Price movements in many commodity markets exhibit significant seasonal patterns. However, given an observed futures price, a deterministic seasonal component at the price level is not relevant for the pricing of commodity options. In contrast, this is not true for the seasonal pattern observed in the volatility of the commodity price. Analyzing an extensive sample of soybean, corn, heating oil and natural gas options, we find that seasonality in volatility is an important aspect to consider when valuing these contracts. The inclusion of an appropriate seasonality adjustment significantly reduces pricing errors in these markets and yields more improvement in valuation accuracy than increasing the number of stochastic factors.

This article investigates four models for minimizing the tracking error between the returns of a portfolio and a benchmark. Due to linear performance fees of fund managers, we can argue that linear deviations give a more accurate description of the investors’ risk attitude than squared deviations. All models have in common that absolute deviations are minimized instead of squared deviations as is the case for traditional optimization models. Linear programs are formulated to derive explicit solutions. The models are applied to a portfolio containing six national stock market indexes (USA, Japan, UK, Germany, France, Switzerland) and the tracking error with respect to the MSCI (Morgan Stanley Capital International Index) world stock market index is minimized. The results are compared to those of a quadratic tracking error optimization technique. The portfolio weights of the optimized portfolio and its risk/return properties are different across the models which implies that optimization models should be targeted to the specific investment objective. Finally, it is shown that linear tracking error optimization is equivalent to expected utility maximization and lower partial moment minimization.

Many commodity markets contain a strong seasonal component not only at the price level, but also in volatility. In this paper, the importance of seasonal behavior in the volatility for the pricing of commodity options is analyzed. We propose a seasonally varying long-run mean variance process that is capable of capturing empirically observed patterns. Semi-closed-form option valuation formulas are derived. We then empirically study the impact of the proposed Seasonal Stochastic Volatility Model on the pricing accuracy of natural gas futures options traded at the New York Mercantile Exchange (NYMEX) and corn futures options traded at the Chicago Board of Trade (CBOT). Our results demonstrate that allowing stochastic volatility to fluctuate seasonally significantly reduces pricing errors for these contracts.