Pricing Derivative Securities by T.W Epp (World Scientific Publishing 2000)

According to the preface, “The aim in writing [this book] was to help fill the gap between books that offer a theoretical treatment [of equity derivatives] without much application and those that simple present the pricing formulas without deriving them”. Moreover, “…understanding is not complete without practice at application and…applying resulting one doesn’t understand is risky and unsatisfying. This book presents the theory but directs it toward the goals of producing practical pricing formulas.. and implementing them empirically’. This is a tall order to satisfy, but does the author deliver on his promises?

*Pricing Derivative Securities *is divided into three parts. The first part introduces basic concepts in finance and derivatives pricing. Chapter 1 provides an overview of the various derivative products available in different markets. This chapter also explains derivative pricing, arbitrage pricing and static versus dynamic replication. The Black-Scholes-Merton models are dealt with difference or differential equations depending on whether the pricing is modeled in discrete or continuous time. This chapter is an excellent introduction and accessible to all intelligent mortals- in contrast to the rest of the book which is really meant for immortal financial whizzes or mathematicians!

Chapters 2 and 3 present the mathematical background which is the basis for the second and third parts of the book. The main topics covered are measurable functions, integration, special functions, and integral transforms, probability spaces, random variables, mathematical expectations, and stochastic processes. Chapter 3 concentrates on the mathematics required for continuous-time models in finance: Wiener processes, martingale pricing, and discontinuous process. Sixty-eight examples are given in these two chapters which make the theory easier to follow. The presentation would have been spicier if some of the examples had been finance-related. There are some good recommendations for literature to help to digest these two chapters.

The second part of the book, which is its core, deals with derivatives pricing. Chapter 4 is about dynamics-free pricing. Valuation of forwards and futures contracts and options pricing are derived from static application arguments. There is also material on pay-off distribution for European and American option, and a discussion of how option prices vary with changes in expiration date, strike price and the current market price of the underlying item.

Chapter 5 discusses pricing under Bernoulli dynamics and the related binomial approach in discrete time. Two interpretations of this binomial estimation are given: (i) the partial difference equation and (ii) the risk-neutral (Martingale). These are applied to both European and American style options. In both Chapters 4 and 5, there are helpful illustrative numerical examples. References for further reading are also provided.

Chapters 6 and 7 deal with the valuation of derivative assets when the price of the underlying assets evolve in continuous time. The Black-Scholes model for pricing European-style derivatives under geometric Brownian motion is introduced. In Chapter 7, the same theory is applied to American and to “exotic” options which have more complex pay-off structures. These chapters use sophisticated mathematics, and unfortunately not one concrete example is provided to illustrate the theory.

Chapters 8 and 9 discuss the pricing of derivatives when the Brownian model does not fit the behavior of prices of the underlying assets, and so the Black-Scholes model has to be adjusted. Alternative models are introduced, for example, those in which volatility itself is an Ito process. Chapter 9 discusses derivatives with discontinuous processes, e.g Poisson processes. The second part of the chapter deals with derivatives on assets that are subject to jumps, e.g. random pay-off times. This chapter is illustrated with concrete examples.

Chapter 10 provides an extensive review of the literature on the pricing of interest-sensitive assets and derivatives. The centerpiece of the chapter in the Heath, Jarrow and Morton models which compare forwards prices under stochastic rates.

The third part of this book is devoted to computation methods. Chapter 11 is an overview of some of the literature on simulation methods for problems in finance. Chapter 12 presents three procedures for the numerical solution of partial differential equations applying finite difference methods. The author uses specific numerical examples to compare the accuracy of the methods. The chapter is one of the most useful in the book. The final chapter presents computer programs in FORTRAN and C++ for the implementation of the principal methods discussed in the book. But this chapter, like many before it lacks specific numerical examples.

*Pricing Derivative Securities *has some excellent features for the sophisticated student of finance. However, a future edition could improve upon these further: there should be more solved examples, and unsolved ones too, with a website where readers can go for a solution. A web site would also be useful so that readers can dialogue with the author, and where the author can update material and methods as this field continues to change.