The Mathematics of Arbitrage

162 9 Fundamental Theorem of Asset Pricing

A={lim inft→∞(H·S)t<β<γ<lim supt→∞(H·S)t}. Since the Boolean
algebra

⋃

0 ≤tFtis dense in theσ-algebraF we have that there ist^1 and
A 1 ∈Ft 1 such thatP[A A 1 ]<ε 1 .Forω=A 1 we putU 1 =V 1 =t 1 and we
concentrate onω∈A 1.
First define

U 1 ′=inf{t|t≥t 1 and (H·S)t<β}forωinA 1

V 1 ′=inf{t|t≥U 1 ′and (H·S)t>γ}forωinA 1.

The variablesU 1 ′ andV 1 ′are clearly stopping times and take values in
[0,∞]. By construction ofA 1 we have that

P[V 1 ′<∞]≥P[A∩A 1 ]>α−ε 1.

Takes 1 >t 1 so thatP[V 1 ′≤s 1 ]>α−ε 1 and define

U 1 =min(U 1 ′,s 1 ),

V 1 =min(V 1 ′,s 1 ).

The setB 1 ={(H·S)U 1 ≤β<γ≤(H·S)V 1 }is inFs 1 andP[B 1 ∩A]>

α−ε 1 .PutK^1 =H (^1) ]]U 1 ,V 1 ]]. We claim thatK^1 is (1 +β)-admissible. Indeed
onAc 1 clearly (K^1 ·S)t=0forallt.Forω∈A 1 andt≤U 1 we also have
(K^1 ·S)t(ω)=0.Forω∈A 1 andU 1 <t≤V 1 we have
(K^1 ·S)t=(H·S)t−(H·S)U 1 ≥− 1 −β=−(1 +β).
Let us putL^1 =K^1. We now apply the same reasoning on the set (B 1 ∩A)
i.e. we taket 2 ≥s 1 ,A 2 ∈Ft 2 such thatA 2 ⊂B 1 ,P[A 2 (B 1 ∩A)]>α−ε 1 −ε 2.
On the setA 2 we define
U 2 ′=inf{t|t≥t 2 and (H·S)t<β}
V 2 ′=inf{t|t≥U 2 ′and (H·S)t>γ}.
P[V 2 ′<∞]>α−ε 1 −ε 2 and we selects 2 >t 2 so thatP[V 2 ′≤s 2 ]>
α−ε 1 −ε 2 .Take
U 2 =min(U 2 ′,s 2 )
V 2 =min(V 2 ′,s 2 )
K^2 =H (^1) ]]U 2 ,V 2 ]].
The integrand is (1 +β)-admissible, but outside the setB 1 the process
(K^2 ·S) is zero. On the setB 1 , however, (L^1 ·S)t 2 =(L^1 ·S)s 2 ≥γ−β>0. The
integrandL^2 =L^1 +K^2 remains therefore (1 +β)-admissible. Furthermore
P[(L^2 ·S)t 2 ≥2(γ−β)]>α−ε 1 −ε 2. This permits us to continue the
construction and to defineLnby induction.

The rest of this section is devoted to some results giving a better under-
standing of the property(NFLVR)of no free lunch with vanishing risk and
relating this property to previous results of [D 92, S 94].