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

and
∑K ν
ν(t) = (ν k cos (2πkt) + νk ∗ sin (2πkt)) (8)
k=1
where K α and K ν determine the number of terms in the sums and α 0 , α k , αk ∗, k = 1, . . . , Kα ,
and ν k , νk ∗, k = 1, . . . , Kν are constant parameters to be estimated. The specific functional
form of the deterministic seasonal components were originally suggested by Hannan, Terrell,
and Tuckwell (1970) as an alternative to standard dummy variable methods in econometric
modeling of seasonality. Due to the continuous time feature of our analysis this is a flexible
and natural choice of modeling the seasonal aspects of commodity price behavior which is also
applied in Sørensen (2001).
2.2 Futures prices
Following Cox, Ingersoll and Ross (1981), the futures price at time t on a futures contract
for delivery at time T ≥ t can be obtained by taking the relevant expectations of the future
spot price, F t (T ) = E Q t [P T ]. Equivalently, by using the Feynman-Kac formula, F t (T ) =
F (P t , δ t , v t , t; T ) where F (·) can be found as the solution to a partial differential equation on
the form
1
2 e2ν(t) vP 2 ∂2 F
+ 1 ∂P 2 2 σ2 δ e2ν(t) v ∂2 F
∂δ 2
+ ρ 23 σ δ σ v e ν(t) v ∂2 F
∂F
∂δ∂v
+ (r − δ)P
∂P
+ 1 2 σ2 vv ∂2 F
∂v 2
+ ρ 12 σ δ e 2ν(t) vP ∂2 F
∂P ∂δ + ρ 13σ v e ν(t) vP ∂2 F
∂P ∂v
+ (α(t) − βδ)
∂F
∂δ
+ (θ − κv)
∂F
∂v + ∂F
∂t
= 0
(9)
with terminal condition F (P, δ, v, T ; T ) = P .
For a given and fixed expiration date T , the solution to the partial differential equation in
(9) is given by
F (P, δ, v, t; T ) = P e A(t;T )+B(t;T )v+D(t;T )δ (10)
where
D(t; T ) = − 1 β
(
1 − e −β(T −t)) (11)
and A(t; T ) and B(t; T ) are the solutions to the ordinary differential equations
1
2 σ2 δ e2ν(t) (D(t; T )) 2 + ρ 12 σ δ e 2ν(t) D(t; T ) + 1 2 σ2 v (B(t; T )) 2
+ (ρ 13 σ v e ν(t) + ρ 23 σ δ σ v e ν(t) D(t; T ) − κ)B(t; T ) + B ′ (t; T ) = 0 , B(T ; T ) = 0
(12)
and
r + α(t)D(t; T ) + θB(t; T ) + A ′ (t; T ) = 0 , A(T ; T ) = 0 (13)
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