The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 414 Wednesday, February 4, 2004 1:08 PM

414 The Mathematics of Financial Modeling and Investment Management

0.25 × 38 × π 2 () 1 +0.25 × 115 × π 2 () 2 – 69.12 ×0.5 × πA 11 , = 0

0.25 × 42 × π 2 () 1 +0.25 × 130 × π 2 () 2 – 75.27 ×0.5 × πA 11 , = 0

( 55 ×0.5 +0.5 × 15 ) × πA 11 , +(70.25 ×0.5 +0.5 × 20 ) × πA 21 ,

• 67.125 × πA 10 , = 0

It can be verified that this system, obviously, is solvable and returns the

same state-price deflators as in the previous example.

Equivalent Martingale Measures

We now introduce the concept and properties of equivalent martingale

measures. This concept has become fundamental for the technology of

derivative pricing. The idea of equivalent martingale measures is the fol-

lowing. Recall from Chapter 6 that a martingale is a process Xt such

that at any time t its conditional expectation at time s, s > t coincides

with its present value: Xt = Et[Xs]. In discrete time, a martingale is a

process such that its value at any time is equal to its conditional expec-

tation one step ahead. In our case, this principle can be expressed in a

different but equivalent way by stating that prices are the discounted

expected values of future payoffs. The law of iterated expectation then

implies that price plus payoff processes are martingales.

In fact, assume that we can write

T

St =Et ∑ dj

j =t + 1

then the following relationship holds:

T T

St =Et ∑ dj =Et dt + 1 +Et + 1 ∑ dj =Et[dt + 1 +St + 1 ]

j =t + 1 j =t ++ 11

Given a probability space, price processes are not, in general, martin-

gales. However it can be demonstrated that, in the absence of arbitrage,

there is an artificial probability measure in which all price processes,

appropriately discounted, become martingales. More precisely, we will see

that in the absence of arbitrage there is an artificial probability measure Q

in which the following discounted present value relationship holds: