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Option valuation under stochastic volatility. With Mathematica code. (English) Zbl 0937.91060

Newport Beach, CA: Finance Press. vii, 350 p. (2000).
This book is a research monograph and is for readers with a preknowledge in stochastic calculus and differential equations at the level it is typically applied in finance. That means that the mathematics is pretty informal. The author is physicist but active in option valuation and related research for over 20 years. This book is interesting for financial academic and traders. In this book option valuation are studied when security prices evolve with stochastic volatility. The goal of this book is to create a uniform and fairly comprehensive theoretical treatment of the case where the stock price and the volatility are described by a two-dimensional diffusion process. Further, a number of particular stochastic processes for volatility are discussed, especially, where volatility is described as the diffusion limit of a GARCH type process.
This book has 11 chapters. Chapter 1 describes the Brownian motion and some examples of stochastic volatility models as for example the GARCH diffusion model. Chapter 2 regards the transformed-based approach. Specially, the formula for option pricing are developed. The first is the usual-style or expected value formula. The second yields the desired arbitrage free value. In Chapter 3, the valuation partial differential equation associated with the risk-adjusted process is solved. In Chapter 4, “mixed theorems” are discussed. In Chapter 5, the theory of smile is taken up. The term structure of implied volatility is the subject of Chapter 6. Chapter 7 presents an utility-based equilibrium theory. In Chapter 8, the change-of-numeraire transformation or duality for short is discussed. This transformation generalizes a known put-call symmetry under constant volatility. Chapter 9 gives a detailed account of the affect of volatility explosions and the failure of the martingal pricing formula. In Chapter 10, both, the fundamental transform and the option pricing at large volatility are studied. In Chapter 11, the closed-form solution for a fundamental transform in the running examples (the square root model, the 3/2 model, and a special case of the GARCH diffusion (geometric Brownian motion) is developed.

MSC:

91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91G20 Derivative securities (option pricing, hedging, etc.)
91-04 Software, source code, etc. for problems pertaining to game theory, economics, and finance
60G35 Signal detection and filtering (aspects of stochastic processes)

Software:

Mathematica
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