I looked at engineers and saw how they became mostly support staff and I viewed them as a negative role model. I wanted to become a philosopher, understand the world, be a decision-maker; I never wanted to accept that I was bound to have a technical career. After an MBA at Wharton, I became obsessed with convex payoffs and became a option trader very quickly. It was a great compromise between decision-making and technical and mathematical work. And from day one I saw that much of these models was severely grounded in the Gaussian, and that it was nonsense. From day one I thought this application of the Gaussian was nonsense. It was nonsense because the Gaussian was not an approximation to real randomness, but something qualitatively different. Now two decades later, I still believe it is nonsense, nothing has changed.

Haug : But aren’t there a lot of other people looking into this through stochastic volatility models?

Taleb : At the time, my biggest mistake was that I started looking at stochastic volatility. I no longer believe in stochastic volatility, because I think it is a fudge but you forget it when you spend too much time with models. A given distribution has four components: first it has a centrality parameter — in other words, for the Gaussian it would be the mean: secondly it has what you call a scale parameter — for a Gaussian, it would be the variance; and it has also a symmetry component — the symmetry attribute is skewness which for a Gaussian is zero; and finally I think the distribution has what I would call the asymptotic tail exponent which for a Gaussian is not relevant because it is infinite. If it does not apply to the Gaussian, it does apply for other classes of distributions, which is where the qualitative difference starts. Building models off the Gaussian do not remedy the lack of tail exponent.

So, given that the Gaussian is not a good representation of the world, the easiest way is to start fudging with it and the mistake I made early on was to use stochastic volatility. Stochastic volatility does the job only up to some degree of out-of-moneyness of the options. If it does a good job sometimes with the body; the problem is that some people believe that stochastic volatility is a real model but it’s not. Stochastic volatility is a simple trick to price out-of-the-money options without getting too much into trouble, you see.

Let me summarize my idea. There are two classes of distributions. There is what I call scalable or scale free (they have a tail exponent or no characteristic scale) and there are what I call non-scalable (no tail-exponent). Scalable distribution can have a constant scale parameter, yet they can perfectly mimic stochastic volatility, without your noticing it. Rama Cont and Peter Tankov, in their book1, made the observation that a student T, with three degrees of freedom (which has a tail exponent alpha of three) will mimic stochastic volatility. You look at it and it resembles what we know. Yet it has infinite stochastic volatility, literally, since it will have an infinite fourth moment. So early on I thought that to fix option models, stochastic volatility was a good patch and luckily realized that it was only a good patch to price some out-of-money options up to fifteen or twenty delta. It did not work really beyond that, in the tails — and the tails are of monstrous importance. Note that the further you go out-of-the money, the higher your model error will be. Also the smaller the probability, the higher the sampling error you’re going to have.

So since then I started hunting for models until I found the fractal model. Most people talking mathematics don’t fully understand the applications of probability distributions; they don’t understand central limit; they don’t understand how something becomes Gaussian and they talk about it as a statistical property that hold asymptotically as if it were for real. We don’t live in the asymptote. For some parts of the distribution we get to the asymptote very slowly, too slowly for any comfort. The body of the distribution becomes Gaussian, not the tails. We live in the real world so I specialise in out-of-the-money based on that.

Haug : Let’s come back to that. One thing is to attack current theories but another thing is to have a good alternative. Do scaling laws and fractal models makes us able to value derivatives?

Taleb : Likewise it is foolish to say “OK I want to have a better theory than the one we have now”. That would be similar to saying let’s take this medicine because it is the best we have. You do not compare drugs to other drugs; you compare drugs to nothing! But it took us a long time to get the FDA to monitor charlatans. Likewise you endorse a model only if it is better than no model. We need to worry about the side effects of models, to see if you are better off having nothing because you should not trade products under unreliable theories.

You should only trade instruments for which you have some degree of comfort — not fall into the trap of back-fitting a theory so you create such unjustified comfort. You should only trade instruments where you are comfortable with the risk because you are sure that you have a good model. It’s exactly the opposite of what people seem to do. They take their models and say this is the best model we have and let people trade and labor under the belief that they have the right model.

Here I would like to phrase this in a different framework which is what I call top down versus bottom up. I am a bottom up empiricist. And I would like to live my life as theory free as I can, because I think that theory can be a very dangerous tool, particularly in social science where you don’t have good standards of validations. The exact opposite thing to that which is held in the quant world, or in academia, particularly in finance academia: they come with some theory that is very tight, based on some arbitrary assumption. And ludicrously, they are very precise and very coherent in the way they calibrate things to each other, but they never think that their assumption can be bogus. And this is what we see, for example, in Black-Scholes.

So let’s discuss Black-Scholes. Black-Scholes makes the assumptions of Gaussianism — that you have a normal distribution, continuous trading. All these assumptions. But then based on that they tell you that you have a rigorous way to derive an option formula. That reminds me of Locke’s statement that a madman is someone who reasons tightly and rigorously off wrong premises. Well I care not so much about the precision and the “rigor” with which you derive conclusions, I care about how robust your assumptions are and how your model tracks empirical reality. But much of modern finance does not have robust assumptions and tracks nothing.

Much of this is based on another problem: belief in our knowledge about the probability distribution. In real life, you don’t observe probability, so even before we talk about fractal or alternative models, we don’t observe those probabilities, therefore I want something that does not depend too much on probabilities. The probabilities you observe are uncertain for out-of-the money events. The smaller the probability, the less we know what’s going on. So you want to have trading strategies that do not depend too much on these probabilities.

That was the first statement. The second statement is that I want to use the techniques to try to rank portfolios based on their sensitivity to model error. That is central to risk management. And they don’t allow you to do that. For me a portfolio that is sensitive to model error is not as of high quality as another portfolio that’s more insulated from such model error. We do not seem to have a good rigorous method coming from quantitative finance — to the contrary they invent theories and try to turn you into a sucker instead of making you aware of the epistemic opacity of the world.

These people fall into the biggest trap called reification. Reification is when you take completely abstract notions and invest them with concreteness by dint of talking about them. They keep talking about risk as something tangible, they talk about variance, they talk about standard deviations. These things are completely abstract notions that are severely embedded in the Gaussian. If you don’t believe in Gaussian you cannot believe in these notions. They don’t exist.

Haug : Back to the Black-Scholes and your background in hedging. Black-Scholes and it’s risk-neutral valuation is based on continuous dynamic delta hedging, what is your experience with this?

Taleb : First of all we don’t use the concepts behind the Black-Scholes derivation and I showed with Derman that we really use a version of the Bachelier-Keynes argument. Black-Scholes is not an equation, the equation existed before them, Black-Scholes is the justification of using that risk neutrality argument, owing to the disappearance of risk for an option operator under continuous time hedging as the risk completely disappears. Now this is grounded in four or five assumptions, and let me read through these assumptions.

Assumption number 1: Gaussian.Assumption number 2: Continuous trading,Assumption number 3: No transaction cost,Assumption number 4: No price impact.Assumption number 5: Knowledge of the parameters of the distribution.

This is not counting sets of other assumptions concerning interest rates and all of that. That the interest rates; etc. also are non-stochastic. No credit risk and so on. I leave these aside.

Now, the fact that all these assumptions are very idealized, I can understand. But at the core, the severe disturbing notion that the Gaussian is not an approximation to other distribution — the risks do not disappear in tails where the payoffs become explosive.

Haug : Merton early on seem to realize this problem by switching to jump diffusion?

Taleb : Jump diffusion still does not enter the class of scalable models and fails in the tails. Jump diffusion is a Poisson jump. Also there is inconsistency in Merton’s attitude. Number 1: He said we don’t use Bachelier’s equation, we use continuous trading, therefore we can take out the risk neutral argument because continuous trading eliminates it, etc. Dynamic hedging eliminates it. Which is, ok, an argument that “if you have no jumps, and believe in Santa Claus”. He later said. Well in some cases we have jumps in which case continuous trading doesn’t work so we use the Bachelier equation and remove the risk by diversification so we can fudge it by saying jumps are not correlated with the market. If they are not correlated with the market then we can diversify them away. So in other words nobody realized that he went back and said okay, now we are using Bachelier’s equation — but in cases the jumps are uncorrected with the market. Well, if you think about it you would realize that up to 97% of options trading today are in instruments like fixed income commodities, Forex, and not necessarily correlated to the market. So if I follow his logic we use Bachelier’s expectation-based option method (adjusted for Log) 97% of the time, or perhaps even 100% of the time since all underlying securities are exposed to jump.

So what I am saying is that all these top down ideas sometimes break down and they patch them with arguments that I’ve been using all along — that we price options as an expectation under some probability distribution. What we presented, Derman and I, is a very simple statement saying it is rigorous art not defective science. All I need is a distribution that has a finite first moment and some arbitrage constraint and the arbitrage constraint can be put-call parity — and I can produce a number I am comfortable with. If you introduce put-call parity constrains you then end up with something like the Black-Scholes equation under some reasonable assumptions, and that’s it. And you don’t need to assume continuous trading, all that bogus stuff. And actually that’s the way we all traded. I traded for a long time knowing that it was not the continuous trading argument that was behind my pricing. All I knew is put-call parity, which makes time value of the put, equal to the time value of the call. And that you discover yourself, that everything else leads to some arbitrages. I’m looking for something that works, and maybe not 100% tight, but close enough, rather than something that is perfectly tight off of crazy assumptions and play Locke’s madman.

So therefore Black-Scholes is first of all not an equation, it’s an argument to be able to remove the risk-free from the Bachelier equation. Bachelier plus some alteration to logs and risk-neutral drift. About 7 or 8 people had it before them. And number 2, Black-Scholes, the argument itself is bogus so we don’t use it but people don’t notice that they don’t use it. There’s a universality of the Gaussian distribution that makes people unnecessarily fall into it as a benchmark.

Haug : So the problem is mainly for out-of-money options?

Taleb : Even at-the-money options have problems, but the problem is most severe for out-of-the-money options because they are very nonlinear to model error. Now anyone who has traded options and managed a book knows that you end up with a portfolio loaded with wings because the market moves away from the strikes over time, therefore your model error increases even when you only trade initially at-the-money options. So therefore the dynamic hedging doesn’t really work. I know that, I’ve traded in dynamically hedged portfolio all my life, and I know what it means.

Finance theory has this art of wanting to be married to a top-down paradigm. They also have something nonsensical portfolio theory that they are married to, so they have all these paradigms, they want everything to be consistent with each other and there’s no check on them. First of all, when using a non-Gaussian, even if you try to fudge it as Merton did with his jump diffusion, introducing the Poisson, you’ve got severe misfitness. This is another version of stochastic volatility. It is not scalable in the tails. Furthermore, what type of Poisson jump are you going to use? There are a lot of Poissons you can use, you can use a scalable fractal jump, why did’t they choose these fractal jumps? Now empirical evidence shows, that the tails are scalable — up to some limit that is not obvious to us. When tails are scalable, you don’t use a nonscalable Poisson that overfits from the past jumps.

Haug : So your alternative is that we hedge options with options?

Taleb : That’s exactly how you trade. You traded. Isn’t that how you trade?

Haug : Yes

Taleb : Go tell them! We do not rely on delta rebalancing except residually. We trade option against option. I don’t know of any operator who dynamically hedges his or her risk. And we told these guys that option dealers have ten billion long, 9 billion and 999 short or vice versa and you’ll dynamically hedge a little residual which disappears or increases within a bound. This is not how we trade options.

Finance theory holds mat everything is based on dynamic delta hedging. They have not revised it after the failure of Leland O’Brien Rubinstein in the stock market crash of 1987 as they were hedging portfolios by replicating an option synthetically. Somehow finance theory is allergic to empiricism.

Haug : But there are option traders out there relying on dynamic hedging selling options, but they tend to blow up in the long run.

Taleb : In the short run as well they tend to blow up. First of all I don’t know of any naked option trader except if the total position size is minimal. Option traders always end up spreading something. Book runners, people who run books — they spread something. Except if you do small amounts. When you sell out-of-the-money options you are very likely to blow up, because as I keep explaining, a 20 sigma events can cost you some 5/6 thousand years of time decay. So I tiled to explain the definition of blow up. So if it works for ten years or a thousand years it doesn’t even mean anything. Furthermore, out-of-the-money options are more difficult to dynamically hedge than at-the-money options.

Haug : But Black, Scholes and Merton also originally make a connection between their model and CAPM.

Taleb : CAPM is nonsense; empirically, conceptually, mathematically. It is a reverse engineered story where the exact assumptions are found that help produce an “elegant” model. It’s top down approach. It relies heavily on Gaussianism. That mean over standard deviation or variance. You think that we know the future return. But on Planet Earth we don’t know the mean. We don’t know the sigma, and the sigma is not representative of risk. So, this is the kind of stuff that is not compatible with my respect for empirical reality and my awareness of fat-tails. Its grounding in the variance is bothersome.

Mandelbrot and I do not think that variance means anything. The central idea is that if you have what we see the tail exponent equals 3 in the markets2. Alpha equals 3 means that the sigma exists but the sampling error is infinite so I don’t see the difference between an infinite sigma or a sigma that exists but the sampling error is infinite.

Haug : Is this also related to what you describe as wild randomness and mild randomness?

Taleb : You have type 1 and type 2 randomness. Type 1 is mild in which case you can use a Gaussian. And because of something I call the Ludic fallacy, you cannot use a Gaussian except in very sterilized environments under very strictly narrow conditions. Type 2 is wild randomness, and in wild randomness typically your entire properties are dominated by a very small number of very large moves.

We showed that fifty years of S&P was dominated by the ten most volatile days, so if you have the dominance of the largest moves to the total variance of the portfolio then you have a problem, the problem is that these conventional methods don’t apply because these focus on the regular, and the regular is of small consequence for derivation of the moments. So this is why CAPM is nonsense.

Haug : According to Mandelbrot fat-tails were observed as early as 1915 by Mitchell and Mandelbrot who focused strongly on fat-tails in the sixties and then people went back to Gaussian, why did it take so much time before people started to focus on this?

Taleb : I taught in a business school and I’ll tell you one tiling, a lot of academic finance is intellectually dishonest, because these people are not interested in the truth, they are interested in tools that allow them to keep their jobs and teach students. They’re not interested in the truth, or they don’t know what it means. If after the stock market crash of 87, they still use sigma as a measure of anything, clearly you can’t trust these people. And we’ve known since Mandelbrot about scalability.

To see the problem, go and buy a book on Investments. Try to evaluate its tools if the sigma does not work, see if any of the conclusions and techniques in it hold once you remove the sigma. See if the book is worth a dollar after that. It will not even be worth a dollar.

Entire finance careers are cancelled if you remove the Gaussian notion of sigma.

Haug : What about risk management? Sharpe and VaR seems to be still very popular measures.

Taleb : Sharpe ratio does not mean anything. Sigma is not a measure of anything so you can’t use the Sharpe ratio (which has been known for a long time as the coefficient of variation).

Haug : But people seem to understand it’s because they are using stress testing?

Taleb : I remember one person at a conference gave a talk in which she said Value at Risk is necessary but not sufficient. This is nihilism. You can stand up and say astrology is necessary but not sufficient.

To me stress testing is dangerous because it can be arbitrary and unrigorous — based on the largest past move which may not be representative of the future. You take a time series, you say what was the worst drop — well it was 10%. Then you stress test for that — this is the Poisson problem. It means you would have missed the stock market crash in 1987 because it’s not part of the sample. You would have stress tested for only 10%. So the point is, stress testing is a backward looking way not forward looking. With a fractal you can generate far more sophisticated scenarios. The scalables give you a structure of the stress testing — you extrapolate outside your sample set. So there is an intelligent method to stress-test using different tail alphas.

Another problem is institutions use stress testing as an adjunct method not as a central method. You should work backwards. And if you stress test as the central method then you would start writing a portfolio differently. There are non-Poisson ways to do so.

Now let me come back once again to what I said about finance and why I tell you that these guys are dishonest. Any single discipline that is new is starting to use good ideas from statistical physics. Now we know that deaths from terrorist wars are fractal power laws. Power laws are pretty much everywhere. Now why don’t they use them in finance? They find obscure reasons not to use them. If you use power laws then you will be able to stress test very well and you can ignore variance because it’s totally irrelevant. But if you use power laws then you have to skip portfolio theory. And I show that you have to skip portfolio theory because all your ideas equations are meaningless. And if you use power laws then the Black-Scholes derivation by dynamic replication is not worth even wallpaper. You get the idea.

Haug : You published a book “Fooled by Randomness” and are now coming out with a book entitled “Black Swan”, how are these similar and dissimilar?

Taleb : Very dissimilar. “The Black Swan” is a far deeper book; it goes as far as I could into the problem of fat-tails and knowledge. It is about the philosophy of history and epistemology; it’s written by the same person, but they are very dissimilar. “Fooled by Randomness” is about randomness, and “The Black Swan” is about extreme wild uncertainty. It’s the second level up. It is the one I wanted to write initially but I was mired down by talking about randomness. “The Black Swan” is mostly about the dynamics of history being dominated by these large scale events about which we know nothing and have trouble figuring out their properties. It is also about the social science theories that decrease our understanding of what’s going on but are packaged in great pomp.

Haug : Are Black Swans related to fat-tails?

Taleb : The Black Swan is a fat-tail event. Except that it’s not a Black Swan as you use a power law, sometimes. If you use a power law of the stock market crash of 87 it is not a Black Swan, or less of a Black Swan. It’s perfectly in line with what you can expect. But it was a Black Swan because we don’t use power laws.

Haug : Is Extremistan something you are writing about in your new book?

Taleb : Yes. I am discussing the following confusion. We think we live in lime with Mediocristan with mild randomness, but we live in Extremistan with wild randomness. We use methods of Mediocristan and applying them to Extremistan is exactly like using tools made for water to describe gas. You can’t!

Haug : Many Wall Street traders still seem to be long some types of positive carry trades, believing that they get some signal to get out in time, what’s your view on this?

Taleb : I was approached by one guy, a not very intelligent person, who told me well, why don’t you sell volatility, all you have to do is buy it back before the event. He was a serious hotshot with authority over large investments. He was also educated. I told him yeah very good, why do we waste our time buying all these lottery tickets? Let’s just buy the winning one and save ourselves a lot of money. So you have embedded in the culture this idea that events, before they show up give you a phone call. You get an urgent email telling you what’s going to happen. People have that impression, that the next event is going to be preceded by a warning but in the past we have not seen that. The hindsight bias makes us think so.

My problem goes beyond — I’m becoming more and more what I call an academic libertarian. An academic libertarian is this — just like libertarians distrusts Governments. I distrust academia because I think the role of academia is not so much to deliver the truth but self-perpetuation by a guild. And just like civil servants and politicians, they are not there to help you, a politician is out there to find an angle to get power. So this is why I am very suspicious of the academic world in social science, because what we have seen in the last 110 years in economics is quite shocking.

Haug : But how can we test out your own theories, can we back test them against historical data?

Taleb : Of course. You just look at the graph. When I say you don’t have to back test the Black Swan, one single example would suffice to tell you that someone is criminal. It’s like saying — you don’t have to do a lot of empiricism to show someone is criminal. All you have to do is prove one day that you committed a crime. Likewise for a distribution. It’s much easier to reject the Gaussian based on these grounds than to accept it. To say a theory is wrong you need one instance. Here we have thousands, right?

Haug : Is the bonus system also affecting how people take risk? People get bonus once a year typically, doesn’t this encourage positive carry trading?

Taleb : The bonus system, giving people a yearly bonus based on strategies that take five or ten years to show their properties is foolish. But it’s practised everywhere. And banks practise it with their managers. They should wait until the end of the cycle before they pay their managers and the chairmen of their companies. You are paying people in a wild randomness type of environment using tools of mild randomness.

Haug : You also studied the Black Swans in art and literature how is this related to what we have talked about?

Taleb : It is very similar. Actually art and literature are far more interesting for me than finance, because people in finance academia are usually dull, uncultured, lacking in conversation and intellectual curiosity, and these people are more colourful. The problem is that everything in art has fat-tails, everything in literature has fat-tails, everything in ideas has fat-tails. So you have to see how movies, for example, become blockbusters. It does not happen from putting special skills into the movie. Or a good story. All these movies that are competing against each other seem to have pretty much the same calibre of actors and the same quality of plot. What we have is a very arbitrary reason creating a contagious effect, an epidemic that blows things out of proportion. And you have a winner take all. Movies — anything that has the media involved in it are dominated by “the winner take all” effects.

Haug : Going back to finance, is it possible to predict what you call Black Swan risk?

Taleb : I used to think no but now I believe that you can tell simply from the number of positions betting against the Black Swan that these people will be in so much trouble that it’s going to make it worse and worse and worse and worse. And the more reliance we’re going to have on tools of portfolio theory, the heavier these effects will be. We saw it with LTCM. We will see it again with hedge funds.

Haug : So the construction of the portfolio is maybe more important than looking at all the statistical properties and the standard risk measures?

Taleb : That’s the best thing, just seeing how many people are short options — today as we are talking Warren Buffet is short options so visibly you know there’s some problem in the offing.

Haug : You spent some time both in academia and on Wall Street, what is the main difference?

Taleb : I spent no lime in academia. I ran away in disgust. As I told you academia in finance … I find it intellectually offensive; in mathematics it’s beautiful. Wall street — I like trading because traders say things the way they are. And they understand things. I can communicate with traders. It’s more fun. Academia bores me to tears — partly because I don’t like captive students shooting for grades. I like communicating with researchers, though. Perhaps I might join some research institute—or create one.

Haug : In your new book you are talking about anthropic bias and survivorship bias. How are these related?

Taleb : It’s all a wrong reference class problem, in both cases you take the beginning cohort and instead of taking the computing probability for survival based on the beginning cohort you compute them based on the surviving cohort. So you are missing out on some statistical property, in other words you are missing out on a large part of your sample in both cases. It is the confusion of conditional and unconditional probability — or the wrong conditioning.

Haug : Where do you think we are in the evolution of quantitative finance and research?

Taleb : If we start using power laws as risk management tools we’ll do very well. If we stay Gaussian-Poisson, I don’t think so. But I don’t think anyone cares about academic finance. Their idea is to look good, to teach MBAs but they are quite irrelevant. We practitioners do well without them, much like birds do not need onithologists.

Haug : Can you tell us more about your new book?

Taleb : The whole idea is that that out-of-the-money events, regardless of distribution are things we know so little about. And if out-of-the-money events are the ones that dominate in the end then we have a problem, then we know very little about the world. So that’s what I’m focusing on currently.

Haug : And GIF what is that?

Taleb : GIF: The Great Intellectual Fraud. It’s that Gaussian. The more I think about it the more I realize how people find solutions to the problems of existence, by discovering top down fudges, as Merton did to prove his dynamic hedging argument — he found the assumptions that allow him to produce a proof. What are the problems we face most in real life, what we face is the problem of induction and the fact that going from to the individual to the general is not an easy proposition, it is very painful. It is fraught with errors. It is very hard to derive a confidence level because you do not know how much data you need.

Say I walk into the world and I see a time series, just a series of some points on a page. Say I have a thousand points. How do I know what the distribution is from looking at the data — from these thousand points? How do I know if I have enough points to accept a given distribution? So if you need data to derive the distribution and you then need the distribution to tell if you have enough data then you have what philosophers would call a severe regress argument.

You are asking the data to tell you what distribution it is and the distribution to tell you how many data points you need to ascertain if it is the right distribution. It’s like asking someone — are you a liar?

You can’t ask the distribution to give its error rate. Likewise you cannot use the same model for risk management as you do for trading. I put it in “Fooled by Randomness”. But I went beyond that with a philosopher, Avital Pilpel when we wrote that long paper, calling it “The Problem of Non-Observability of Probability Distribution”. Your knowledge has to pre-suppose some probability distribution, otherwise you don’t know what’s going on. It so happens, very conveniently, that if you have an a priori probability distribution called the Gaussian, then everything becomes easy. So this is why it is was selected, it sort of solves all these problems at one stroke, the Gaussian takes care of everything, and that is what I call the Great Intellectual Fraud, GIF. This is a severe circularity.

Haug : Academics agree and disagree with you, because there are thousands of papers talking about fat-tails?

Taleb : There are thousands of papers on jump or GARCH — the wrong brand of fat-tails. Moreover, when you see a fat-tail, you don’t know which model to fit. With a fractal — all I know is that we have a fractal distribution, I just don’t know how to parameterize it very well. But I personally don’t look at one distribution. I look at a family of distributions of fat-tails, and I just make sure that I’m insulated from them. See unlike other people I don’t bet against the tail. If I knew the distribution, I would know where to sell out-of-money options. But my knowledge of the properties of the tails is fundamentally incomplete — even with a fractal. I really don’t know the upper bound, I know the lower bound. And the lower bound is higher than the Gaussian. And this is what people fail to understand. Yes there have been thousands of papers trying to go into precision, fitting fat-tails, and not realizing the fundamental problem, and this is what the fundamental problem of knowledge is. The Gaussian, guess what? The Gaussian gives you its own error rate. But if you have other distributions, saying I’m going to fatten the tail is not trivial. Because a sampling error of other distribution is very high. So if you select Cauchy you’re never going to see anything. Cauchy tells you that you cannot parameterize me. Likewise if you have distribution of infinite variance you have a huge error rate in measuring the variance. So the problem that you have with fat-tails is that these distribution do not deliver their properties easily from data. Assume I have a combination distribution, with a very severe, say jump diffusion with a very severe fat-tail. Nothing moves and boom you have a huge event. Now deriving the probability of that very huge event, deriving it from finite set is not possible because by it’s very nature it’s very rare.

Haug : But do we have good enough statistical tools and mathematical tools to use power laws and scaling laws to price an option?

Taleb : For scaling you can price an option on scale distribution, you can very easily price it. It’s done all the time. It is very trivial in fact to do these mathematics of options on that. I wrote this paper saying you don’t need variance to price an option; most people don’t realise that all you need is finite mean absolute deviation. An option is not sensitive to variance. It’s sensitive to MAD.

Haug : Have you published much on this topic?

Taleb : I don’t like the process of publishing finance stuff — it is too perishable and I feel I’m wasting time away from more profound issues. There is the time wasted with the referee whose intellectual standards are very low. The best referee is time, not some self-serving academic who thinks he understands the world. I, as a risk-taker have much, much higher standards of rigor, relevance and a no-nonsense need to focus on the bottom line than finance academics. History has treated my work very well. I never submit directly and my dogma is against writing for submission; I usually post on the web and if a journal requests it I still keep it on the web after that. So I wrote a paper called “Who Cares about Infinite Variance” and it is on the web though I may change the title.

Haug : What about variance swaps?

Taleb : Variance swaps are not a real product, try to de-compose a variance swap into real options — you can’t. A variance swap is a contract delivering the squares of moves, and an option dose not depend on square of moves, an option is a piecewise linear product you see, the problem we have with variance swaps is that they can deliver an infinite payoff. But delivering infinite payoff means that you don’t know how high you can go if you have a company going bankrupt or very large moves.

Haug : Is this why traders or market makers often cap their payout?

Taleb : If you truncate the variance swap then it ends up as finite properties and it is easier to de-compose into regular options, but it has no longer anything to do with variance. The mere fact of capping the tail cuts a lot of it.

Haug : But this seems like market makers are aware of this problem?

Taleb : Market makers are implicitly aware of it. But my point is that we don’t need variance to price options, you need variance to price a variance swap. An uncapped variance swap. A capped variance swap becomes very similar to a portfolio of options. But variance is a square. You multiply large moves by large moves. It’s dominated in a very small number of observations.

Haug : So incidentally we could maybe change to MAD swaps?

Taleb : MAD is much more stable and is what naturally goes into to the pricing of an option. People don’t realise MAD is what an option is priced on. An at-the-money straddle delivers MAD (risk-neutral by put-call parity). It is trivial to show that an option depends on mean absolute deviation, not on variance.

Haug : You are very well aware of tail events, but some year’s back we went skiing and we always went in double black diamond slopes, and you never wore a helmet. How is this consistent with your philosophy and spending most of your time understanding tail events?

Taleb :