Stochastic Volatility Models - Computational Methods for Quantitative Finance - page 120

Fig. 9.2 Option price ( top )
and convergence rate
( bottom ) for the Heston
model

is the infinitesimal generator of the process Z and G Y

where A denotes the state

space of Y . Then, V(T − t,z) = v(t,z) .

We perform the same transformations as described in Sect. 9.3 and obtain the
following system of inequalities for w v 0 )e − η := (v − in the Heston model

H

f κ ∂ t w − A

κ w

+ rw ≥ in J × R × R + ,

w(t,x) ≥ 0i J × R × R + ,

(9.35)

H

(∂ t w − A

κ w + rw)w =

0i J × R × R + ,

w( 0 ,x,y) = 0in R × R + .

The set of admissible solutions in the Heston model for the variational form of ( 9.35 )
is the convex set K 0 given as

K 0 := v

∈ R × R + ,

∈ V | v ≥ 0a.e. z