14-Arbitrage Page 419 Wednesday, February 4, 2004 1:08 PM
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Arbitrage Pricing: Finite-State Models 419
Q ()R 0 ,T
(Et [])Akt = ∑
qω
ω
pω πT ω
xj -------------------xj()= ∑ ------------------- ----------------------------xj() ω
ω ∈AktQA(^ kt) ω ∈AktQA( kt) π^0 () ω
(^1) , pωπT ω ω
= ------------------- ∑ ∑
R 0 ,jRjT ()xj()
QA( kt)A π 0 ()ω
hj ⊂Akt ω ∈Ahj
1 xAhjR^0 ,j
= ------------------- ∑ --------------------- ∑ RjT, pωπT ()ω
( π 0 ()ω ω ∈ (^) A
hj
QAkt)A
hj ⊂Akt
1 xAhjR^0 ,jπAhjPA( hj)
= ------------------- ∑ -------------------------------------------------
QA( (^) kt)A π 0 ()ω
hj ⊂Akt
1
= ------------------- ∑ [xAhj ξAhjPA( hj)]
QA( kt)A
hj ⊂Akt
1 xAhj ξAhjPA( hj)^1 P
= ---------- ∑ --------------------------------------= ----------[Et (ξjxj )Akt]
ξAkt A PA( kt) ξAkt
hj ⊂Akt
Let’s now apply the above result to the relationship:
T
1 π 0 ,
T π
jRtjdj
i
Sti = -----Et ∑ πjdji = ------Et ∑ -------------- ---------
πt j = t + 1 πt j = t + 1 π 0 Rtj,
π 0 T πjR 0 ,j dji Qdji
= ---------------Et ∑ --------------- --------- = Et ---------
πtR 0 ,j j = t + 1 π 0 Rtj, Rtj,
We have thus demonstrated the following results: There is no arbitrage
if and only if there is an equivalent martingale measure. In addition, πt
is a state-price deflator if and only if an equivalent martingale measure
Q has the density process defined by
πtR 0 ,t
ξt = ----------------
π 0
In addition, it can be demonstrated that, if there is no arbitrage,
markets are complete if and only if there is a unique equivalent martin-
gale measure.