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.
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.