Empirical study of methods to complete the swaption volatility cube from the caplet volatility surface

Abstract: Fixed income markets are vast markets, involving a large number of actors including financial institutions, state actors, asset managers and corporations. An import part of these markets are contracts written on the xIBOR rates. This report is concerned with the trying to provide prices for options written on these rates, in particular for swaptions that are not at-the-money (atm) utilizing prices in the cap market. Different methods have been suggested in the literature for solving this problem. In particular we study the method suggested by Hagan et al where one calibrates a SABR model to the caplet surface with the same expiry as the swaption. One then assumes that the swaption contract with the same expiry follows the same SABR dynamics as the caplet, but with a recalibrated initial volatility to fit the atm point. We also study the approach suggested by Rebonato and Jäckel. They derive a model for swaption prices based on the individual volatilities of the forward rates that the underlying interest rate swap consists of, as well as the correlation between the forward rates.  Both of these approaches are studied empirically for the STIBOR market. The data set span between 2016 and 2021 and consists of the yield curve, flat cap volatilities and swaption volatilities. We use the 1Y1Y and 5Y5Y swaption surfaces, where the prices are not only quoted atm, to verify our model. We conclude that despite the SABR model being able to fit the caplet prices well, the method suggested by Hagan does not capture the swaption smile. The Rebonato and Jäckel approach also falls short of capturing the smile and produces similar results as the Hagan et al method. This is suggested to be due to the Hagan method capturing the caplet smile well, and the constant correlation assumption made in this thesis.