This book is an introduction to financial mathematics. It is intended for graduate students in
mathematics and for researchers working in academia and industry. The focus on stochastic
models in discrete time has two immediate benefits. First the probabilistic machinery is
simpler and one can discuss right away some of the key problems in the theory of pricing and
hedging of financial derivatives. Second the paradigm of a complete financial market where
all derivatives admit a perfect hedge becomes the exception rather than the rule. Thus the
need to confront the intrinsic risks arising from market incomleteness appears at a very early
stage. The first part of the book contains a study of a simple one-period model which also
serves as a building block for later developments. Topics include the characterization of
arbitrage-free markets preferences on asset profiles an introduction to equilibrium analysis
and monetary measures of financial risk. In the second part the idea of dynamic hedging of
contingent claims is developed in a multiperiod framework. Topics include martingale measures
pricing formulas for derivatives American options superhedging and hedging strategies with
minimal shortfall risk. This fourth newly revised edition contains more than one hundred
exercises. It also includes material on risk measures and the related issue of model
uncertainty in particular a chapter on dynamic risk measures and sections on robust utility
maximization and on efficient hedging with convex risk measures. Contents: Part I: Mathematical
finance in one period Arbitrage theory Preferences Optimality and equilibrium Monetary measures
of risk Part II: Dynamic hedging Dynamic arbitrage theory American contingent claims
Superhedging Efficient hedging Hedging under constraints Minimizing the hedging error Dynamic
risk measures
