*Financial Calculus*is a presentation of the mathematics behind derivative pricing, building up to the Black-Scholes theorem and then extending the theory to a range of different financial instruments. It is clearly presented, with a systematic build up of the necessary results, and with extensions separated from the core ideas.

Chapter one explains the limitations of expectation pricing, introducing instead the use of "no arbitrage" constructions to derive prices. Beginning with the discrete case, chapter two introduces a simple binomial tree model. The approach is based around martingales, or processes whose expected future value, given the past history, is the same as the current value.

Chapter three extends this to the continuous realm, using basic stochastic calculus, Ito's formula and stochastic differential equations. The Radon-Nikodym derivative, the Cameron-Martin-Girsanov theorem, and the martingale representation theorem allow a similar construction to that of chapter two, coming together in the Black-Scholes theorem.

This covers basic options. Chapter four applies and extends this to other kinds of securities: foreign exchange, dividend-paying equities, bonds, and quantos (derivatives denominated in one currency but settled in another). And chapter five, which I only glanced over, builds progressively more complex models for interest rates. Some of this involves clever constructions, but it doesn't add that much to the core theory. More interestingly, chapter six extends the basic model: to variable drift and volatility, general log-normal models, multiple stocks, and the notion of an arbitrage-free complete market.

One strength of *Financial Calculus* is that, while it is rigorous and
the approach is quite abstract — it assumes familiarity with calculus
and a general competence with formal mathematics — concrete worked
examples are used to anchor the theory and assist intuition. There are
also a few exercises, with solutions, which mostly test understanding
of basic concepts and the ability to use the formal machinery.

The models presented in *Financial Calculus* are abstractions, and
obviously any real-world application would need to address a whole range
of issues not considered: the assumption of liquidity, counter-party
risks, and so forth.

One concern I have is with the assumption of Brownian price movements, for which Baxter and Rennie offer no more than hand-waving support — but where, given the number of times they wave their hands, they clearly realise there is a problem. This is a "widely accepted model", "sophisticated enough to produce interesting models and simple enough to be tractable", "at least a plausible match to the real world", and "a respectable stochastic model". The only evidence provided is a comparison of two small and vaguely similar graphs, one of the UK FTA index from 1963 to 1992 and the other generated using exponential Brownian motion.

Now "interesting and tractable" is a fine basis for doing mathematics,
but not a strong basis for applying the results to reality. If most
real-world markets are not Brownian, as Mandelbrot and others have argued,
that doesn't undermine any of the mathematics in *Financial Calculus*
but does make its utility entirely unclear.

Paradoxically, I also worry about the very elegance and rigour of the
results in *Financial Calculus*. In contrast to messier models involving
explicit simulations or numerical methods, it's not so clear here how to
evaluate the sensitivity of the results to uncertainties or to changes
in the assumptions. And a reluctance to lose the beauty of the analytic
formalism may make it harder to face up to empirical ugliness.

In any event, there's probably too much detail in *Financial Calculus*
for anyone who isn't actually planning to work in the finance industry.
Other readers are likely to be less interested in the various elaborations
and want more philosophical and empirical background.

May 2010

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