Published in RISK, 10(6) Jun. 1997, p. 75

I often interview job candidates with degrees in science or finance. I `m always glad if they show that, whatever time decay has done to their knowledge of their subject’s details, they departed their studies with a clear idea of its principles and methods. But, like Keats, sometimes I’ve grown half in love with easeful death at hearing Ode to a Martingale recited as though it were an axiom of options theory. From this perspective, I like Derivative Securities, Profs. Jarrow & Turnbull’s balanced new book, because readers will emerge with a clear understanding of the assumptions, principles and methods of risk-neutral” derivatives valuation. Impressively, it also touches on issues important to practitioners, but commonly ignored by academics.

Derivative Securities is based on courses taught to undergraduates, MBA’s and executives. The authors set themselves very specific goals: to explain options and futures, in theory and practice, for equity, interest-rate, currency and commodity underlyers; to cover exotic options; to base valuation in a unified way on the martingale approach; to cover credit risk; to cover valuation and hedging, including proofs and references; and to include software (an MS-DOS floppy disk, which I didn’t try at all). I think they’ve largely succeeded, especially with regard to understanding valuation methods.

Each chapter starts with the relevant assumptions and their limitations, and concludes with a crisp summary, some problems to solve, and pointers to relevant research. The early chapters provide a lucid introduction to fundamentals: forwards, futures, options, bonds, compounding, and trading rules, and the model-independent arbitrage bounds on security values. They are especially clear on important practitioner notions such as the implied repo rate, on the theoretical difference between futures and forwards, and on the Merton inequalities.

Chapter 4 appropriately begins the long journey into derivatives’ modeling by tackling asset price dynamics. The aim is well put: find a model simple enough to facilitate analysis but complex enough to be approximately realistic. This is the balance sought daily by any theorist working with derivatives traders. Jarrow and Turnbull try to deliver both the analytic benefits of continuous time evolution and the intuitive beauty of discrete binomial steps, though I would (quibblingly) have liked a little more explicit detail on the connection between them.

Chapters 5 and 6 present the binomial model and martingale pricing as the foundation for all future derivations, identifying synthetic replication as the valuation strategy, risk-neutrality as a consequence, and backward induction as the method. They also clarify self-financing and dynamical completeness, and show how to replicate futures as well as options, something rarely described.

The right way to handle martingales is to demystify them. If, at each instant, the stock cannot dominate or be dominated by the riskless bond, then stochastic stock returns must bracket the riskless return, which can therefore be expressed as an average over the stock returns. This makes the stock a martingale. Since options can be replicated by self-financing positions in the stock, options too are martingales.

Jarrow and Turnbull do a good job of explaining these key points, and appropriately mention the importance and difficulty of making interest rate evolution a martingale, covered later.

Some variables treated as deterministic are in fact stochastic, which makes all models incomplete (a term I prefer to the authors’ use of “misspecified”, which somehow suggests the Platonic existence of a perfectly specified model). The authors tackle delta, gamma, theta, vega and rho hedging in this light.

The chapters continues systematically through indexes, currencies, commodities and interest rates, presenting a careful introductory treatment of one-factor interest rate models, including the essence of HJM. Here, I appreciated their nice distinction between mean reversion in the martingale world and the real world. As promised, they culminate by extracting martingale default probabilities in the Jarrow-Turnbull treatment of credit risk, and conclude with a couple of chapters on exotics.

Some disadvantages to the book. First, it’s too long, almost 700 pages. This may make it a better reference than a textbook. Some of the excess length is taken up with repetitive numerical examples that are eventually summarized in a simple algebraic formula that comprises all the individual cases. Second, to my eye the pages are unattractively typeset, with a large, bland, textbookish look that’s symbol-heavy and off-putting. The authors’ publishers, as well as their own meticulous use of symbols, haven’t quite done full justice to their clear thinking. But these are small matters of taste regarding an ambitious presentation that’s principled, coherent, and broad.