American Options - Computational Methods for Quantitative Finance - page 67

5.2 Variational Formulation

The set of admissible solutions for the variational form of ( 5.2 ) is the convex set
K g ⊂

H 1 (

R )

H 1 ( K g := { v ∈ R ) : v ≥ g a . e .x } . (5.3)

The variational form of ( 5.2 ) reads:

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

∈ ; R ∩ ; R · ∈ K g and

Find u )) )) such that u(t, )

(∂ t u, v − u) + a BS (u, v − u) ≥ 0 , ∀ v ∈ K g , a.e. in J, (5.4)

u( 0 ) = g.

Since the bilinear form a BS (

· , · ) is continuous and satisfies a Gårding inequality in

H 1 ( R ) by Proposition 4.2.1, problem ( 5.4 ) admits a unique solution for every payoff

L ∞ ( g ) by Theorem B.2.2 of the Appendix B.

As in the case of plain European vanilla contracts, we localize ( 5.4 ) to a bounded
domain G

∈ R

= ( − R,R) , R> 0, by approximating the value v in ( 5.1 )

e − r(τ − t) g(e X τ )

x ,

v(t,x) := sup

∈ T t,T E

| X t =

τ

by the value of a barrier option

e − r(τ − t) g(e X τ ) 1 { τ<τ G } |

x . v R (t, x) =

τ ∈ T t,T E

sup X t = (5.5)

Repeating the arguments in the proof of Theorem 4.3.1, we obtain the estimate for
the localization error: there exist constants C(T,σ),γ 1 ,γ 2 > 0 such that

C(T,σ)e − γ 1 R + γ 2 | x | , | v(t,x) − v R (t, x) | ≤

valid for payoff functions g : R → R satisfying the growth condition (4.10). Thus,

we consider problem ( 5.1 ) restricted on G

BS v R +

∂ t v R − A rv R ≥

×

0 in J G,

g(e x )

v R (t, x) ≥ | G ×

in J G,

BS v R +

(∂ t v R − A rv R )(g | G − v R ) = 0 in J × G, (5.6)

g(e x ) v R ( 0 ,x) = | G in G,

g(e ± R ) v R (t, ± R) = in J.

We cast the truncated problem into variational form. To get a simple convex set for
admissible functions and to facilitate the numerical solution, we introduce the time
value of the option (also called excess to payoff ) w R := v R − g | G and consider

H 0 (G) K 0 ,R := { v ∈ : v ≥ 0a . e .x ∈ G } .