We address two empirical issues related to the long end of the yield curve based on euro swap rates. First, for maturities longer than 20 years we find evidence for an `excess' downward slope that cannot be explained by convexity. Second, volatility at the very long end of the yield curve is...

We extrapolate interest rate yield curves for the purpose of discounting very long-dated pension liabilities and insurance contracts. The extrapolation uses a no-arbitrage term structure model estimated on liquid euro swap instruments with maturities between 5 and 20 years. The extrapolation...

In this paper we derive a market value for Guaranteed Annuity Optionusing martingale modeling techniques. Furthermore, we show how to construct a static replicating portfolio of vanillainterest rate swaptions that replicates the Guaranteed Annuity Option. Finally, we illustrate with historical...

In this paper we derive a market value for Guaranteed Annuity Optionusing martingale modeling techniques. Furthermore, we show how to construct a static replicating portfolio of vanillainterest rate swaptions that replicates the Guaranteed Annuity Option. Finally, we illustrate with historical...

This technical note gives implementation notes for estimating the Koijen-Nijman-Werker model from historical data based on a Kalman filter. We provide an independent derivation of the KNW model. We propose a different implementation of the state-space formulation of the KNW model and we test the...