Copenhagen Business School Copenhagen, Spring 2020

The SABR model in a negative framework Theory and practice

Jökull Ívarsson (125476) Supervisor: Peter Feldhütter

Master thesis, cand.merc. Finance and Investments

Copenhagen Business School

Number of pages: 79 Characters including spaces: 107.504 Date: 15.05.2020

This thesis was written as a part of the Master of Science in Economics and Business Administration at CBS. Please note that neither the institution nor the examiners are responsible – through the approval of this thesis – for the theories and methods used, or results and conclusions drawn in this work. Acknowledgements

I would like to thank my thesis supervisor, Peter Feldhütter, for his encouragement, support, and giving me critical feedback when I needed it.

I would also like to thank my family and friends for their emotional support and a gentle push in the right direction when needed.

Furthermore, I would like to specifically thank my brother, Egill. For his immense help with the final steps and completion of the thesis.

1 2

Abstract

This thesis focused on the stochastic mode, the SABR model. This model is well known and has been used in financial institutions since its inception. However, the model is unable to work if interest rates are negative. Some refinements have been proposed to change the workings of the model to incorporate negative rates. The thesis looked at these refinements and tested whether the Normal SABR model and the Shifted SABR model were able to produce implied volatilities similar to the ones observed in the market for both caplets and swaptions.

The results showed us that, in a negative interest rate environment, these modifications to the SABR model can produce implied volatilities that are very close to the market volatilities for both caplets and swaptions. Furthermore, the thesis looked at Obłój’s refinement, which states that the SABR model is unable to produce accurate volatilities for options with low strikes and long maturities. It was found that this method was unable to produce better fitting volatilities for those options than the Shifted SABR model. Contents 3

Contents

1 Introduction 7 1.1 Background and motivation ...... 7 1.2 Research question ...... 7 1.3 Structure of the thesis ...... 8

2 Theory and literature review 9 2.1 Interest rates and interest rate derivatives ...... 9 2.1.1 Mathematical framework ...... 9 2.1.2 Interest rates concepts ...... 12 2.1.3 Interest rate derivatives ...... 14 2.1.3.1 Forward rate agreements ...... 14 2.1.3.2 Interest rate swaps ...... 14 2.1.3.3 Caps and floors ...... 16 2.1.3.4 Swap options (Swaptions) ...... 17 2.2 Interest rate models ...... 19 2.2.1 Black’s Model (1976) ...... 19 2.2.2 Shifted Black model ...... 21 2.2.3 Bachelier’s (Normal) model ...... 22 2.2.4 Risk management within the constant volatility models ...... 23 2.2.5 Short-rate models (One-factor) ...... 25 2.2.5.1 Vasicek Model ...... 25 2.2.5.2 Hull-White one-factor model ...... 26 2.2.5.3 Swaption pricing with short-rate models ...... 27 2.2.6 Libor Market Model ...... 28 2.2.7 Comparison of the models ...... 30 2.3 Stochastic volatility models ...... 31 2.3.1 Volatility smiles ...... 31 2.3.2 Local volatility model ...... 32 2.3.3 SABR model ...... 33 2.3.3.1 The SABR parameters ...... 34 2.3.3.2 Usage of SABR ...... 37 2.3.3.3 Calibrating a SABR model ...... 38 2.3.4 Shifted SABR model ...... 39 2.3.5 Obłój’s refinement of the SABR model ...... 40 2.3.6 Risk management under the SABR model ...... 41 2.3.7 Further research in the SABR model ...... 43 2.4 Summary of chapter ...... 44

3 Data & methodology 45 3.1 Data ...... 45 3.1.1 Spot rates ...... 45 3.1.2 Implied volatilities ...... 45 3.2 Empirical model ...... 46 3.2.1 Stripping cap volatility ...... 46 4 Contents

3.2.2 Calibrating the SABR model .