Analytically Tractable Stochastic Stock Price Models

This book offers sharp asymptotic formulas with error estimates for distribution densities of stock prices, option pricing functions and implied volatilities in stochastic volatility models. For readers familiar with stochastic analysis and probability theory.

Asymptotic analysis of stochastic stock price models is the central topic of the present volume. Special examples of such models are stochastic volatility models, that have been developed as an answer to certain imperfections in a celebrated Black-Scholes model of option pricing. In a stock price model with stochastic volatility, the random behavior of the volatility is described by a stochastic process. For instance, in the Hull-White model the volatility process is a geometric Brownian motion, the Stein-Stein model uses an Ornstein-Uhlenbeck process as the stochastic volatility, and in the Heston model a Cox-Ingersoll-Ross process governs the behavior of the volatility. One of the author's main goals is to provide sharp asymptotic formulas with error estimates for distribution densities of stock prices, option pricing functions, and implied volatilities in various stochastic volatility models. The author also establishes sharp asymptotic formulas for the implied volatility at extreme strikes in general stochastic stock price models.

The present volume is addressed to researchers and graduate students working in the area of financial mathematics, analysis, or probability theory. The reader is expected to be familiar with elements of classical analysis, stochastic analysis and probability theory.

Comprehensive in scope Results discussed appear for the first time in a mathematical monograph Unique source of information about analytically tractable stochastic volatility models? Includes supplementary material: sn.pub/extras

Autorentext

Archil Gulisashvili received his Ph.D. degree and Doctor of Science degree from the Tbilisi State University in Tbilisi, Georgia. Currently he is a Professor of Mathematics at Ohio University. Prior to joining Ohio University, he has held visiting positions at Boston University, Cornell University, and Howard University. His research interests include financial mathematics, Schrödinger semigroups, Feynman-Kac propagators, and Fourier analysis.

Inhalt
Preface.- Aknowledgements.- 1.Volatility Processes.- 2.Stock Price Models with Stochastic Volatility.- 3.Realized Volatility and Mixing Distributions.- 4.Integral Transforms of Distribution Densities.- 5.Asymptotic Analysis of Mixing Distributions.- 6.Asymptotic Analysis of Stock Price Distributions.- 7.Regularly Varying Functions and Pareto Type Distributions.- 8.Asymptotic Analysis of Option Pricing Functions.- 9.Asymptotic Analysis of Implied Volatility.- 10.More Formulas for Implied Volatility.- 11.Implied Volatility in Models Without Moment Explosions.- Bibliography.- Index.