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Focusing on recent advances in option pricing under the SABR model, this book shows how to price options under this model in an arbitrage-free, theoretically consistent manner. It extends SABR to a negative rates environment, and shows how to generalize it to a similar model with additional degrees of freedom, allowing simultaneous model calibration to swaptions and CMSs.

Since the SABR model is used on practically every trading floor to construct interest rate options volatility cubes in an arbitrage-free manner, a careful treatment of it is extremely important. The book will be of interest to experienced industry practitioners, as well as to students and professors in academia.

Aimed mainly at financial industry practitioners (for example quants and former physicists) this book will also be interesting to mathematicians who seek intuition in the mathematical finance.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
The SABR model introduced in Hagan et al. [39] is widely used by practitioners to capture skew and smile features observed in the interest rates implied volatilities.
Alexandre Antonov, Michael Konikov, Michael Spector

Chapter 2. Exact Solutions to CEV Model with Stochastic Volatility

Abstract
In this chapter, we discuss the classic CEV model and the CEV model with stochastic volatility. We derive PDF and option values for the CEV, based on analysis of solutions close to a singular point. For a general stochastic volatility, we obtain the option price in terms of the moment generating functions, assuming zero-correlation between asset and volatility [11].
Alexandre Antonov, Michael Konikov, Michael Spector

Chapter 3. Classic SABR Model: Exactly Solvable Cases

Abstract
In this chapter, we analyze exactly solvable cases for the SABR model; free Normal SABR, Log-normal SABR, Normal SABR with positive rate and zero correlation between underlying and volatility, and a general zero correlation case.
Alexandre Antonov, Michael Konikov, Michael Spector

Chapter 4. Classic SABR Model: Heat Kernel Expansion and Projection on Solvable Models

Abstract
In this chapter, we discuss the classic SABR model. We review invariant forms of the backward and forward Kolmogorov equations, applying then the heat kernel asymptotic expansion to the SABR model. We improve the approximate results for SABR option pricing, combining asymptotic expansion close to ATM with mapping procedure (Antonov and Misirpashaev in Projection on a quadratic model by asymptotic expansion with an application to LMM swaption, SSRN paper, 2009 [7]) onto the exactly solvable zero correlation SABR.
Alexandre Antonov, Michael Konikov, Michael Spector

Chapter 5. Extending SABR Model to Negative Rates

Abstract
In the low or negative interest rates environment, extending option models to negative rates becomes important. This chapter describes two such extensions of the SABR model: free SABR and mixture SABR. For free SABR, an exact formula is derived for option prices in the case of zero correlation between the rate and its volatility. For nonzero correlation, a mapping procedure onto a mimicking zero-correlation model is applied. Mixture SABR always has a closed-form solution for option prices, and has additional degrees of freedom allowing it to calibrate to a broader set of trades, e.g a set of swaptions and a CMS payment. Analytical results for free and mixture SABR models are compared with the Monte Carlo simulation ones.
Alexandre Antonov, Michael Konikov, Michael Spector

Chapter 6. Conclusion

Abstract
In this book, we have unified modern analytical material on the SABR model. Due to a small size of the book we have selected the shortest and most intuitive proofs of the underlying mathematical results. Moreover, we have presented simple insights for numerous non-trivial concepts in the stochastic processes and the approximation theory. Finally, we have presented numerous numerical results comparing analytics with simulation which can potentially serve as validation benchmarks.
Alexandre Antonov, Michael Konikov, Michael Spector

Backmatter

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