Sensitivities and Greeks - Computational Methods for Quantitative Finance - page 149

Example 11.2.2 (Black-Scholes model) We consider a one-dimensional diffusion
process X with the infinitesimal generator

1
2 σ 2 ∂ x f(x).
For the sensitivity of the price with respect to the volatility σ , the set of admissible
parameters

1
2 σ 2 ∂ xx f(x) −

BS f )(x) =

( A

S η is S η = R + with η = σ .Wehave

( A V ∗ ),

(δσ )f )(x) = δσσ 0 ∂ xx f(x) − δσσ 0 ∂ x f(x) ∈ L ( V ,

with δσ ∈ R = C . The Dirichlet form a(δσ ;· , · ) appearing in the weak formulation

( 11.8 )of u(δσ ) is given by

; =

δσσ 0 (∂ x ϕ,∂ x φ)

+ δσσ 0 (∂ x ϕ,φ).

a(δσ ϕ,φ)

In this setting, V = V = H 0 (G) .

Example 11.2.3 (Tempered stable model) We consider a one-dimensional pure
jump process X with the tempered stable density k as in (10.10) and infinitesimal
generator

J f )(x)

zf (x))k(z) d z. ( A = (f (x + z) − f(x) −

R

For the sensitivity of the price with respect to the jump intensity parameter α of the
Lévy process X ,wehave

S η = =

( 0 , 2 ) with η α and

δα ( A zf (x))k(z) d z ( V , V ∗ ),

(δα)f )(x) = (f (x + z) − f(x) − ∈ L

R

where the kernel k is given by

k(z)

:= − ln | z | k(z).

It is easy to check that ,

|

k(z) d z<

> 1 k(z) d z< 1 z 2

∞ ∞

. In this setting,

| z |≤ z |

V = H α/ 2 + ε (G)

⊂ H α/ 2 (G) = V , ε> 0.

For the discretization of ( 11.7 ), ( 11.8 ), we can use either finite differences or
finite elements. Here, we consider the finite element method and obtain the matrix
form of ( 11.8 )

u m + 1

N

Find ∈ R such that for m = 0 ,...,M − 1 ,

u m + 1 u m t A (θ u m + 1 θ) u m ), ( M + θt A ) = ( M − ( 1 − θ)t A ) − + ( 1 −

u 0 = 0 ,

A is matrix of the Dirichlet form A ij

where a(δη ;· , · ) in the basis of V N , =

N , and u m + 1 , m a(δη ; b j ,b i ) for 1 ≤ i, j ≤ = 0 ,...,M − 1, is the coefficient vector