Stochastic Volatility and Seasonality in ... - Interconti, Limited

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and the longest available futures contract are observed without measurement error. Hence, it is not surprising that short and long contracts are seemingly fitted better by the model than medium maturity contracts, as reflected in the higher standard deviations on the imposed measurement errors for medium maturity contracts. Note that the standard deviations on the individual error terms on futures prices are estimated of the same numerical magnitude as in Schwartz (1997), Schwartz and Smith (2000), and Sørensen (2001) who use only commodity futures data and Kalman filtering estimation approaches. For example, Sørensen (2001) provides results for soybean futures where the measurement errors are assumed uncorrelated and with identical standard deviation which he estimates to be 0.187. As apparent from the correlation matrix in Table 3, especially measurement errors for contracts that have maturities close to each other are highly positively correlated while the correlations decrease systematically for contracts having increasingly dispersed maturities. Finally, the estimated correlations indicate slight co-variation between option prices and futures prices measurement errors. In particular, the measurement errors related to short maturity options have higher correlations with the measurement errors related to short maturity futures while the medium/long maturity option measurement errors are more correlated with measurement errors on futures prices of similar maturities. The correlations are in general positive, however, no clear pattern is evident since the measurement error related to short maturity in-the-money call options is negatively correlated with the measurement error on futures prices of similar maturity while the short maturity out-of-the-money call option measurement error at the same time is positively correlated with the same futures price measurement error. Structural changes In order to provide insights into whether the basic dynamic commodity price and contingent claims pricing model is robust over time, we have estimated the model for two equally long non-overlapping sub-periods. The relevant sub-samples thus cover the periods: October 1984 to November 1991 and November 1991 to March 1999. Rather than purely quantitative econometric considerations with respect to diagnostic checking of the estimated model, this investigation is especially motivated by qualitative considerations such as: Does seasonal patterns change systematical over time? And, are correlation structures qualitatively robust over time? Seasonal patterns may vary over time due to especially production and storage innovations 24
as well as changes in government policies with respect to intervention in grain markets. For example, Frechette (1997) analyzes the “flattering” of the soybean futures profile up through the 1970s where the Brazilian soybean production increased rapidly and changed the seasonal pattern in world soybean production. However, such dramatic changes in world soybean production patterns have not occurred in the time periods relevant for our investigation. Also, although US government policies prevailing in the 1970s and 1980s provided a floor under grain prices, soybeans had a relatively low support level and, thus, also no dramatic changes in government intervention policies with respect to soybeans seem to have occurred in the relevant estimation periods. In sum, it is not obvious on beforehand what kind of parameter instability one would expect to find due to changes in soybean markets conditions when splitting the data into sub-samples. The sub-period estimation results are tabulated in Table 4 and Table 5, respectively. [ INSERT TABLE 4 AND TABLE 5 ABOUT HERE ] We will be brief in our discussion of the differences between the estimates in the two subperiods and will mainly point out and focus on differences in the persistence of shocks, seasonal patterns, and correlations of relevance to the above full-sample discussion of the relation to the theory of storage and the “inverse leverage effect.” It may first be noted that volatility shocks and convenience yield shocks seemingly are more persistent in the first sub-period than the second sub-period, as reflected in the estimates of the mean-reversion parameters κ and β. Thus, the estimates of κ indicate that, under the risk-neutral measure, the half-lives of shocks to volatility equal 0.940 years in the first subperiod and 0.160 years in the second sub-period (while the full-sample analog is 0.305 years, as discussed earlier). While this is true under the risk-neutral measure, as reflected in options and futures prices, the relevant mean-reversion parameter under the true probability measure is given by κ − λ v σ v . Hence, under the true probability measure the persistence of shocks to the volatility is much more similar across the two sub-periods, and relevant half-lives are 0.141 years and 0.093 years, respectively, as opposed to 0.137 years for the full-sample estimation results. In order to see more clearly the estimated seasonality patterns in volatilities across the two sub-periods, we have graphed the relevant seasonal components in Figure 4. [ INSERT FIGURE 4 ABOUT HERE ] 25
Page 1 and 2: Stochastic Volatility and Seasonali
Page 3 and 4: 1 Introduction This paper sets up a
Page 5 and 6: The estimated commodity price model
Page 7 and 8: where β, κ, θ, λ P , λ δ , λ
Page 9 and 10: Solutions to the ordinary different
Page 11 and 12: The measurement equation relates th
Page 13 and 14: and, N ˆΓ ∼ W m (N, Γ) where W
Page 15 and 16: where T s and T l refer to the rele
Page 17 and 18: prices seems lower than that of sho
Page 19 and 20: [ INSERT FIGURE 2 ABOUT HERE ] As s
Page 21 and 22: where adverse weather conditions ma
Page 23 and 24: Inverse leverage effects The correl
Page 25: (with the relevant estimated standa
Page 29 and 30: Besides estimating model parameters
Page 31 and 32: Chambers, M. & Bailey, R. (1996).
Page 33 and 34: Kaldor, N. (1939). “Speculation a
Page 35 and 36: Appendix Lemma 1 Let p t = log P t
Page 37 and 38: Table 2: Summary statistics for soy
Page 39 and 40: Table 4: Parameter estimates for so
Page 41 and 42: 1100.00 Futures prices in cents per
Page 43 and 44: 1.00 0.75 0.50 0.25 0.00 - 0.25 α(
Page 45: 2.00 1.50 1.00 0.50 0.00 - 0.50 α

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