Exotic Options - Computational Methods for Quantitative Finance - page 80

Applying Theorem 4.1.4, we can obtain the value V 1 of the underlying option by
solving the partial differential equation

∂ t V 1 + A V 1 − rV 1 = 0in ( 0 ,T 1 ) × R + ,V 1 (T 1 ,s) = g 1 (s) in R + ,

where denotes the Black-Scholes generator (4.5). In a second step, we get the

price of the compound option by solving

A

∂ t V + A V − rV = 0in ( 0 ,T) × R + ,V T, ) = g(V 1 (T , s)) in R + .

As in the European plain vanilla case, we change to log-price x and localize the
problem to a bounded domain G R,R) . The corresponding value of the barrier

option for the underlying option is denoted with v 1 ,R ,i.e.

= ( −

e − r(T 1 − t) g 1 (e X T 1 ) 1

x ,

v 1 ,R (t, x) = E

T 1 <τ G } |

X t = (6.7)

{

and the value of the barrier option for the compound option by v R ,i.e.

e − r(T − t) g(v 1 ,R (T , X T )) 1

x . v R (t, x) = E

} |

X t = (6.8)

{ T<τ G

Again, the compound barrier option price v R converges to the compound option
price v exponentially fast as R →∞ in a Black-Scholes model.

Theorem 6.3.1 Suppose the payoff functions g,g 1 : R → R of a compound contract

in a Black-Scholes market satisfy (4.10) and that g is Lipschitz continuous . Then ,
there exist C(T,T 1 ,σ),γ 1 ,γ 2 > 0, such that

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

Proof Applying Theorem 4.3.1 to the underlying option value v 1 , we obtain
v 1 (t, x)

v 1 ,R (t, x) ≤ C 1 (T 1 ,σ)e − γ 3 R + γ 4 | x | , −

for some C 1 (T 1 ,σ),γ 3 ,γ 4 > 0. If we denote by v R the barrier compound option

on v 1 ,

e − r(T − t) g(v 1 (T , X T )) 1 { T<τ G } |

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

we have with Lipschitz constant C 2

C 2 E v 1 (T , X T ) v 1 ,R (T , X T ) 1 { T ≥ τ G } | x | v R (t, x) − v R (t, x) | ≤ − X t =

e γ 4 | X T | 1 { T<τ G } |

x C 1 C 2 (T 1 ,σ)e − γ 3 R

≤ E X t =

C 3 (T , T 1 ,σ)e − γ 3 R + γ 4 | x | .

≤