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

Hence, using the arguments and approach in Heston (1993), pp. 330-331 and his Appendix,
one obtains the following option pricing formula on the form of Black (1976),
C(F, v, t) = e −r(τ−t) [F P 1 − KP 2 ] (17)
where (see equation (18) in Heston (1993))
P j = 1 2 + 1 ∫ [
]
∞
e −iφ ln[K] f j (x, v, t; φ)
Re
dφ , j = 1, 2 (18)
π
iφ
0
and where x = ln F and f j (·; φ), j = 1, 2, denote characteristic functions that are obtained
as solutions to PDE’s with terminal condition e iφx . Since coefficients in the particular PDE’s
are affine and the terminal conditions are exponential-affine, the solutions for f j , j = 1, 2, are
exponential-affine and on the form
f j (x, v, t; φ) = e a j(t)+b j (t)v+iφx , j = 1, 2 (19)
where a j (t) and b j (t) are solutions to the ordinary differential equations
b ′ j (t) = ( 1
2 φ2 − u j iφ ) σ 2 F (t; T ) + (k j(t) − iφσ F v (t; T )) b j (t) − 1 2 σ2 v (b j (t)) 2 , b j (τ) = 0 (20)
and
a ′ j(t) = −θb j (t) , a j (τ) = 0 (21)
with u 1 = 1 2 , u 2 = − 1 2 , k 1(t) = κ − σ F v (t; T ), and k 2 (t) = κ.
2.4 The estimation approach
The estimation approach is based on formulating the model in state space form so that the
model is represented by a measurement equation, and a transition equation. In our case
the state space model is non-linear and the specific approach relies on a general state space
estimation approach suggested originally by Chen and Scott (1993) for estimation of models
of the term structure of interest rates.
The data are in the following observed at equidistant time points, t n , n = 0, . . . , N, and
∆ = t n+1 − t n . The transition equation describes the dynamics of the basic unobserved
state-variables. In our case, there are three state-variables which can be summarized by the
three-dimensional state-vector X n = (p tn , δ tn , v tn ) where p t = log P t . The transition equation
is given by the discrete-time solution to the stochastic differential system in (1), (2), and (3)
between the observation time points.
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