Dr. Alexandre Antonov received his PhD from the Landau Institute for Theoretical Physics in 1997. Based at Numerix from 1998 to 2017 he recently joined Standard Chartered Bank in London as a director. His work concentrates on modeling and numerical methods for interest rates, cross currency, hybrid, credit and CVA/FVA/MVA. Dr. Antonov has multiple publications in mathematical finance and is a frequent speaker at financial conferences. He received a Quant of the Year Award from Risk magazine in 2016.Dr. Michael Konikov is an Executive Director and Head of Quantitative Development at Numerix, where he manages a team responsible for the development and delivery of models in Numerix software.  Previously, he worked at Citigroup, Barclays and Bloomberg in quantitative research and desk quant roles.  He completed his PhD in mathematical finance at the University of Maryland College Park, concentrating in particular on the application of pure jump processes to option pricing.  Dr. Konikov's publications cover diverse asset classes ranging from equity to interest rates and credit.Dr. Michael Spector, Director of Quantitative Research, received his PhD in theoretical physics from the Budker Institute of Nuclear Physics, Novosibirsk. He has worked at research centers and universities in Russia, Israel, and the US and is the author of multiple publications on plasma physics, hydrodynamics, turbulence, nonlinear wave dynamics and mathematical finance. He joined the Numerix quantitative research team in 2006, working on the valuation of various exotic options (Asians, lookbacks, barriers), and has lately concentrated on the development of stochastic volatility models for interest rates and equity.

• 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 OptionValue 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.

Dr. Alexandre Antonov received his PhD from the Landau Institute for Theoretical Physics in 1997. Based at Numerix from 1998 to 2017 he recently joined Standard Chartered Bank in London as a director. His work concentrates on modeling and numerical methods for interest rates, cross currency, hybrid, credit and CVA/FVA/MVA. Dr. Antonov has multiple publications in mathematical finance and is a frequent speaker at financial conferences. He received a Quant of the Year Award from Risk magazine in 2016.Dr. Michael Konikov is an Executive Director and Head of Quantitative Development at Numerix, where he manages a team responsible for the development and delivery of models in Numerix software.  Previously, he worked at Citigroup, Barclays and Bloomberg in quantitative research and desk quant roles.  He completed his PhD in mathematical finance at the University of Maryland College Park, concentrating in particular on the application of pure jump processes to option pricing.  Dr. Konikov's publications cover diverse asset classes ranging from equity to interest rates and credit.Dr. Michael Spector, Director of Quantitative Research, received his PhD in theoretical physics from the Budker Institute of Nuclear Physics, Novosibirsk. He has worked at research centers and universities in Russia, Israel, and the US and is the author of multiple publications on plasma physics, hydrodynamics, turbulence, nonlinear wave dynamics and mathematical finance. He joined the Numerix quantitative research team in 2006, working on the valuation of various exotic options (Asians, lookbacks, barriers), and has lately concentrated on the development of stochastic volatility models for interest rates and equity.

• 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 OptionValue 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.

Dr. Alexandre Antonov received his PhD from the Landau Institute for Theoretical Physics in 1997. Based at Numerix from 1998 to 2017 he recently joined Standard Chartered Bank in London as a director. His work concentrates on modeling and numerical methods for interest rates, cross currency, hybrid, credit and CVA/FVA/MVA. Dr. Antonov has multiple publications in mathematical finance and is a frequent speaker at financial conferences. He received a Quant of the Year Award from Risk magazine in 2016.Dr. Michael Konikov is an Executive Director and Head of Quantitative Development at Numerix, where he manages a team responsible for the development and delivery of models in Numerix software.  Previously, he worked at Citigroup, Barclays and Bloomberg in quantitative research and desk quant roles.  He completed his PhD in mathematical finance at the University of Maryland College Park, concentrating in particular on the application of pure jump processes to option pricing.  Dr. Konikov's publications cover diverse asset classes ranging from equity to interest rates and credit.Dr. Michael Spector, Director of Quantitative Research, received his PhD in theoretical physics from the Budker Institute of Nuclear Physics, Novosibirsk. He has worked at research centers and universities in Russia, Israel, and the US and is the author of multiple publications on plasma physics, hydrodynamics, turbulence, nonlinear wave dynamics and mathematical finance. He joined the Numerix quantitative research team in 2006, working on the valuation of various exotic options (Asians, lookbacks, barriers), and has lately concentrated on the development of stochastic volatility models for interest rates and equity.

• 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 OptionValue 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.