In the current LIBOR transition to overnight–rate benchmarks, it is important to understand theoretically and empirically what distinguishes actual term rates from overnight benchmarks or 'synthetic' term rates based on such benchmarks. The well–known 'multi–curve' phenomenon of tenor basis spreads between term structures associated with different payment frequencies provides key information on this distinction. This information can be extracted using a modelling framework based on the concept of 'roll–over risk', i.e., the risk a borrower faces of not being able to refinance a loan at (or at a known spread to) a market benchmark rate. Separating the roll–over risk priced by tenor basis spreads into a credit–downgrade and a funding–liquidity component, the theoretical modelling and the empirical evidence show that proper term rates based on the new benchmarks remain elusive and that a multi–curve environment will persist even for rates secured by repurchase agreements.

The Secured Overnight Funding Rate (SOFR) has become the risk‐free rate benchmark in US dollars, thus term structure models should reflect key features exhibited by SOFR and forward rates implied by SOFR futures. We construct a multifactor, stochastic volatility term structure model which incorporates these features. Calibrating to options on SOFR futures, we achieve a reasonable fit to the market across maturities and strikes in a single model. This also provides novel insights into SOFR term rate behavior (and implied volatilities) within their accrual periods, and a model mechanism by which interest rate mean reversion arises from monetary policy.

A joint model of commodity price and interest rate risk is constructed analogously to the multi-currency LIBOR Market Model (LMM). Going beyond a simple ‘re-interpretation’ of the multi-currency LMM, issues arising in the application of the model to actual commodity market data are specifically addressed. Firstly, liquid market prices are only available for options on commodity futures, rather than forwards, thus the difference between forward and futures prices must be explicitly taken into account in the calibration. Secondly, we construct a procedure to achieve a consistent fit of the model to market data for interest options, commodity options and historically estimated correlations between interest rates and commodity prices. We illustrate the model by an application to real market data and derive pricing formulas for commodity spread options.

The present paper analyses a broad range of one- and multifactor models of the term structure of interest rates. We assess the influence of the number of factors, mean reversion, and the factor probability distributions on the term structure shapes the models generate, and use spread options as an aggregate measure of the relative importance assigned to rising and falling forward rate curves by the models considered. We derive valuation formulas for these contingent claims in the multifactor Gaussian and CIR- models. Our main result is that the specification of mean reversion and the number of factors are both much more important for the relative movements of interest rates than the distributional characteristics of the factors. To the extent that interest rate risk depends on the movements of different parts of the term structure relative to one another rather than on shifts of its absolute level, the distributional assumption on the factor dynamics is found to be essentially irrelevant.