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Finance provides a dramatic example of the successful application of advanced mathematical techniques to the practical problem of pricing financial derivatives. This self-contained text is designed for first courses in financial calculus aimed at students with a good background in mathematics. Key concepts such as martingales and change of measure are introduced in the discrete time framework, allowing an accessible account of Brownian motion and stochastic calculus: proofs in the continuous-time world follow naturally. The Black-Scholes pricing formula is first derived in the simplest financial context. The second half of the book is then devoted to increasing the financial sophistication of the models and instruments. The final chapter introduces more advanced topics including stock price models with jumps, and stochastic volatility. A valuable feature is the large number of exercises and examples, designed to test technique and illustrate how the methods and concepts can be applied to realistic financial questions.
Preface; 1. Single period models; 2. Binomial trees and discrete parameter martingales; 3. Brownian motion; 4. Stochastic calculus; 5. The Black-Scholes model; 6. Different payoffs; 7. Bigger models; Bibliography and further reading; Notation; Index.
Interesting modification of the definitions of calculus for the non-differentiable context
Some areas of finance can be easily and accurately modeled using basic mathematics. We offer a course in finite mathematics at the college where I teach and we cover the basic formulas of interest and annuity computation. However, when you explore markets, it becomes a problem in multiple and complex stochastic processes. The equations are not differentiable the best mathematical model uses the random Brownian motion of molecular action. The depth and complexity of the probability models is what makes the large markets such as the commodity, bond, stock and currency, so unpredictable. Yet, mathematicians, being an optimistic lot, have developed a set of models that can be used to at least partially model the behavior. Some of the more successful models are described in this book. While knowledge of differential and integral calculus are necessary for understanding the material, it is largely secondary. The primary knowledge needed is that of complex probability models. Chapter 4 covers the concepts of stochastic calculus, which starts with the subheading ¿Stock prices are not differentiable.¿ However, this being said and clearly true, the model makers, in the true spirit of mathematicians, alter the definitions enough so that the functions can be partitioned so that a stochastic integral is defined and used. It is most unlikely that I will ever teach a course that would cover the material in this book. Nevertheless, I found the material very interesting the Black-Scholes model of securities valuation is something that all mathematicians should look at. Even if you don¿t understand it, you can appreciate it as a demonstration of the power of mathematics.
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