Construct Smith-Wilson risk-free interest rate curves with endogenous and positive ultimate forward rates

We propose several methods for obtaining endogenous and positive ultimate forward rates (UFRs) for risk-free interest rate curves based on the Smith-Wilson method. The Smith-Wilson method, which is adopted by Solvency II, can both interpolate the market price data and extrapolate to the UFR. However, the method requires an exogenously-chosen UFR. To obtain an endogenous UFR, de Kort and Vellekoop (2016) proposed an optimization framework based on the Smith-Wilson method. In this paper, we prove the existence of an optimal endogenous UFR to their optimization problem under the condition that the cash flow matrix is square and invertible. In addition, to ensure the positivity of the optimal endogenous UFR during extreme time periods such as the COVID-19 pandemic, we extend their optimization framework by including non-negative constraints. Furthermore, we also propose a new optimization framework that can not only generate endogenous and positive UFRs but also incorporate practitioners' prior knowledge. We prove the feasibility of our frameworks, and conduct empirical studies for both the Chinese government bonds and the EURIBOR swaps to illustrate the capabilities of our methods.