Lévy Models - Computational Methods for Quantitative Finance - page 141

Fig. 10.3 Option prices for
the VG model ( top )andthe
exercise boundary ( bottom )

Find u m + 1

N

∈ R such that for m = 0 ,...,M − 1 ,

B u m + 1 F m , ≥

u m + 1 g m ,

( u m + 1 ) B u m + 1

≥

F m =

− 0 .

Example 10.7.1 As in Example 10.6.3 we use the variance gamma model with pa-
rameters σ 100, we compute

the price of a European and American put option and compare these with the corre-
sponding Black-Scholes prices. It can be seen in Fig. 10.3 that the smooth pasting
condition (see Remark 5.1.2) does not hold for Lévy models. We also observe that in
contrast to the Black-Scholes model the exercise boundary values in a Lévy model
never reach the option's strike price, i.e. s ∗ (t) < K , as proven in [110, Theorem 4.4].

= 0 . 3, ϑ = 0 . 25 and θ =− 0 . 3. For T = 1 and K =