The Mathematics of Financial Modelingand Investment Management

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

∑