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Managing Swaption Risk with a Dynamic SABR Model Amsterdam School of Economics Faculty of Economics and Business Managing Swaption Risk with a Dynamic SABR Model MSc in Econometrics Financial Econometrics track Frank de Zwart 10204245 supervised by Dr. S.A. Broda and supervisor at Abn Amro Ms Hiltje Bijkersma July 28, 2017 ABN AMRO Bank N.V. CRM | Regulatory Risk | Model Validation Frank de Zwart Abn Amro Model Validation Statement of Originality This document is written by Student Frank de Zwart who declares to take full responsibility for the contents of this document. 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 Economics and Business is responsible solely for the supervision of completion of the work, not for the contents. Frank de Zwart Abn Amro Model Validation Abstract This thesis focuses on models that can be used to estimate risk measures, like Value at Risk and Expected Shortfall. The displaced Black’s model and the displaced SABR volatility model are used to price a portfolio of swaptions. The aim here is to capture the dynamics of the SABR parameters in a time series model to obtain more accurate swaption risk estimates. Hence, this time series model is used to simulate the one-day-ahead profit and loss distribution and is then compared to the Historical Simulation method. In an empirical study, we compute the Value at Risk and Expected Shortfall estimates based on the Historical Simulation method as well as the time series model. These models are analyzed with several backtests and diagnostic tests to be able to answer the following research question. Can one outperform the Historical Simulation Value at Risk and Expected Shortfall forecasts by fitting a time series model to the calibrated SABR model parameters instead? A vector autoregressive model is used as well as a local level model. Based on these two models we are not able to outdo the Historical Simulation estimates of the risk measures. Diagnostic tests show remaining significant autocorrelation as well as heterogeneity in the residuals of the vector autoregressive model. Also the backtests that are carried out show that the vector autoregressive model performs worse than the Historical Simulation method. Frank de Zwart Abn Amro Model Validation Contents 1 Introduction 1 2 Preliminaries on financial notation2 2.1 Interest rate instruments.....................................2 2.2 Bootstrapping the zero curve..................................3 2.3 Swaptions.............................................6 2.4 Martingales and Measures....................................7 3 Literature review 9 4 Models and method 12 4.1 Option pricing models...................................... 12 4.2 Time series analysis....................................... 16 4.3 Risk measurement........................................ 18 4.4 Backtests............................................. 20 5 Data 26 5.1 Calculating the implied volatilities............................... 27 5.2 Leaving out of some strikes................................... 28 6 Empirical study and results 29 6.1 Calibrating the SABR model parameters............................ 29 6.2 Fitting a model through the SABR parameters time series.................. 32 6.3 Risk measurement........................................ 35 6.4 Backtests............................................. 38 6.5 Robustness check: Local level model.............................. 42 7 Conclusion 43 References 45 A Appendix 47 Frank de Zwart Abn Amro Model Validation 1 Introduction The Basel Committee(2013) has introduced the Fundamental Review of the Trading Book (FRTB). To contribute to a more resilient banking sector, they have decided to change the current framework’s reliance on Value at Risk (VaR) to the Expected Shortfall (ES) measure to estimate market risk. On the other hand Pérignon and Smith(2010) state that most banks use Historical Simulation (HS) to estimate their VaR. This Historical Simulation method computes the VaR by using past returns of the portfolio’s present assets so that one obtains a distribution of price changes that would have realized, had the current portfolio been held throughout the observation period. The decision described in the FRTB shows that it is getting even more important for financial institutions to estimate their market risk accurately. However, we also see that a relatively simple method is still used to obtain these risk measures. One of the main drawbacks of this Historical Simulation method is that it does not take the decreasing predictability of older returns into account. Derivatives are traded extensively these days, and one of these products is a swap option, or swaption. A swaption is an option on an interest rate swap. Swaptions are traded over-the-counter, so compared to derivatives that are traded on an exchange, the information is more scarce and not publicly available. This makes it an interesting challenge to find an accurate method to assess the risk of holding these derivatives. Besides this, the negative interest rates also affect almost all the valuation methods for these options. In the current interest rate environment, the Historical Simulation method that is used to produce the VaR and ES estimates of market risk may not be reliable. Hence, finding a method to get more reliable estimates for the VaR and ES, based on historical swaption data, is of interest. This leads to the following research question, which defines the main purpose of this thesis. Can one outperform the Historical Simulation Value at Risk and Expected Shortfall forecasts by fitting a time series model to the calibrated SABR model parameters instead? An empirical study is performed to be able to answer this question. This study is based on an ICAP data set of swaption premiums, interest rate deposits, and interest rate swaps. The time series, with a time grid of approximately 2.5 years, of the displaced SABR volatility model parameters will be analyzed to obtain an one-day-ahead forecast of the price of a portfolio of swaptions. Finally, a backtesting procedure will assess the quality of this new method compared to the well-known Historical Simulation method. The remainder of this research report is structured as follows. First in Section2, the necessary background theory for this research will be discussed. This includes theory on interest rate instruments in general as well as a description of an interpolation method called bootstrapping to obtain the zero and discount curves. We will then give a description of a swaption and some of its relevant trading strategies are discussed. This section will be concluded with the a description of martingales and measures. Then we will briefly discuss the relevant literature for this research in Section3 and subsequently we will continue in Section4 with the theory that is used to price the swaptions. In this section, the well-known model of Black(1976) is described in detail. Besides this, we will focus on the SABR volatility model of Hagan et al.(2002) and the correction to their work from Obłój(2008). We will then discuss the implications of negative interest rates on these models. In Section 4.2, some basic time series models that are used in this research will be discussed. We will then continue with the risk measurement concepts, like Value at Risk and Expected Shortfall. Finally, several different backtests will be elaborated. Different backtests are used to be able to assess the quality of our model estimates as thoroughly as possible. The data set will be described in Section5. Not only the raw data will be described, but also the different pre-processing techniques will be explained. This section also contains information on some of the limitations and 1 Frank de Zwart Abn Amro Model Validation argumentation on why some adjustments are made. In the next section, Section6, the empirical study and results are described. This section follows the structure of Section4 and starts with the calibrated SABR parameters and continues with the time series analysis, risk measurement, and concludes with the backtesting procedure. There are also some diagnostic tests carried out, besides the backtests themselves, to assess the quality of the fit of the time series analysis. The results will be elaborated and discussed in every single step. Finally in Section7, the main findings are summarized and a conclusion is drawn. The research question will be answered and some limitations and recommendations for further research will be provided. 2 Preliminaries on financial notation Trading in derivatives has become an indispensable part of the financial industry. There are multiple different derivatives for every type of investment asset. The magnitude of this market shows that it is of great importance to have an understanding of how these derivatives work. Consequently a lot of researchers have focused on all these derivatives. Numerous papers and books describe how derivatives work and what risks the holder of an open position in them is taking. We will first explain some basic but crucial concepts of interest rate instruments. Then in Section4, we will describe the models and methods that are applied in the empirical analysis of this research. 2.1 Interest rate instruments Interest rates are crucial in the valuation of derivatives. Especially the ’risk-free’ rate is of concern when evaluating derivatives. Hull(2012) explains that the interest rates implied by Treasury bills are artificially
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