Preface This monograph is intended as an introduction to some elements of mathe- matical finance. It begins with the development of the basic ideas of hedg- ing and pricing of European and American derivatives in the discrete (i.e., discrete time and discrete state) setting of binomial tree models. Then a general discrete finite market model is defined and the fundamental theo- rems of asset pricing are proved in this setting. Tools from probability such as conditional expectation, filtration, (super)martingale, equivalent martin- gale measure, and martingale representation are all used first in this simple discrete framework. This is intended to provide a bridge to the continuous (time and state) setting which requires the additional concepts of Brownian motion and stochastic calculus. The simplest model in the continuous set- ting is the Black-Scholes model. For this, pricing and hedging of European and American derivatives are developed. The book concludes with a descrip- tion of the fundamental theorems of asset pricing for a continuous market model that generalizes the simple Black-Scholes model in several directions. The modern subject of mathematical finance has undergone considerable development, both in theory and practice, since the seminal work of Black and Scholes appeared a third of a century ago. The material presented here is intended to provide students and researchers with an introduction that will enable them to go on to read more advanced texts and research pa- pers. Examples of topics for such further study include incomplete markets, interest rate models and credit derivatives. For reading this book, a basic knowledge of probability theory at the level of the book by Chung [10] or D. Williams [38], plus for the chapters on continuous models, an acquaintance with stochastic calculus at the level of the book by Chung and Williams [11] or Karatzas and Shreve [27], is vii
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