The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 417 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 417

Y = dQ

---------

dP

It is clear from the definition that P and Q are equivalent probabil-

ity measures as they assign probability zero to the same events. Note

that in the case of a finite-state probability space the new probability

measure is defined on each state and is equal to

qω = Y ()ωpω

Suppose πt is a state-price deflator. Let Q be the probability measure

defined by the Radon-Nikodym derivative:

πTR 0 ,T

ξT = -------------------

π 0

The new state probabilities under Q are the following:

πT ()ω R 0 ,T

qω = ----------------------------pω

π 0 () ω

Define the density process ξt for Q as ξt= Et[ξT]. As ξt = Et[ξT] is an

adapted process, we can write:

pω ()R 0 ,T

ξT A

kt

ω ∈AktPAkt)

(Et[]) = ξAkt = ∑ -----------------ξT ()ω = ∑ ----------------- ----------------------------

pω πT ω

( ω ∈A ( ()

kt

PAkt) π 0 ω

πA

kt

R 0 ,t 1 pω

= --------------------- ---------- ∑ -----------------πT[π 0 ()ω]RtT, = ---------------------

πAktR 0 ,t

π 0 ()ω πAkt ω ∈A (

kt

PAkt) π 0

As Rt,s = (dtdt + 1...ds – 1) is the payoff at time s of one dollar invested in

a risk-free asset at time t, s > t, we can then write the following equations:

1

1 = -----Et[πsRts, ]

πt

Therefore,