Pricing of Long-Dated Commodity Derivatives with Stochastic Volatility and Stochastic Interest Rates

Abstract

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.

Empirical Pricing Performance on Long-Dated Crude Oil Derivatives: Do Models with Stochastic Interest Rates Matter?

“…By comparing the pricing errors of the two models (deterministic interest rates against stochastic interest rates), we find that the RMSEs from the stochastic interest rates counterpart are lower than the deterministic interest rates counterpart, an effect that holds as the maturity of the crude oil futures options increases. These results are more pronounced during the period of relatively high volatility of the interest rates, consistent with the numerical findings in Cheng et al (2015). During the period of very low interest rates and interest rate volatility, there are no noticeable differences between the stochastic and deterministic interest rate models.…”

“…This section presents a brief description of the HJM term structure models used in the empirical analysis. The proposed models are both stochastic volatility models but one model features stochastic interest rates (see Cheng et al (2015) for more details) and the other model is restricted to deterministic interest rate specifications.…”

“…Note that i 2 = −1 and ℑ( + i ) = . The logarithm of the futures prices at time with maturity can be expressed in terms of 6 state variables, namely ( ), ( ), ( ), ( ), ( ) and ( ) (see Proposition 1.2 in Cheng et al (2015)):…”

“…Motivated by the limited literature dedicated to empirical research on pricing long-dated commodity derivatives by incorporating stochastic interest rates as well as the increasing importance of long-dated commodity derivative contracts to the financial markets, we consider an empirical investigation of the impact of stochastic interest rates to pricing models of long-dated crude oil derivative contracts. We employ two multi-factor stochastic volatility Heath et al (1992) type models, one model with deterministic interest rate specifications and one model with stochastic interest rates as proposed in Cheng, Nikitopoulos, and Schlögl (2015). The models feature a full correlation structure and fit the entire initial forward curve by construction rather than generating it endogenously from the spot price as required in the spot price models.…”

“…To assess the impact of stochastic interest rates under different market conditions, we consider two subsamples; the period 2005 − 2007 that was characterised by relatively volatile interest rates and the period 2011 −2012 that exhibited very low interest rates (consequently a very stable interest rate market). According to the numerical analysis performed in Cheng et al (2015), the volatility of the interest rates itself, rather than the level of the interest rates plays important role to the pricing of long-dated commodity derivatives. By comparing the pricing errors of the two models (deterministic interest rates against stochastic interest rates), we find that the RMSEs from the stochastic interest rates counterpart are lower than the deterministic interest rates counterpart, an effect that holds as the maturity of the crude oil futures options increases.…”