I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the BlackScholes (1973) model. The solution technique is based on characteristi c functions and can be applied to other problems.

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A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Convex Analysisの二,三の進展について

Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.

Functional Analysis, Sobolev Spaces and Partial Differential Equations

Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.

https://www.springer.com/productFlyer_978-0-387-98463-6.pdf?SGWID=0-0-1297-1590487-0

One-Parameter Semigroups for Linear Evolution Equations

Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.

Singular Integrals and Differentiability Properties of Functions.

PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular Integral Operators: Fourier Transform228VIIPseudo-Differential and Singular Integral Operators: Almost Orthogonality269VIIIOscillatory Integrals of the First Kind329IXOscillatory Integrals of the Second Kind375XMaximal Operators: Some Examples433XIMaximal Averages and Oscillatory Integrals467XIIIntroduction to the Heisenberg Group527XIIIMore About the Heisenberg Group587Bibliography645Author Index679Subject Index685

Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals

The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces.

Introduction to Fourier Analysis on Euclidean Spaces.

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H p spaces of several variables

We study the stock price distributions that arise when prices follow a diffusion process with a stochastically varying volatility parameter. We use analytic techniques to derive an explicit closed-form solution for the case where volatility is driven by an arithmetic Ornstein-Ublenbeck (or AR1) process. We then apply our results to two related problems in the finance literature: (1) options pricing in a world of stochastic volatility, and (2) the relationship between stochastic volatility and the nature of "fat tailes" in stock price distributions. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

Elias M. Stein

Papers

Singular Integrals and Differentiability Properties of Functions.

Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals

Introduction to Fourier Analysis on Euclidean Spaces.

H p spaces of several variables

Stock Price Distributions with Stochastic Volatility: An Analytic Approach