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The aim of this book is to present the mathematical tools underlying the valuation of American-style derivative securities, i.e. contracts that can be exercised prior to their maturity date at the option of the holder. The book is divided into 6 chapters: the first 2 provide background mathematical results, the next 4 present applications to the valuation of derivative securities. Chapter 1 contains standard material on stochastic processes, martingales and stochastic calculus. Some of the topics, such as the martingale representation theorem or the Girsanov change of measure, are essential for later development. Others, such as the discussion of Levy processes, appear less relevant. Chapter 2 covers key results from the Theory of Optimal Stopping. The results are presented in a condensed form with short but explicit proofs. The presentation is informative and easy to follow. Much of the discussion is centered around the Dirichlet and Snell Envelopes. These notions are extended to the case in which the stopping region is a subset of the time domain. The classic notions and their properties extend in a natural way to this setting. The application to finance is mainly based on the martingale approach to valuation (risk neutralization procedure). The foundations of the approach for the complete market case are detailed in Chapter 3. That material has now become standard as it is presented in a number of books (see for example Karatzas and Shreve (1988)). Some of the roots of this procedure can be found in an early article of Cox and Ross (1976). Chapter 4 presents a very short introduction to options. The classification adopted distinguishes (i) options with constrained exercise times, (ii) options with constrained portfolios and (iii) path-dependent options. Although the reasons behind this classification are understood it appears odd in certain respects. Indeed, constraints on the exercise time or path-dependency are properties of the option payoff, i.e. of the option contract. Constrained portfolios are usually properties of the financial market in which one operates, and not of the option contract itself. Thus under (ii) the author

Generalized optimal stopping problems and financial markets, by Dennis Wong