The Mathematics of Financial Modelingand Investment Management

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 Qdji 

= ---------------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.