The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 415 Wednesday, February 4, 2004 1:08 PM

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Arbitrage Pricing: Finite-State Models 415

d

i

St

i Q

= Et j

T

∑

j = t + 1 Rtj,

We can rewrite this equation explicitly as follows:

S Q

t

i = E

t

= EQ

t

dj i

Rtj,

• j t += 1

T

∑ Et

Q dt + 1

i

Rtt , + 1

• 1
  Rtt , + 1


• 
  

dj i

j t += 2 Rt + 1 ,j

T

= + ∑

dt i + 1

Rtt , + 1

• 
  

Et Q + 1

Rtt , + 1

• 
  

dj i

Rtj,

• j t += 2

T

+ ∑ Et

Q dt + 1

i S

t + 1

+ i

Rtt , + 1

=

which shows that the discounted price plus payoff process is a martin-

gale. The terms on the left are the price processes, the terms on the right

are the conditional expectations under the probability measure Q of the

payoffs discounted with the risk-free payoff.

The measure Q is a mathematical construct. The important point is

that this new probability measure can be computed either from the real

probabilities if the state-price deflators are known or directly from the

price and payoff processes. This last observation illustrates that the con-

cept of arbitrage depends only on the structure of the price and payoff

processes and not on the actual probabilities. As we will see later in this

chapter, equivalent martingale measures greatly simplify the computa-

tion of the pricing of derivatives.

Let’s assume that there is short-term risk-free borrowing in the sense

that there is a trading strategy able to pay for any given interval (t,s) one

sure dollar at time s given that (dtdt + 1...ds – 1)–1 has been invested at

time t. Equivalently, we can define for any time interval (t,s) the payoff

of a dollar invested risk-free at time t as Rt,s = (dtdt + 1...ds – 1).

We now define the concept of equivalent probability measures.

Given a probability measure P the probability measure Q is said to be

equivalent to P if both assign probability zero to the same events. An

equivalent probability measure Q is an equivalent martingale measure if

all price processes discounted with Ri,j become martingales. More pre-

cisely, Q is an equivalent martingale measure if and only if the market

value of any trading strategy is a martingale:

j

θ

Rtj,

• 
  

T

∑

d

θt× St = EQt

j = t + 1