Negative Rates in a Multi Curve Framework - Cap Pricing and Volatility Transformation

The SABR model has for a long time been an invaluable tool for capturing the volatility smile and to price nancial derivatives not quoted in the market. However, the current negative rate environment in the EUR market has led to numerous challenges for nancial institutions. One of the most problematic issues is that the SABR model used for volatility interpolation and extrapolation fails when rates are negative. A related issue is that SABR based techniques for transformation of cap volatilities between dierent tenors, do not work anymore. This thesis describes how to use an extension of the SABR model, known as the shifted SABR, to solve these issues. Using three dierent methods, the shifted SABR model is calibrated to EUR cap volatilities based on 6 month EURIBOR. However, market standard is to quote a mixture of 3 month and 6 month cap volatilities, thus the volatilities have to be transformed to a common tenor before calibration. To this end we have developed a technique for volatility transformation in a multi curve framework when rates are negative. The concept is derived by applying It^o's formula on an arbitrage relation between shifted forward rates. The results show that two of the methods for calibrating the shifted SABR model have a good t to liquid contracts. However, the methods vary in performance capturing far OTM caps. Our developed volatility transformation technique also works well, and the sensitivity to the potentially unknown correlation between forward rates is low. The implications are that the extension of the SABR model to the shifted SABR model works ne, both in terms of pricing of caps and volatility transformation in a multi curve framework. Although, it comes to the cost of some additional complexity of extending formulas and arbitrage relations, that used to hold in a positive rate environment.
Popular Science Summary Negative Rates when dealing with Caps Negative rates have become a more common view on the leading markets. This creates problems for nancial institutions, as some of their existing models break down. We have analyzed three methods, used for calibration to caps, and developed a new technique that deals with some of the problems related to negative rates. In June 2014 the European Central Bank (ECB) did something extremely unusual seen from a historical perspective: they applied negative deposit rates. For nancial institutions and investors this was an entirely new situation. Historically, negative rates have been regarded as impossible. The economic intuition behind this is that negative rates actual means you have to pay for lending money. The motivation for negative interest rates diers depending on the national bank. The main reason in the Eurozone and Sweden is to ght the growing threat of de ation. Policymakers want to make it less attractive to save money. As such lowering the interest rate on savings should potentially make it more attractive to invest the money instead. Financial institutions use statistical models in dierent areas such as pricing and risk validation. An example is the SABR model which for a long time has been an invaluable tool for capturing the volatility smile and to price cap contracts not quoted in the market. However, the current negative rate environment in the European market has led to numerous challenges for nancial institutions. One of the most problematic issues is that the SABR model used for volatility interpolation and extrapolation fails when rates are negative. A related issue is that SABR based techniques for transformation of cap volatilities, do not work anymore. Our work1 focuses on the pricing of cap contracts available on the European market. Caps are contracts based on a set of simpler contracts known as caplets. Each caplet provides the holder an insurance guaranteeing that the rate connected