14-Arbitrage Page 405 Wednesday, February 4, 2004 1:08 PM
Arbitrage Pricing: Finite-State Models 405
P({}ω ∩Akt) P({}ω )
P({}ω Akt)= ------------------------------------ = -------------------, if ω ∈Akt , 0 if ω∉Akt
PA( kt) PA( kt)
Given that the probability space is finite,
PA( jt)= ∑ pω
ω ∈Ajt
As we defined P({ω}) ≡pωthe previous equation becomes
P({}ω ∩Akt) P({}ω ) pω
P({}ω Akt )= ------------------------------------ = ------------------- = ----------------------------
PA( kt) PA( kt)
∑ pω
ω ∈Akt
if ω ∈Akt , 0 if ω ∉Akt.
Pricing Relationships
We can now write the pricing relationship as follows:
T i
ω
i 1
SAkt = ---------- ∑ P({}ω Akt) ∑ πj()ωdj()
πAkt ω ∈A j = t + 1
kt
1 pω T
= ---------- ∑ ---------------------------- ∑ πj()di ω
π ω ∈A
kt
∑
ω j()
j = t + 1
Akt pω
ω ∈A
kt
Akt ∈It , 1 ≤k ≤Mt
The above formulas generalize to any trading strategy. In particular,
if there is a state-price deflator, the market value of any trading strategy
is given by
θθθθ = -----^1 E
t ×St
πt
πjdj θ
j t += 1
T