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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.

Capital assets pricing model. --- Global analysis (Mathematics). --- Stocks -- Prices -- Mathematical models. --- Business & Economics --- Economic Theory --- Stock price forecasting --- Stock price indexes. --- Mathematical models. --- Averages, Stock --- Indexes, Stock --- Stock averages --- Stock indexes --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Approximation theory. --- Applied mathematics. --- Engineering mathematics. --- Economics, Mathematical. --- Probabilities. --- Quantitative Finance. --- Analysis. --- Probability Theory and Stochastic Processes. --- Approximations and Expansions. --- Applications of Mathematics. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Economics --- Mathematical economics --- Econometrics --- Engineering --- Engineering analysis --- Mathematical analysis --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- 517.1 Mathematical analysis --- Math --- Science --- Methodology --- Price indexes --- Finance. --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Funding --- Funds --- Currency question --- Economics, Mathematical . --- Social sciences --- Mathematics in Business, Economics and Finance. --- Probability Theory.

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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.

Quantitative methods (economics) --- Mathematical analysis --- Operational research. Game theory --- Mathematics --- Financial analysis --- Computer science --- analyse (wiskunde) --- toegepaste wiskunde --- stochastische analyse --- informatica --- financiële analyse --- kansrekening

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Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts. Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour. Graduate students, researchers and practitioners will find this book very useful, and the diversity of topics will appeal to people from mathematical finance, probability theory and differential geometry.

Mathematics. --- Quantitative Finance. --- Probability Theory and Stochastic Processes. --- Approximations and Expansions. --- Differential Geometry. --- Finance. --- Global differential geometry. --- Distribution (Probability theory). --- Mathématiques --- Finances --- Géométrie différentielle globale --- Distribution (Théorie des probabilités) --- Economics. --- Finance -- Mathematical models. --- Finance -- Statistical methods. --- Business & Economics --- Economic Theory --- Finance --- Mathematical models. --- Statistical methods. --- Economic theory --- Political economy --- Approximation theory. --- Economics, Mathematical. --- Differential geometry. --- Probabilities. --- Social sciences --- Economic man --- Distribution (Probability theory. --- Funding --- Funds --- Economics --- Currency question --- Geometry, Differential --- Math --- Science --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Economics, Mathematical . --- Differential geometry --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Mathematical economics --- Econometrics --- Methodology