Multi-asset Options - Computational Methods for Quantitative Finance - page 95

8.3 Localization

d

The unbounded domain R of the log-price is truncated to a bounded domain G =

( − R,R) d

d , R> 0. This again corresponds to approximating the option price
by a knock-out barrier option

⊂ R

e − r(T − t) g(e X T ) 1 { T<τ G } |

x ,

v R (t, x) = E X t = (8.11)

(X 1 ,...,X d ) is now a d -dimensional Brownian motion.
We show that the barrier option price v R again converges to the option price expo-
nentially fast in R .

d

where x ∈ R and X =

d

Theorem 8.3.1 Suppose the payoff function g : R → R ≥ 0 satisfies ( 8.10 ). Then ,

Q

there exist C(d,T, ) , γ 1 ,γ 2 > 0, such that

| v(t,x) − v R (t, x) |≤ C(d,T, Q )e − γ 1 R + γ 2 x ∞ .

Proof Let M T = sup τ ∈[ t,T ] X τ ∞ . Then, with ( 8.10 )

|≤E g(e X T ) 1 x | v(t,x) − v R (t, x)

τ G } |

X t =

{ T ≥

E e qM T 1 { M T >R } | x .

≤ C(d) X t =

It suffices to show that there exist a constant C(d,T, Q )> 0 such that

E e q X T ∞ 1 x ≤ )e − γ 1 R + γ 2 x ∞ .

} |

X t = C(d,T, Q

{ X T ∞ >R

We have, following the proof of Theorem 4.3.1,
E e q X T ∞ 1 { X T ∞ >R } | x =

e q z + x ∞ 1 { z + x ∞ >R } p T − t (z) d z X t =

d

R

d

e (q + Q − 1 μ ∞ ) | z i | 1 { z + x ∞ >R } e − z Q − 1 z/( 2 (T − t)) d z )e q x ∞ ≤ C(d,T, Q

d

R

i = 1

≤ C(d,T, Q )e q x ∞

d

e − (η − q − Q − 1 μ ∞ )(R − x ∞ ) e η | z i | e − z Q − 1 z/( 2 (T − t)) d z

×

d

R

i = 1

d

e η | z i | e − z i /( 2 σ i (T − t)) d z i ,

)e − γ 1 R + γ 2 x ∞ ≤ C(d,T, Q

R

i = 1

, and γ 2 = γ 1 + q . Since e η | z i | e − z i /( 2 σ i (T − t)) d z i < with γ 1 = η − q − Q − 1 μ ∞

R

∞ 1 ,...,d , and any η> 0, we obtain the required result by choosing

η>q + Q − 1 μ ∞

for all i =

.

The price v R of t he barrier option is, under the smoothness assumption v R ∈
C 1 , 2 (J

d )

C 0 (J

d ) , and after switching to time-to-maturity t

× R ∪ × R → T − t ,

a solution of the PDE