European Options in BS Markets - Computational Methods for Quantitative Finance - page 52

To show coercivity, note that with
R 2

ϕ ϕ d x

1

(ϕ 2 ) d x

= =

0wehave

R

1
2 σ 2

1
2 σ 2

a BS (ϕ, ϕ) ϕ

2
L 2 (

2
L 2 (

2
H 1 (

σ 2 / 2 )

2
L 2 (

=

) +

r ϕ

) =

ϕ

) +

(r − ϕ

R R R R )

1
2 σ 2

2
H 1 (

σ 2 / 2

2
L 2 (

≥ ϕ

) −|

r − | ϕ

) . R R

H 1 ( Referring to the abstract existence result Theorem 3.2.2 in the spaces V = R )

L 2 ( and ) , we deduce that the variational problem ( 4.8 ) admits a unique

weak solution u

H = R

L 2 (J H 1 ( H 1 (J L 2 ( L 2 (

∈ ; R ∩ ; R

)) for every u 0 ∈

R

) . Since
u 0 (x) = g(e x ) , u 0 ∈ L 2 ( R ) implies an unrealistic growth condition on the pay-
off g . In the next section, we reformulate the problem on a bounded domain where
this condition can be weakened. In particular, we require the following polynomial
growth condition on the payoff function: There exist C> 0, q

))

≥ 1 such that

1 ) q , g(s) ≤ C(s + for all s ∈ R + . (4.10)

This condition is satisfied by the payoff function of all standard contracts like, e.g.
plain vanilla European call, put or power options.

4.3 Localization

R =

The unbounded log-price domain log s is truncated to a

bounded domain G . In terms of financial modeling, this corresponds to approxi-
mating the option price by a knock-out barrier option. Let G = ( − R,R) , R> 0be
an open subset and let τ G :=

of the log price x

G c inf { t ≥ 0 | X t ∈ } be the first hitting time of the

complement set G c G by X . Then, the price of a knock-out barrier option in

log-price with payoff g(e x ) is given by

= R\

e − r(T − t) g(e X T ) 1

T<τ G } | X t = x .

v R (t, x) = E (4.11)

{

We show that the barrier option price v R converges to the option price

e − r(T − t) g(e X T )

x ,

v(t,x) = E | X t =

exponentially fast in R .

Theorem 4.3.1 Suppose the payoff function g : R → R ≥ 0 satisfies ( 4.10 ). Then ,

there exist C(T,σ),γ 1 ,γ 2 > 0, such that
| v(t,x) − v R (t, x) | ≤ C(T,σ)e − γ 1 R + γ 2 | x | .

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

| ≤ E g(e X T ) 1 { T ≥ τ G } |

x ≤ E e qM T 1 { M T >R } | x . |

v(t,x) − v R (t, x) X t = C X t =