Sensitivities and Greeks - Computational Methods for Quantitative Finance - page 147

We assume that a(η 0 ;·

· : V × V → R

satisfies continuity (3.8) and the Gårding
inequality (3.9) for all η 0 ∈ S η . In general, the space

, )

V

may depend on the parameter

V η 0 . For notational simplicity, we drop the subscript η 0 .For

η 0 , and we should write

L 2 (J

; V ∗ ) and u 0 ∈ H

f ∈ , the weak formulation of the problem ( 11.3 ) is given by:

L 2 (J H 1 (J Find u ∈ ; V ) ∩ ; H ) such that

a η 0 ; u, v = (∂ t u, v)

H +

f, v V ∗ , V , ∀ v ∈ V , (11.4)

u( 0 , · ) = u 0 .

Under the assumption that (3.8)-(3.9) hold for every model parameter η 0 ∈ S η ,the
problem ( 11.4 ) admits a unique solution. We assume that

V is a Sobolev-type space

with smoothness index r ,i.e.

V = H r (G), H = H 0 (G) L 2 (G).

= (11.5)

Note that r depends on the order of the operator A (η 0 ) . We also assume that the

solution u(η 0 ) to ( 11.4 ) has higher regularity in space, u(η 0 )(t) ∈ H s (G) ⊂ H r (G)
for t ∈ J , where

H s (G) is again a Sobolev-type space with smoothness index s .

11.2 Sensitivity Analysis

For a parametric Markovian market model X in the sense of Definition 11.1.1 ,we
distinguish two classes of sensitivities:

sδη , s> 0, of
an input parameter η 0 ∈ S η . Typical examples are the Greeks Vega ( ∂ σ u ), Rho
( ∂ r u ) and Vomma ( ∂ σσ u ). Other sensitivities which are not so commonly used
in the financial community are the sensitivity of the price with respect to the
jump intensity or the order of the process that models the underlying.
(ii) The sensitivity of the solution u to a variation of arguments t,x . Typical exam-
ples are the Greeks Theta ( ∂ t u ), Delta ( ∂ x u ) and Gamma ( ∂ xx u ).

(i) The sensitivity of the solution u to a variation S η η s := η 0 +

11.2.1 Sensitivity with Respect to Model Parameters

d .

Let C be a Banach space over a domain G ⊂ R C is the space of parameters or

coefficients in the operator is the set of admissible coefficients. We

denote by u(η 0 ) the unique solution to ( 11.4 ) and introduce the derivative of u(η 0 )
with respect to η 0 ∈ S η as the mapping D η 0 u(η 0 ) : C → V

A and S η ⊆ C

s u(η 0 +

u(η 0 ) ,δη

1

u(δη) := D η 0 u(η 0 )(δη) := lim sδη) − ∈ C .

0 + s →