The Mathematics of Arbitrage

3.3 The Binomial and the Trinomial Model 45

H∈H}, the constant function 1, and (f^1 ,...,fk) linearly spanL∞(Ω,F,P).
Define thekprocesses

Sdt+j=EQ̂(y)[fj|Ft],j=1,...,k, t=0,...,T. (3.33)

Now extend theRd+1-valued processS=(S^0 ,S^1 ,...,Sd)totheRd+k+1-
valued processS=(S^0 ,S^1 ,...,Sd,Sd+1,...,Sd+k) by adding these new co-

ordinates. By (3.33) we still have thatSis a martingale underQ̂(y), which, by
our choice of (f^1 ,...,fk) and Corollary 2.2.8, is now the unique probability
under whichSis a martingale, by our choice of (f^1 ,...,fk) and Corollary
2.2.8.
Hence we find ourselves in the situation of Theorem 3.1.3. By comparing
(3.20) and (3.29) we observe that the optimal pay-off functionX̂T(x)hasnot
changed. Economically speaking this means that in the “completed” market
Sthe optimal investment may still be achieved by trading only in the first
d+ 1 assets and without touching the “fictitious” securitiesSd+1,...,Sd+k.
In particular, we now may apply formula (3.25) toQ=Q̂(y)toobtain
(3.32).

Finally we remark that the pricing rule induced byQ̂(y) is precisely such

that the interpretation of the optimal investmentX̂T(x) defined in (3.29)
(given in Remark 3.1.4 in terms of marginal utility and the ratio of Arrow
priceŝqn(y) and probabilitiespn) carries over to the present incomplete set-
ting. The above completion of the market by introducing “fictitious securities”
allows for an economic interpretation of this fact.

3.3 The Binomial and the Trinomial Model

Example 3.3.1 (The binomial model (one-period)).To illustrate the
theory we apply the results to the (very) easy case of a one-period binomial
model, as encountered in Chap. 1. The probability measurePassignsP[g]=
P[b]=^12 to the two statesgandbof Ω ={g, b}. In order to make the constants
obtained below more easily comparable to the usual notation in the literature
(e.g. [LL 96]), we refrain for a moment from our usual condition that we are
working with a model in discounted terms. Let

Ŝ 00 =1, Ŝ 10 =1+r,

and Ŝ 01 =1, Ŝ 11 =

{

1+u,forω=g,

1+d,forω=b,

wherer>−1 denotes the riskless rate of interest andu>rstands for “up”
and− 1 <d<rstands for “down”. In discounted terms (see Sect. 2.1 above)
the model then becomes

S^00 =1,S^01 =1

andS^10 =1,S^11 =

{

1+u, ̃forω=g,

1+d, ̃forω=b,

where 1 + ̃u=1+1+ur>1and1+d ̃=1+1+dr<1.