Steven E. Shreve 的经典著作

Summary of Volume I

Volume I presents the binomial asset pricing model. Although this model is
interesting in its own right, and is often the paradigm of practice, here it is
used primarily as a vehicle for introducing in a simple setting the concepts
needed for the continuous-time theory of Volume II.

Chapter 1, The Binomial No-Arbitrage Pricing Model, presents the noarbitrage
method of option pricing in a binomial model. The mathematics is
simple, but the profound concept of risk-neutral pricing introduced here is
not. Chapter 2, Probability Theory on Coin Toss Space, formalizes the results
of Chapter 1, using the notions of martingales and Markov processes. This
chapter culminates with the risk-neutral pricing formula for European derivative
securities. The tools used to derive this formula are not really required for
the derivation in the binomial model, but we need these concepts in Volume II
and therefore develop them in the simpler discrete-time setting of Volume I.
Chapter 3, State Prices, discusses the change of measure associated with riskneutral
pricing of European derivative securities, again as a warm-up exercise
for change of measure in continuous-time models. An interesting application
developed here is to solve the problem of optimal (in the sense of expected
utility maximization) investment in a binomial model. The ideas of Chapters
1 to 3 are essential to understanding the methodology of modern quantitative
finance. They are developed again in Chapters 4 and 5 of Volume II.

The remaining three chapters of Volume I treat more specialized concepts.
Chapter 4, American Derivative Securities, considers derivative securities
whose owner can choose the exercise time. This topic is revisited in
a continuous-time context in Chapter 8 of Volume II. Chapter 5, Random
Walk, explains the reflection principle for random walk. The analogous reflection
principle for Brownian motion plays a prominent role in the derivation of
pricing formulas for exotic options in Chapter 7 of Volume II. Finally, Chapter
6, Interest-Rate-Dependent Assets, considers models with random interest
rates, examining the difference between forward and futures prices and introducing
the concept of a forward measure. Forward and futures prices reappear
at the end of Chapter 5 of Volume II. Forward measures for continuous-time
models are developed in Chapter 9 of Volume II and used to create forward
LIBOR models for interest rate movements in Chapter 10 of Volume II.