Lévy Models - Computational Methods for Quantitative Finance - page 142

Fig. 10.4 Loss of the smooth pasting condition, CGMY model with μ ≈− 0 . 0018, C = 0 . 5,

G = 5, M = 5 . 4, Y = 0 . 1, r = 0 . 13 ( left ) and CGMY model with μ ≈ 0 . 0478, C = 1, G = 6,

M = 5 . 4, Y = 0 . 1, r = 0 . 01 ( right )and h = 0 . 01 in both cases

For American style contracts on underlyings whose log-returns are described
by pure jump Lévy models, the analytic behavior of the American put price and
the free exercise boundary can be considerably more involved than in the Black-
Scholes model. For prices of American style contracts on underlyings modeled by
pure jump Lévy processes, it is shown in [111, Theorem 4.2] that the smooth pasting
principle fails for admissible market models if the model is of finite variation and
μ = (e z 0. The smooth pasting principle holds for admissible

market models (in particular, for pure jump models) of infinite variation and for pure
jump Lévy models of finite variation which satisfy

− 1 ) + ν( d z) − r ≤

R

μ =
R

(e z

r> 0,
see [111, Theorem 4.1] and Fig. 10.4 . For an admissible market model, the free
boundary t

− 1 )ν( d z) −

s ∗ (t) satisfies s ∗ (t) > 0for t 0 ,T) and is a continuous mapping

[110, Proposition 4.1 and Theorem 4.2]. We get the following result:

→ ∈[

s ∗ (t) lim = K for μ ≤ 0 ,

t → T

s ∗ (t) K ∗ lim = for μ> 0 ,

t → T

where K ∗ <K . Note that μ> 0 holds for all infinite variation market models, and
hence there is an instant jump of the exercise boundary from the strike price K to
K ∗ . We refer to [110, Theorem 4.4] for a proof. The early exercise of an American
put if the interest rate is zero is not optimal, i.e. for the free boundary we have
s ∗ (T ) → 0for r → 0[110, Remark 4.7].