Theory of Financial Risk and Derivative Pricing : From Statistical Physics to Risk Management

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Summary: An Unconventional and Engaging Treatment of Risk
Review: In 'More Heat Than Light', Philip Mirowski observed that the expertise brought to economics by the " ... influx of engineers, physicists manqués, and mathematicians during the Great Depression and after ... did not get parlayed into novel physical/economic metaphors." In the literature of the new field of "econophysics" there are promising indications that the recent influx into finance following the end of the cold war will not repeat this. An exciting addition to this literature is the recent publication of this augmented and English version of Théorie des Risques Financiers.

In this monograph Drs. Bouchaud and Potters present much of their research together with related contemporary and previous work including that of Bachelier. Their "physicists viewpoint" of comparing theory to observed data appears early in the first chapter where time-series data illustrating 3 market crashes motivates their review of the basic notions of probability with an emphasis on non-Gaussian probability densities. This is followed by an interesting data-intensive comparison of these notions to the statistics of real prices including, as examples, the S&P 500 index, the DEM/USD exchange rate, and the Bund futures contract. The results of this comparison between theory and observation are then applied in the chapters that follow in which portfolio optimization, risk management, and the valuation of derivative securities are discussed.

The authors' approach in general, and to derivative securities in particular, is both unconventional and refreshing. It will appeal to those who have wondered if stochastic calculus is really required to price options. They demonstrate how a number of well-known results can be recovered in the appropriate (usually Gaussian) limits and provide an even-handed discussion of the risk associated with failing to include non-Gaussian effects.

This book is readily accessible to readers who have/can read either McQuarrie's 'Statistical Mechanics' or Ingersoll's 'Theory of Financial Decision Making'. I enjoyed this book because of the authors' unconventional approach, stat-mech style, interesting comparisons of theory and data, and the important implications of their approach for risk measurement and management

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Summary: Can do more harm than good
Review: Simply the best book written on mathematical finance. Bouchaud brings a physicist's clarity, insight and deft use of mathematical tools to finance. There has never been a better generalisation of the Black-Scholes hedging recipe. I know of no book where equal attention is paid to data and to the model used to describe it. Practical concerns are what drive the book from front to end.

Furthermore, Bouchaud writes superbly and elegantly, in sharp contrast to standard finance books, which are unrivalled for their pedantry. Mathematical finance was for years hijacked by individuals with no experience of, and interest in, real life modelling. We desperately need the insights and methods of applied science that an approach like Bouchaud's brings. A first in this field.

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Summary: Can do more harm than good
Review: This book is a supposedly new approach to financial modeling from the viewpoint of "statistical physics". In fact, it is far from being that. First, there is little or no content really related to statistical physics in it. Apart from the fact that random variables and stochastic processes are also used in physics, the only feature in common between statistical physics and this book is some notational similarities and a lack of rigour which, justified in the case where it is supplemented by physical intuition, leads here to numerous mistakes and sloppy reasoning.

The title, while promising, is quite arrogant: not only there is no "theory of financial risks" in the book but many of the main issues of risk management are not even mentioned: Value at Risk receives less than a page at the end, while hedging of exotic options is not even an issue.

Also, while the first part of the book insists on choosing the correct distribution for price returns, the chapter on options exclusively gives computations for the case of ...Brownian motion (not even exponential Brownian motion)! One is left wondering whether these fancy models presented in the first part were worth mentioning?

Another point is the readership of this book: given the notational complexity of the book and the analogies with physics, only a PhD in theoretical physics can possibly find this book readable. In fact, a finance student will find it too light on the finance side while a math-minded student will find it too sloppy and imprecise.

The surprisingly low level of mathematical rigour - one confuses regularly "uncorrelated" with "independence"- is nevertheless accompanied by an incredibly sophisticated set of tools such as random matrix theory, which are exotic even for professional researchers. Perhaps it would be better to spend more time explaining the concept of stochastic volatility or nonstationarity than rocketing the reader into unknown grounds...

I come to the conclusion that the aim of the book is more to impress the reader about the technical sophistication of the authors than to teach anything in a clear manner.

Although OK as a bedtime reader, this book certainly does not contain anything one can practically implement: in fact the presentation is so imprecise that one is lost in the successive and uncontroled approximations, not knowing at the end what is the algorithm proposed to solve a given problem.

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Summary: Fat tails and more
Review: This text has a nice discussion of Levy distributions and (important!) discusses why the central limit theorem does not apply to the tails of a distribution in the limit of many independent random events. An exponential distribution is given as an example how the CLT fails. I was first happy to see a chapter devoted to portfolio selection, but the chapter (like most of the book) is very difficult to follow (I gave up on that chapter, unhappily, because it looked interesting). The notation could have been better (to be quite honest, the notation is horrible), and the arguments (many of which are original) could have been made sharper and clearer. For my taste, too many arguments in the text rely on uncontrolled approximations, with Gaussian results as special limiting cases. The chapters on options are original, introducing their idea of history-dependent strategies (however, to get a strategy other than the delta-hedge does not not require history-dependence, CAPM is an example), but the predictions too often go in the direction of showing how Gaussian returns can be retrieved in some limit (I find this the opposite of convincing!). For an introduction to options, the 1973 Black-Scholes paper is still the best (aside from the wrong claim that CAPM and the delta-hedge yield the same results). The argument in the introduction in favor of 'randomness' as the origin of macroscopic law left me as cold as a cucumber. On page 4 a density is called 'invariant' under change of variable whereas 'scalar' is the correct word (a common error in many texts on relativity). The explanation of Ito calculus is inventive but inadequate (see instead Baxter and Rennie for a correct and readable treatment, one the forms the basis for new research on local volatility). Also, utlility is once mentioned but never criticized. Had the book been more pedagogically written then one could well have used it as an introductory text, given the nice choice of topics discussed.

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Summary: Reply to the previous reviewer
Review: Unfortunately, but not surprisingly, the previous reviewer prefered to remain anonymous. Otherwise, we would happily have argued with him privately. But his review contains so many erroneous and obnoxious statements that we feel we have to reply publicly, at least on the most important points.

a) After spending a full chapter (2) on empirical data and faithful models to describe them, we only price options using...the Brownian motion, says our reviewer (not even the Black-Scholes model, adds he). Well, either the reviewer has only casually browsed through our book, or this is total bad faith and disinformation. After discussing a general option pricing formula, we indeed illustrate it first (4.3.3) with the Black-Scholes model, then with Bachelier's (Brownian) model which, as we explain, is actually a better model for short term options. But the rest of the chapter is entirely devoted to non-Gaussian effects: a theory of the smile, its relation with kurtosis and long-ranged correlation in the volatility, and comparison with actual market smiles (4.3.4), and more importantly, the hedging strategies and residual risk (4.4), alternative hedging strategies for Value-at-Risk control (4.4.6), etc. The emphasis on risk, absent in the Black-Scholes world, is our main message, and partly justifies the title of our book.

b) "There is no statistical physics" in our book, moans the reviewer. Our aim was not to draw phoney analogies, but to present this field in the spirit of statistical physics, with what we feel is an interesting balance between intuition and rigour. (Many physicists feel stranded when reading standard mathematical finance books, where data is scarce, and rigour hides the inadequacies of the models). However, there are several genuine inputs from statistical physics, e.g. data processing, approximations, simple agent based models (2.8-9), functional derivatives to obtain optimal hedges (4.4), saddle point estimates of the Value at Risk for complex portfolios (5.4) and finally, Random Matrices that the reviewer finds unduly complex -- perhaps only because new to him. However, this is contained in "starred" section, indicating that it can be skipped at first reading, as many more advanced sections.

Two more details. We indeed sometimes consider independent random variables, sometimes only uncorrelated, hopefully not confusing the two. If the reviewer spotted incorrect statements, we would be grateful to him if we can correct them in further editions. Second, our book is not meant to provide ready to implement recipes but to present a different way of thinking about finance. Nevertheless, many of the ideas have already been implemented and are used by several (open minded?) financial institutions.

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Summary: Summarises new advances in quantifying financial risk
Review: `Econophysics' (the application of techniques developed in the physical sciences to economic, business and financial problems) has emerged as a newly active field of interdisciplinary research. `Theory of Financial Risks' (written by two of the pioneers of this field) highlights very clearly the contribution that physicists can make to quantitative finance.

From the outset the point of view of the book is one of empirical observation (of the statistical properties of asset price dynamics) followed by the development of theories attempting to explain these results and enabling quantitative predictions to be made. This philosophy is reflected in the structure of the book. After a brief account of relevant mathematical concepts from probability theory the statistics of empirical financial data is analysed in detail. A key result from this analysis is the observation that the correlation matrix (measuring the correlation in asset price movements between pairs of assets) is dominated by measurement noise (which, as the authors observe, has serious consequences for the construction of optimal portfolios). Chapter 3 begins the core theme of the book with a discussion of measures of risk and the construction of optimal portfolios. A central result of this chapter is that minimisation of the variance of a portfolio may actually increase its Value-at-Risk.

The theme of improved measures of risk continues in chapters 4 and 5 which focus on futures and options. A new theory for measuring the risk in derivative pricing is presented. In the appropriate limit (continuous-time, Gaussian statistics) this model reproduces the central results of the Black-Scholes model - namely that one can construct a portfolio of options and assets such that the residual risk is identically equal to zero. However as the book has constantly highlighted, these market conditions are simply not observed in practice. Moreover the new theory presented allows one to calculate the residual risk which exists under more general and realistic market conditions (allowing the development of improved trading strategies).

In summary this book highlights very clearly many of the inadequacies of current financial theories and presents a number of new approaches, based upon concepts developed in statistical physics, to overcome these problems. It is to be recommended to both students of finance as well as to professional analysts as a good example of how an interdisciplinary approach to financial engineering may yield improved measures of risk.