This is the first of the two-volume series evolving from the author’s mathematics courses in M.Sc. Computational Finance program at Carnegie Mellon University (USA). The contents of this book is organized such as to give the reader precise statements of results, plausibility arguments, mathematical proofs and, more importantly, the intuitive explanations of the financial and economic phenomena. Each chapter concludes with summary of the discussed matter, bibliographic notes, and a set of really useful exercises.
The present Volume I introduces the (discrete-time) binomial asset pricing model as a paradigm of practice and a prologue, necessary for the much more complex concepts and results needed in the continuous-time theory of stochastic calculus exposed in Volume II (see below). Volume I is a prerequisite for Computational Finance courses based on Volume II, and prepares the reader for the more general setting in the latter volume by treating several different concepts in a simpler, discrete-time, and also less technical style, e.g. martingales, Markov processes, change of measure, risk-neutral pricing etc. Here it is a brief inspection on the contents of the six (6) chapters in Volume I: Chapter 1 (The Binomial No-Arbitrage Pricing Model) presents the no-arbitrage method of option pricing in a binomial model, and the fundamental concept of risk-neutral pricing in a first mathematical setting. Chapter 2 (Probability Theory on Coin Toss Space) introduces martingales, Markov processes, and the risk-neutral pricing formula for European derivative securities. Chapter 3 (State Prices) discusses the change of measure associated with risk-neutral pricing of European derivative securities, and its application to solving the problem of optimal investment in a binomial model. Chapter 4 (American Derivative Securities) investigates derivative securities whose owner can choose the exercise time. Chapter 5 (Random Walk) explains the reflection principle for random walk, while Chapter 6 (Interest-Rate-Dependent Assets) considers models with random interest rates, examines the difference between forward and futures prices, and introduces the concept of a forward measure. The Appendix provides the Proof of Fundamental Properties of Conditional Expectations.