Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 223

Fig. 14.10 Barrier option
price in d = 1( top )and d = 2
with barrier B = 80 and strike
K = 100

E (g(x + X T )) − E (g(x + Y 1 ,T )) ≤ C 1 ε 2 − max { α 1 ,...,α d } ,

d , 0 <α j < 2 ,

∀ x ∈ R

E

Y 2 ,T )) ≤ C 2 ε 3 − max { α 1 ,...,α d } ,

X T ))

d , 0 <α j < 1 .

(g(x + − E (g(x + ∀ x ∈ R

C 4 (

d ) and

These convergence rates can also be shown numerically even for g/ ∈ R

α> 1.

Example 14.7.6 Let d = 1 and consider the tempered stable density (10.10),

c e − β + | z |
|

c e − β − | z |
|

k(z) = 1

} +

1 .

{ z> 0 { z< 0 }

1 + α 1 + α

| |

z z

We compute the price of a put option with maturity T = 0 . 5, strike K = 100 and

1, β − = 10, β + = 15 and compute for Y 1 , Y 2 interest rate r = 0 . 01. We set c =