Functional extrapolation is a modelling framework for yield-curve extrapolation that draws on basic techniques from functional analysis. It is designed to meet the requirements of extrapolation methods that are actually used in practice, while also offering clear direction for risk management. Notably, we show how to fully decouple the choice of a particular extrapolation model from the implications for downstream risk management tasks such as valuation and hedging. This allows for much clearer analysis and comparisons across different extrapolation methodologies. We derive formulae for stress testing, scenario analysis, and immunisation; including static duration–convexity hedging and key rate duration. We discuss recent extrapolation proposals from the EIOPA and particular examples in the case of long-dated liabilities.
Virtually all methods for yield-curve extrapolation demand that the long forward rate—or infinite-maturity yield—be set as an exogenous parameter. We survey and study model-based extrapolation and empirical approaches to estimate the long forward rate based on historical data. Using a two-factor affine term-structure model and a Kalman filtering approach, we attempt maximum-likelihood estimation of the long forward rate, along with other model parameters. In a comprehensive study, we document the weak identifiablity non-robustness of such estimation methods both to the choice of cross section or sample period. We discuss implications for policy and proposals from the EIOPA.
Interest-rate benchmark reform has revived short-rate modelling. One reason is that short-rate models provide a consistent framework in which different benchmarks, and contracts linked to them, can be compared. Another reason is that new benchmarks can be directly dependent on very short-term rates; the key example is a backward-looking compounding of overnight rates, a prominent alternative to forward-looking Libor. Indeed, under Libor, one can often safely ignore aspects of short-rate behaviour, especially jumps. At least partially for this reason, jumps are inadequately treated in the interest-rate literature, particularly expected jumps (jumps with known timing). We estimate a model with expected and unexpected jumps, which involves separating their effect on term rates. We then price forward- and backward-looking caplets, quantifying the spread exhibited by the latter over the former. Expected jumps lead to significantly time-inhomogeneous option behaviour, particularly for short-term options linked to a backward-looking benchmark.
We propose a heteroscedastic generalisation of the (homoscedastic) regression-based term-structure model of Adrian et al. (2013). Interest rates are subject to various shocks, such as monetary policy announcements, which contribute additional volatility at specific cross sections, resulting in an uneven distribution of risk premia over time; for instance, jumps in the short rate. Bond prices, in the absence of arbitrage, are risk-adjusted expectations of future short rates, and are also subject to this heteroscedasticity. Our generalised model allows volatility to vary with time, capturing heteroscedasticity whilst maintaining the near-instantaneous speed of a regression-based approach. We implement a three-factor specification, using the first three principal components of UK bond yields. Further, we show both theoretically and empirically, that the homoscedastic model consistently underestimates the market price of risk when the data are heteroscedastic.