# Seasonality in Practice

The seasonal behavior of commodity prices has been well-documented, but the literature is relatively sparse on the impact of seasonal volatility on the prices of commodity options. Back, Prokopczuk, and Rudolf (2010)* address this lacuna by testing two pricing models calibrated to be sensitive to the seasonality of volatility. They analyze the soybean and heating oil markets, and find that including the seasonality of volatility substantially improves the performance of option pricing models.

It is intuitive that the volatility of some commodities should be related to seasonal factors. The authors provide one example:

"A good example is provided by most agricultural markets, where the harvesting cycles determine the supply of goods. Shortly before the harvest, the price uncertainty is higher than after the harvest when crop yields are known to the market participants, resulting in a seasonal pattern in volatility in addition to the price level seasonality. (3)"

Soybeans are one such supply-driven product--their seasonal patterns are determined by factors like the perishable nature of the product, changes in the weather, and harvesting cycles. Heating oil and other energy products are, by contrast, more vulnerable to changes in demand. By analyzing markets with such different seasonal pressures, the authors reduce the chances that the inclusion of seasonal volatility in an options pricing model will merely reflect idiosyncratic fundamental factors. Figures 1 and 2, from the paper, show the seasonality of soybean and heating oil futures prices and historical volatility, respectively, from roughly 1990-2006.**

Soybean futures seasonal volatility. Source: Back et al.

Heating oil futures seasonal volatility. Source: Back et al.

** Improved Pricing **

Two models are developed to assess the value of seasonality as a pricing factor. The first model assumes a deterministic long-term equilibrium price and includes a factor for seasonal volatility; the second model keeps the seasonal volatility factor, but adds a second factor allowing for uncertainty about the long-term equilibrium price. The authors concede that other pricing models will include jumps, stochastic volatility, regime switching, etc., but exclude these considerations in order to focus on the impact of seasonal volatility alone.

The results are promising, both in- and out-of-sample. The consistency of the price improvement achieved by factoring in seasonal volatility is particularly notable:

"Incorporating seasonal volatility reduces the pricing error in every instance, i.e. for both markets, both models, in-sample and out- of-sample, for every maturity bracket, and for every moneyness category at a 1% significance level. ... The overall pricing errors of the one- and two-factor models are reduced by 10.26% and 12.47% for the soybean options, and by 18.37% and 11.95% for the heating oil options in the out-of-sample test, respectively. The greatest improvements are observed for short term heating oil contracts, with a maximal improvement of 37.96% for the ATM options and the one-factor model. (20)"***

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The most surprising outcome was that in some cases the one-factor model adjusting for seasonality alone offered lower pricing errors than even the two-factor model: "thus, allowing for seasonally varying volatility seems to be more important than adding additional stochastic factors." (18) Other research has arrived at similar conclusions after analyzing seasonal effects in the volatility of options on wheat futures.**** Given this data, it is hard to imagine a justification for trading commodity options without accounting for seasonal volatility.

Seasonality in Practice

In addition to using the seasonal volatility of commodities to modify theoretical option prices in general, traders could use observed implied volatility seasonality as the justification for a trade. If, for example, implied volatility in the options for some agricultural commodity tends to rise in advance of reports of crop yields (with attendant volatility in the underlying occurring immediately thereafter), a trader might enter long vega positions before the bid-up in implied volatility and hedge directional exposure to remove unwanted price risk.

The commodity traders I know tend, as a group, to at least be cognizant of prevailing seasonality when structuring trades, even if their pricing models don't make explicit allowances for it. Sometimes, academic research provides confirming evidence for theories that are already well-established among market practitioners. This may be just such a case.

** Back, Janis, Prokopczuk, Marcel and Rudolf, Markus, Seasonality and the Valuation of Commodity Options (March 09, 2010). Available at SSRN: http://ssrn.com/ abstract=1514803 *

* ** A good resource for similar charts plotting the seasonal tendencies of various stock indexes and commodities is http://www. seasonalcharts.com/index.html. *

* *** The percent changes quoted are in reference to the root mean squared pricing error, not percent revisions in the option contract price. *

* **** Koekebakker, Steen and Lien, Gudbrand, "Volatility and Price Jumps in Agricultural Futures Prices--Evidence from Wheat Options," American Journal of Agricultural Economics 86:4 (November 2004). *

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