Modern SABR Analytics

1 Introduction.- 1.1 Introduction.- 1.2 Wide popularity of the SABR.- 1.3 Simple derivation.- 1.4 Modifications and extensions of the SABR.- 1.5 CMS and the SABR.- 1.6 Approximation accuracy and its improvements.- 1.7 About this book.- 2 Exact Solutions to CEV Model with Stochastic Volatility.- 2.1 Introduction.- 2.2 Transforming CEV Process into the Bessel One.- 2.3 Solution behavior near singular point x = 0, integrability, flux.- 2.4 Laplace Transform.- 2.5 Probability distributions.- 2.6 Back to CEV model.- 2.6.1 Option pricing through Chi Square distributions.- 2.7 Alternative expressions for CEV option values.- 2.8 CEV Model with Stochastic Volatility.- 2.9 Conclusion.- 3 Classic SABR Model: Exactly Solvable Cases.- 3.1 Introduction.- 3.2 Probability Density Functions for the Free Normal and Log-Normal SABR, Probabilistic Approach.- 3.3 Deriving PDFs using Kolmogorov equations.- 3.4 Option Value for the Free Normal SABR.- 3.5 Option Value for the Lognormal SABR.- 3.6 The Zero Correlation case.- 4 Classic SABR Model: Heat Kernel Expansion and Projection on Solvable Models.- 4.1 Introduction.- 4.2 Invariant forms of Diffusion Equations.- 4.3 Heat Kernel Expansion.- 4.4 Non-Zero Correlation General Case.- 4.5 Conclusion.- References.