Analytically Tractable Stochastic Stock Price Models

Transcription

1 Archil Gulisashvili Analytically Tractable Stochastic Stock Price Models 4Q Springer

2 Contents 1 Volatility Processes Brownian Motion s Geometric Brownian Motion Long-Time Behavior of Marginal Distributions Ornstein-Uhlenbeck Processes Ornstein-Uhlenbeck Processes and Time-Changed Brownian Motions Absolute Value of an Ornstein-Uhlenbeck Process Squared Bessel Processes and CIR Processes Squared Bessel Processes and Sums of the Squares of Independent Brownian Motions Chi-Square Distributions Noncentral Chi-Square Distributions Marginal Distributions of Squared Bessel Processes. Formulations Laplace Transforms of Marginal Distributions Marginal Distributions of Squared Bessel Processes. Proofs Time-Changed Squared Bessel Processes and CIR Processes....., Marginal Distributions of CIR Processes Ornstein-Uhlenbeck Processes and CIR Processes Notes and References 35 2 Stock Price Models with Stochastic Volatility Stochastic Volatility Correlated Stochastic Volatility Models Hull-White, Stein-Stein, and Heston Models Relations Between Stock Price Densities in Stein-Stein and Heston Models Girsanov's Theorem Risk-Neutral Measures Risk-Neutral Measures for Uncorrelated Hull-White Models... 52

3 Contents 2.8 Local Times for Semimartingales Risk-Neutral Measures for Uncorrelated Stein-Stein Models Risk-Neutral Measures for Uncorrelated Heston Models Hull-White Models. Complications with Correlations Heston Models and Stein-Stein Models. No Complications with Correlations Notes and References 65 Realized Volatility and Mixing Distributions Asymptotic Relations Between Functions Mixing Distributions and Stock Price Distributions Stock Price Densities in Uncorrelated Models as Mixtures of Black-Scholes Densities Mixing Distributions and Heston Models \ Mixing Distributions and Hull-White Models with Driftless Volatility Mixing Distributions and Hull-White Models Mixing Distributions and Stein-Stein Models Notes and References 75 Integral Transforms of Distribution Densities Geometric Brownian Motions and Laplace Transforms of Mixing Distributions Bougerol's Identity in Law Squared Bessel Processes and Laplace Transforms of Mixing Distributions CIR Processes and Laplace Transforms of Mixing Distributions Ornstein-Uhlenbeck Processes and Laplace Transforms of Mixing Distributions Hull-White Models with Driftless Volatility and Hartman-Watson Distributions Mixing Density and Stock Price Density in the Correlated Hull-White Model Mellin Transform of the Stock Price Density in the Correlated Heston Model.., Mellin Transform of the Stock Price Density in the Correlated Stein-Stein Model Notes and References 107 Asymptotic Analysis of Mixing Distributions i Asymptotic Inversion of the Laplace Transform Asymptotic Behavior of Fractional Integrals Asymptotic Behavior of Integral Operators with Log-Normal Kernels Asymptotic Formulas for Mixing Distribution Densities Associated with Geometric Brownian Motions 122

4 Contents Hypergeometric Functions Dufresne's Theorems Exponential, Beta, and Gamma Distributions Proof of Formula(5.77) for r^o Dufresne's Recurrence Formula Equivalent Formulation of Duresne's Recurrence Formula Completion of the Proof of Theorem Asymptotic Behavior of Mixing Distribution Densities Near Zero Asymptotic Formulas for Mixing Distribution Densities Associated with CIR Processes Asymptotic Formulas for Mixing Distribution Densities Associated with Ornstein-Uhlenbeck Processes Constants in Asymptotic Formulas. Simplifications Notes and References 165 Asymptotic Analysis of Stock Price Distributions Asymptotic Formulas for Stock Price Densities in Heston Models Heston Models as Affine Models and Moment Explosions Saddle Point Method and Mellin Inversion Finding the Saddle Point Local Expansion Around the Saddle Point Saddle Point Approximation of the Density Tail Estimates Explicit Formula for the Constant A \ Asymptotic Formulas for Stock Price Densities in Uncorrelated Heston Models The Constants A\, A 2 and A3 Obtained by Different Methods Are Equal Asymptotic Formulas for Stock Price Densities in Stein-Stein Models Asymptotic Formulas for Stock Price Densities in Uncorrelated Hull-White Models Comparison of Stock Price Densities The Constants A3 and fi Notes and References 198 Regularly Varying Functions and Pareto-Type Distributions Regularly Varying Functions Class R-] and Regularly Varying Majorants of Integrable Monotone Functions Fractional Integrals of Regularly Varying Functions Slowly Varying Functions with Remainder 214

5 xvi ^ Contents 7.5 Smoothly Varying Functions Pareto-Type Distributions Pareto-Type Distributions in Stochastic Volatility Models Notes and References Asymptotic Analysis of Option Pricing Functions Call and Put Pricing Functions in Stochastic Asset Price Models The Black-Scholes Model Black-Scholes Formulas Derivatives of Option,Pricing Functions Asymptotic Behavior of Pricing Functions in Stochastic Volatility Models Notes and References Asymptotic Analysis of Implied Volatility Implied Volatility in General Option Pricing Models Implied Volatility Surfaces and Static Arbitrage Asymptotic Behavior of Implied Volatility Near Infinity Corollaries Extra Terms: First-Order Asymptotic Formulas for Implied Volatility Extra Terms: Higher-Order Asymptotic Formulas for Implied Volatility 258" 9.7 Symmetries and Asymptotic Behavior of Implied Volatility Near Zero Symmetric Models Asymptotic Behavior of Implied Volatility for Small Strikes Notes and References More Formulas for Implied Volatility Moment Formulas Tail-Wing Formulas Tail-Wing Formulas with Error Estimates Regularly Varying Stock Price Densities and Tail-Wing Formulas...', Implied Volatility in Stochastic Volatility Models Asymptotic Equivalence and Moment Formulas Implied Volatility in Mixed Models Asset Price Models with Jumps Volatility Smile : Gatheral's SVI Parameterization of Implied Variance Notes and References Implied Volatility in Models Without Moment Explosions General Asymptotic Formulas in Models Without Moment Explosions 315

6 Contents».- xvii 11.2 Constant Elasticity of Variance Model Displaced Diffusion Model Finite Moment Log-Stable Model Piterbarg's Conjecture Asymptotic Equivalence and Piterbarg's Conjecture SV1 and SV2 Models of Rogers and Veraart Notes and References 345 References 347 Index r 357 \ '" - - -