This work is aimed at an audience with a sound mathematical background
wishing to learn about the rapidly expanding field of mathematical finance.
Its content is suitable particularly for graduate students in mathematics
who have a background in measure theory and probability.
The emphasis throughout is on developing the mathematical concepts
required for the theory within the context of their application. No attempt
is made to cover the bewildering variety of novel (or ‘exotic’) financial instruments
that now appear on the derivatives markets; the focus throughout
remains on a rigorous development of the more basic options that lie
at the heart of the remarkable range of current applications of martingale
theory to financial markets.
The first five chapters present the theory in a discrete-time framework.
Stochastic calculus is not required, and this material should be accessible
to anyone familiar with elementary probability theory and linear algebra.
The basic idea of pricing by arbitrage (or, rather, by non-arbitrage)
is presented in Chapter 1. The unique price for a European option in a
single-period binomial model is given and then extended to multi-period
binomial models. Chapter 2 introduces the idea of a martingale measure
for price processes. Following a discussion of the use of self-financing trading
strategies to hedge against trading risk, it is shown how options can
be priced using an equivalent measure for which the discounted price process
is a martingale. This is illustrated for the simple binomial Cox-Ross-
Rubinstein pricing models, and the Black-Scholes formula is derived as the
limit of the prices obtained for such models. Chapter 3 gives the ‘fundamental
theorem of asset pricing’, which states that if the market does not
contain arbitrage opportunities there is an equivalent martingale measure.
Explicit constructions of such measures are given in the setting of finite
market models. Completeness of markets is investigated in Chapter 4; in a
complete market, every contingent claim can be generated by an admissible
self-financing strategy (and the martingale measure is unique). Stopping
times, martingale convergence results, and American options are discussed
in a discrete-time framework in Chapter 5.
The second five chapters of the book give the theory in continuous time.
This begins in Chapter 6 with a review of the stochastic calculus. Stopping
times, Brownian motion, stochastic integrals, and the Itˆo differentiation