reading

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

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

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

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

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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).

“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.

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.

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”.