Abstract

AbstractCalibrating Short Rate Models in Negative Interest Rate Environments: An Application to Bermudan SwaptionsThe evolution of the interest rate has been modelled for a long time by a lognormal distribution. Similar to stock prices, it was generally agreed upon that the lognormal distribution provides the benefit of interest rates not falling below zero. However, governmental interventions after the 2008 financial crisis forced interest rates of several currencies to fall below zero. This event opened up a new area of research in mathematical finance. The contemporary negative interest rates can no longer be modelled by the classical lognormal distribution. Rather, distributions that allow for negative values, such as the normal distribution or the shifted lognormal distribution, are being tested. In order to test for model differences in the years between 2016 and 2020, JPY swaption market implied volatilities based on a lognormal and normal distribution are collected. The moment when the JPY interest rate adopts a negative value, implied volatilities for the lognormal distribution go missing. For the normal distribution, all implied volatilities are returned. This observation confirms the failure of the Black formula in a negative interest rate environment. Two well-known one-factor short rate models, the Hull-White Extended Vasicek model and the Black-Karasinski model, are calibrated to market data of JPY swaptions. The OIS rate is chosen because it has proven to be superior to the LIBOR in the pricing of financial derivatives. The Hull-White Extended Vasicek model follows a normal distribution and can therefore accomodate negative values. The Black-Karasinski model follows a lognormal distribution and can therefore only accomodate non-negative values. The interest rate plays a critical role in the pricing of any derivative. The calibrated short-rate models are used to price 1Yx10Y Bermudan style JPY swaptions. The results indicate that calibration with Bachelier implied volatilities is more accurate compared to using Black implied volatilities.