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

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Stochastic Volatility and Seasonality in Commodity Futures and Options: The Case of Soybeans Abstract We estimate a continuous-time stochastic volatility model using panel data of soybean futures and options on soybean futures. The model of commodity price dynamics is within the class of so-called affine asset pricing models, and option prices are determined using a standard inversion of characteristic functions approach. Our modeling acknowledges that commodities exhibit seasonality patterns in both spot price level and volatility and, hence, the involved commodity dynamics involve stochastic differential equations that are inhomogeneous in time. The estimation method is based on a state space formulation of the model and a quasi maximum likelihood approach. Estimation results are obtained based on weekly observations of soybean futures prices and options prices from the Chicago Board of Trade in the period October 1984 to March 1999. Our empirical results support the conceptual ideas in the theory of storage, though not the view that convenience yields are like timing options. JEL Classification: G13, C00
1 Introduction This paper sets up and estimates a continuous-time model of commodity price behavior using panel data of soybean futures prices and soybean option prices using data from the Chicago Board of Trade (CBOT) over the period 1984-1999. The approach is conceptually related to the analysis in Schwartz (1997), who sets up and estimates models of commodity futures prices using the Kalman-filter and a state-space approach. 1 However, the model in this paper includes additional realistic features since we, in particular, allow for stochastic volatility and seasonality in the theoretical modeling of commodity price behavior. These features are especially important for agricultural commodities with a harvesting pattern that induces a seasonal component in the commodity price as well as in the volatility of the commodity price. The seasonality feature in both commodity prices and volatilities is thus known to be one of the empirical characteristics that make commodities strikingly different from stocks, bonds, and other conventional financial assets; see e.g. the discussion in Routledge, Seppi and Spatt (2000). Furthermore, as an innovative feature, our panel data estimation approach is based on both futures prices as well as option prices across different exercise prices and maturities. The commodity price behavior is modeled in continuous-time as part of a system of stochastic differential equations within the class of asset pricing models where the drift and diffusion coefficients are affine functions of the relevant basic state-variables. In our framework there are three basic state-variables: the commodity spot price, the convenience yield, and the volatility of the spot price. Affine models are tractable for asset pricing purposes and, in our case, for the valuation of commodity futures contracts and options written on commodity futures. Due to the tractability of affine models, they are widely used for modeling of term structures of interest rates and pricing options on equity and foreign exchange rates. 2 In particular, it is in general possible to determine relevant conditional characteristic functions within affine models which in turn facilitates the evaluation of option prices through an inversion approach, building on the ideas suggested by Heston (1993) for a specific stochastic volatility model of 1 See also Schwartz and Smith (2000) and Sørensen (2001) for empirical analyzes of commodity price models following this approach. 2 See e.g. Duffie and Kan (1996) for a general specification of affine term structure models, Heston (1993) and Bakshi, Cao and Chen (1997) for affine models of equity returns, and Backus, Foresi and Telmer (1996) and Bates (1996) for affine models of foreign exchange rates. Duffie, Pan and Singleton (2000) provide a discussion of other applications as well as a general treatment of the affine class of models, with the possibility of jumps included. 1
Page 1: Stochastic Volatility and Seasonali
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 and 26: (with the relevant estimated standa
Page 27 and 28: as well as changes in government po
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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