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

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Appendix Lemma 1 Let p t = log P t . The conditional means and conditional variances of the process X t = (p t , δ t , v t ), described by the stochastic differential equation system (1), (2), and (3), are given by where ( E t [X t+∆ ] = Φ(t + ∆) X t + ∫ t+∆ t ) Φ −1 (u)l(u) du (∫ t+∆ ) Var t [X t+∆ ] = Φ(t + ∆) E t [v u ] Φ −1 (u)Σ(u)Φ −1 (u) ′ du Φ(t + ∆) ′ (31) t ⎛ ⎞ ⎛ ⎞ r e ν(u) ρ 12 σ δ e ν(u) ρ 13 σ v e 1 2 ν(u) l(u) = ⎜ α(u) ⎟ ⎝ ⎠ , Σ(u) = ⎜ ρ ⎝ 12 σ δ e ν(u) σδ 2eν(u) ρ 23 σ v σ δ e 1 2 ν(u) ⎟ ⎠ θ ρ 13 σ v e 1 2 ν(u) ρ 23 σ v σ δ e 1 2 ν(u) σv 2 and where Φ is the unique solution to the matrix differential equation with dΦ(u) du (30) = L(u)Φ(u) , Φ(t) = I (32) ⎛ ⎞ 0 −1 − 1 2 + λ P e 1 2 ν(u) L(u) = ⎜ 0 −β λ ⎝ δ σ δ e 1 2 ν(u) ⎟ ⎠ . 0 0 −κ + λ v σ v In particular, the conditional expectations entering the right hand side integral of (31) are described by (30) and given by E t [v u ] = v t e −κ(u−t) + θ κ (1 − e−κ(u−t) ). Proof: The dynamics of X t = (p t , δ t , v t ) can be written in matrix form as dX t = (L(t)X t + l(t)) dt + √ v t σ(t) dW t (33) where σ(t) = diag[e 1 2 ν(t) , σ δ e 1 2 ν(t) , σ v ] is a diagonal matrix with the three arguments in the diagonal and zeros off the diagonal. By using Ito’s lemma, and the definition of Φ in (32), it is seen that X can be characterized as the solution to (s ≥ t) ( ∫ s ∫ s ) X s = Φ(s) X t + Φ −1 √ (u)l(u) du + vu Φ −1 (u)σ(u) dW u . (34) Note that ∫ s t t t √ vu Φ −1 (u)σ(u) dW u , s ≥ t, is a martingale starting at zero at time t. Hence, by taking conditional expectations in (34) (and setting s = t + ∆), one obtains (30). Furthermore, since the two first terms on the right hand side of (34) are F t -measurable, the result in (31) follows by an application of e.g. Karatzas and Shreve (1991), Proposition 2.17, p. 144. 33
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 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: Kaldor, N. (1939). “Speculation a
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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