A Comparative Study of Various Versions of the SABR Model Adapted to Negative Interest Rates
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A Comparative Study of Various Versions of the SABR Model Adapted to Negative Interest Rates IRINA SHADRINA Master's Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam Faculty of Economics and Business Amsterdam School of Economics Author: Irina Shadrina Student nr: 11088397 Email: [email protected] Date: July 23, 2017 Supervisor: Prof. dr. L. J. van Gastel Second reader: Prof. dr. ir. M. H. Vellekoop SABR Volatility Smile Interpolation | Irina Shadrina iii Abstract In this thesis we compare different extensions of the SABR model adapted to negative interest rates. Namely, we introduce and discuss analytical properties of the Shifted SABR, the Normal SABR, the Free Boundary SABR and the Mixture SABR models. To compare numerical outcomes of the models we compute approxima- tion formulae for the implied Normal volatility in terms of the models parameters and compare the quality of the volatility smile approxima- tion using data on swaptions as a benchmark. Computational results show that the approximation quality of the Dis- placed SABR model outperforms the other models. Moreover, in some special cases of the market data the approximation quality is only sufficient for the Displaced SABR model. We also conclude that the Hagan approach can not be used for calibration of the parameters of the Free Boundary SABR and Mixture SABR models. Finally, we conclude, using our quality estimation criteria, that the Displaced SABR model is the most suitable model for modelling the volatility smile in the presence of negative rates. Keywords negative interest rates, Black's model, Bachelier's model, SABR model, Displaced SABR model, Normal SABR model, Free Boundary SABR model, Mixture SABR model, swap- tions Contents Preface vi 1 Introduction1 1.1 Economic background............................1 1.2 Problem statement..............................3 1.3 Thesis outline.................................5 2 Interest rate derivatives6 2.1 Market of interest rate derivatives.....................6 2.2 Interest rates.................................6 2.3 Interest rate derivatives...........................7 3 Mathematical framework9 3.1 Assumptions.................................9 3.2 Ito calculus..................................9 3.3 Pricing under the risk neutral measure................... 11 4 Implied volatility 13 4.1 Black's model................................. 13 4.2 Bachelier's model............................... 13 4.3 Implied volatility............................... 14 4.4 Comparison of Normal and Black volatilities................ 14 4.5 Volatility smile................................ 15 4.6 Local volatility models............................ 15 5 SABR model 17 5.1 Model definition............................... 17 5.2 Risk-neutral probability density function.................. 18 5.3 Results of Hagan et. al............................ 19 5.3.1 Impact of parameters........................ 20 5.3.2 Problems with Hagan et. al. formula................ 20 5.4 Results of Antonov et. al........................... 20 5.5 Parameters calibration............................ 21 5.5.1 Hagan's formula........................... 21 5.5.2 Antonov's formula.......................... 22 5.5.3 Fit to the market........................... 22 6 SABR extensions for negative interest rates 23 6.1 Displaced SABR............................... 23 6.1.1 Advantages of the Displaced SABR model............. 24 6.1.2 Disadvantages of the Displaced SABR model........... 24 6.2 Normal SABR................................ 24 6.2.1 Hagan's approximation formula................... 24 6.2.2 Antonov's explicit solution..................... 25 iv SABR Volatility Smile Interpolation | Irina Shadrina v 6.2.3 Practical computations with Antonov's formula.......... 25 6.2.4 Advantages of the Normal SABR model.............. 26 6.2.5 Disadvantages of the Normal SABR model............ 26 6.3 Free Boundary SABR............................ 26 6.3.1 Antonov's solution.......................... 26 6.3.2 Hagan's approximation formula................... 27 6.3.3 The approximation formula for the At-The-Money volatility... 27 6.3.4 Advantages of the Free Boundary SABR model.......... 29 6.3.5 Disadvantages of the Free Boundary SABR model........ 29 6.4 Mixture SABR................................ 29 6.4.1 Definition of Mixture SABR..................... 29 6.4.2 Analytical formulas for the Mixture SABR model......... 31 6.4.3 Advantages of the Mixture SABR model.............. 32 6.4.4 Disadvantages of the Mixture SABR model............ 33 7 Comparison of the models 34 7.1 Comparison criteria............................. 34 7.2 Market data.................................. 34 7.2.1 Implied normal volatility...................... 34 7.3 Methodology................................. 36 7.4 Computational results............................ 36 7.4.1 Displaced SABR........................... 36 7.4.2 Normal SABR and Free Boundary SABR............. 37 7.4.3 Approximation of different kind of smiles............. 37 7.5 Mixture model................................ 40 8 Conclusions 41 8.1 Summary................................... 41 8.2 Discussion................................... 41 8.2.1 Stickiness at zero........................... 41 8.2.2 Shortcomings of Antonov's approach................ 42 8.3 Conclusion.................................. 42 Appendix 1 44 Appendix 2 48 References 49 Preface Statement of Originality: This document is written by Irina Shadrina who declares to take full responsibility for the contents of this docu- ment. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Eco- nomics and Business is responsible solely for supervision of completion of the work, not for the contents. Acknowledgement: I would like to thank my supervisor Leendert van Gastel for his assistance during the writing of this thesis. vi Chapter 1 Introduction 1.1 Economic background In July 2012 The Central Bank of Denmark reduced the interest rate to −0:02%. Al- though highly unusual and innovative, negative interest rates did not damage the fi- nancial system in any essential way. Moreover, from the Denmark experience it was suggested that "negative rates may deserve to move from taboo to the standard mone- tary policy toolbox" 1. In 2014, the European Central Bank introduced negative interest rates. Other countries that were not a part of the Eurozone, such as Japan, Sweden and Switzerland, followed as well. We provide here an illustration from the European Central Bank. The reasons for introduction negative interest rates The main reason for introduction negative interest rates was the slow economy with the low inflation and the low level of investments. This needed to be speeded up, and was the basic reason to introduce negative interest rates. In simple words, the idea was to motivate people to spend and invest, rather than save. In the situation of negative interest rates banks are encouraged to lend more and penalized for keeping cash. It stimulates investors to borrow money and invest into the economy and customers to spend more money. It increases the inflation and the economic growth rates. Banks and investors are encouraged not to invest in domestic currency. This will depreciate the currency value in the foreign markets and stimulates export. 1https://www.bloomberg.com/news/articles/2016-06-06/denmark-land-below-zero-where-negative- interest-rates-are-normal 1 2 Irina Shadrina | SABR Volatility Smile Interpolation Possible negative consequences We would like to discuss possible negative implications if the negative interest rates will stay on the market for a long time. One of the first concerns that comes to mind is whether the low rates will cause the bubbles on the housing market. Also, banks are encouraged to move from safe assets, like government bonds, to more risky ones. This raises the financial stability risk and the risk of bank runs. Moreover, a possible result of that would by that the rules of banking capital requirements will imply the higher obligatory reserves. Banks are not sure about the reaction of the depositors on the introduction of the negative interest rates for their deposits and are afraid to lose customers. This can make the banks less profitable. Lower profitability means lower earnings, and that will make it harder for banks to build up capital during the next credit cycle. A notable example is described in the article "In the era of negative interest rates, Japans second-largest bank finds itself having to run just to stand still." 1 Negative interest rates put pressure profitability of insurers and pension companies. Negative interest rates reduce the discount on the cash flows from assets. The present value of liabilities of the insurance companies and pension funds is getting bigger, and so do the computed reserves. Moreover, to keep reserves on a bank account, companies would have to pay. If negative interest rates stay for a substantially long period, financial institutions will be forced to change their behaviour. This will cause additional risks and unknown effects. Typical market behaviour could be changed as well. If more countries will introduce negative interest rates, this will also diminish ex- pected positive effects for the economy of the countries that did this first. For instance, the currency depreciation would probably be turned back. Expected timeframe for the