Lévy Models - Computational Methods for Quantitative Finance - page 134

10.5 Localization

As in the Black-Scholes case, the unbounded range R of the log price x = log s

is truncated to a bounded computational domain G = ( − R,R) , R> 0. Let τ G =

G c be the first hitting time of the complement set G c inf { t ≥ 0 | X t ∈ } = R\ G by X .

Note that for diffusion models, the paths of X are continuous and therefore
inf G c { t ≥ 0 | X t ∈ }= inf { t ≥ 0 | X t =± R } .

This does not hold for jump models where the process can jump out of the domain
G instantaneously. As before we denote the value of the knock-out barrier option in
log-price by v R .

Theorem 10.5.1 Suppose the payoff function g : R → R ≥ 0 satisfies (4.10). Let X be

a Lévy process with characteristic triplet (σ 2 ,ν, 0 ) where the Lévy measure satisfies
( 10.11 ) with β + ,β − >q and q as in (4.10). Then , there exist C(T,σ,ν) , γ 1 ,γ 2 > 0,
such that

| v(t,x) − v R (t, x) |≤ C(T,σ,ν)e − γ 1 R + γ 2 | x | .

Proof Let M T = sup τ ∈[ t,T ] | X τ | . Then, with (4.10)

x .
Using [143, Theorem 25.18], similar to the proof of Theorem 4.3.1, it suffices to
show that

|≤E g(e X T ) 1

x ≤

E e qM T 1 | −

τ G } |

X t =

} |

X t =

v(t,x) v R (t, x) C

{ T ≥ { M T >R

E e q | X T | 1

} | X t = x =

e q | z + x | 1 p T − t (z) d z

{| X T | >R {| z + x | >R }

R

e q | x | e − (η − q)(R −| x | ) e η | z | p T − t (z) d z

≤ e − γ 1 R + γ 2 | x |

≤

R

e η | z | p T − t (z) d z,

R

with γ 1 = η − q and γ 2 = γ 1 + q .Using[143, Theorem 25.3], we obtain the equiv-

alence

e η | z | p T − t (z) d z< e η | z | ν( d z) < ∞⇔ ∞ ,

R | z | > 1

which follows from ( 10.11 ).

We define the space
H s (G)

H s ( := { u | G : u ∈ R ), u | R\ G = 0 } .

, the space H s (G) coincides with H 0 (G) , the closure of C 0 (G)
with respect to the norm of H s (G) . For any function u with support in G , we denote
by

For s + 1 / 2 ∈ N

u its extension by zero to all of R . Then, the bilinear form on the bounded domain

G is given by a J R (u, v) a J (

= u, v) , and we have continuity and a Gårding inequality

H ρ (G) . Now we can restate the problem ( 10.21 ) on the bounded domain:

on