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

Abstract: 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.