Does modelling stochastic interest rates, beyond stochastic volatility, improve pricing performance on long-dated commodity derivatives? To answer this question, we consider futures price models for commodity derivatives that allow for stochastic volatility and stochastic interest rates and a correlation structure between the underlying variables. We examine the empirical pricing performance of these models on pricing long-dated crude oil derivatives.Estimating the model parameters from historical crude oil futures prices and option prices, we find that stochastic interest rate models improve pricing performance on long-dated crude oil derivatives, when the interest rate volatility is relatively high. Furthermore, increasing the model dimensionality does not tend to improve the pricing performance on long-dated crude oil option prices, but it matters for long-dated futures prices. We also find empirical evidence for a negative correlation between crude oil futures prices and interest rates that contributes to improving fit to long-dated crude oil option prices.

Aiming to study pricing of long-dated commodity derivatives, this paper presents a class of models within the Heath, Jarrow, and Morton (1992) framework for commodity futures prices that incorporates stochastic volatility and stochastic interest rate and allows a correlation structure between the futures price process, the futures volatility process and the interest rate process. The functional form of the futures price volatility is specified so that the model admits finite dimensional realisations and retains affine representations, henceforth quasi-analytical European futures option pricing formulae can be obtained. A sensitivity analysis reveals that the correlation between the interest rate process and the futures price process has noticeable impact on the prices of long-dated futures options, while the correlation between the interest rate process and the futures price volatility process does not impact option prices. Furthermore, when interest rates are negatively correlated with futures prices then option prices are more sensitive to the volatility of interest rates, an effect that is more pronounced with longer maturity options.

Does modelling stochastic interest rates beyond stochastic volatility improve pricing performance on long-dated crude oil derivatives? To answer this question, we examine the empirical pricing performance of two forward price models for commodity futures and options: a deterministic interest rate-stochastic volatility model and a stochastic interest rate-stochastic volatility model. Both models allow for a correlation structure between the futures price process, the futures volatility process and the interest rate process. By estimating the model parameters from historical crude oil futures prices and option prices, we find that stochastic interest rate models improve pricing performance on long-dated crude oil derivatives, with the effect being more pronounced when the interest rate volatility is relatively high. Several results relevant to practitioners have also emerged from our empirical investigations. With regards to balancing the trade-off between precision and computational effort, we find that two-factor models would provide good fit on long-dated derivative prices thus there is no need to add more factors. We also find empirical evidence for a negative correlation between crude oil futures prices and interest rates.

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This paper presents a simulation study of hedging long-dated futures options, in the Rabinovitch (1989) model which assumes correlated dynamics between spot asset prices and interest rates. Under this model and when the maturity of the hedging instruments match the maturity of the option, forward contracts and futures contracts can hedge both the market risk and the interest rate risk of the options positions. When the hedge is rolled forward with shorter maturity hedging instruments, then bond contracts are additionally required to hedge the interest rate risk. This requirement becomes more pronounced for longer maturity contracts and amplifies as the interest rate volatility increases. Factor hedging ratios are also considered, which are suited for multi-dimensional models, and their numerical efficiency is validated.