The parameter that measures volatility has long caused many problems in financial modeling. The "correct" or "true" value of the parameter is desperately needed for the pricing of derivatives, but it is not so simple to determine it. One can think of statistical methods to estimate the value of the volatility parameter, but it seems it is just not enough. The problem is how to extract the volatility from the market, i.e., the volatility that is actually observed in option prices. Now we talk about implied volatility, and this is what this book is about.
Fengler has written a research monograph. Nevertheless, some basic concepts are explained, such as the Black-Scholes pricing formula, which leads directly into discussion of implied volatility and the implied volatility surface. Local volatility, smoothing techniques and dimension-reduced modeling are presented. Concepts are presented in detail, elegantly connecting the past and current research, mathematical presentation, and numerical output (graphics). The author leaves one final chapter for discussion on future research. The appendices serve primarily for presentation of proofs and some results from stochastic calculus.
This book is suitable for researchers, graduate students, and finance professionals. Some finance knowledge on derivatives is helpful, primarily to be able to place the text in the right area. The mathematics required is at the level of stochastic calculus. Simulations and graphics presented in the book are done in XploRe. The code for the output presented in the book can be downloaded (using the license from the book) for personal use.
Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.
Introduction.- The Implied Volatility Surface.- Smile Consistent Volatility Models.- Smoothing Techniques.- Dimension-Reduced Modeling.