- Discrete Time Market Models 83
Proof
This is an immediate consequence of Exercise 3.18 and Proposition 4.2.
4.1.4 Fundamental Theorem of Asset Pricing
In this section, which can be omitted on first reading, we return to the general
setting under Assumptions 4.1 to 4.5.
We already know that the discounted stock prices in the binomial tree model
form a martingale under the risk-neutral probability, see Proposition 3.5 and
Corollary 3.6. The following result extends these observations to any discrete
model.
Theorem 4.4 (Fundamental Theorem of Asset Pricing)
The No-Arbitrage Principle is equivalent to the existence of a probabilityP∗
on the set of scenariosΩsuch thatP∗(ω)>0 for each scenarioω∈Ωand the
discounted stock pricesS ̃j(n)=Sj(n)/A(n) satisfy
E∗(S ̃j(n+1)|S(n)) =S ̃j(n) (4.3)for anyj =1,...,mandn=0, 1 , 2 ,...,whereE∗(·|S(n)) denotes the
conditional expectation with respect to probabilityP∗computed once the stock
priceS(n) becomes known at timen.
The proof of the Fundamental Theorem of Asset Pricing is quite technical
and will be omitted.
Definition 4.5
A sequence of random variablesX(0),X(1),X(2),...such that
E∗(X(n+1)|S(n)) =X(n)for eachn=0, 1 , 2 ,...is said to be amartingalewith respect toP∗.
Condition (4.3) can be expressed by saying that the discounted stock prices
S ̃j(0),S ̃j(1),S ̃j(2),...form a martingale with respect toP∗. The latter is called
arisk-neutralormartingale probabilityon the set of scenariosΩ.Moreover,E∗
is called arisk-neutralormartingale expectation.