The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 432 Wednesday, February 4, 2004 1:08 PM

432 The Mathematics of Financial Modeling and Investment Management

The above discussion can be illustrated in the case of multiple

assets, each following a binomial model. If there are N linearly indepen-

dent assets, the price paths in the interval (0,T) will form a total of 2NT

states. In a binomial model, we can limit our considerations to one time

step as the other steps are identical. In one step, each price Sti at time t

can go up to Si idi

tu

i or down to S

t at time t + 1. Given the prices

{}Sit ≡ {S^1 t ,St^2 , ...,St N} at time t, there will be, at the next time step, 2N

possible combinations {S^1 22 N N i i

t w

(^1) ,S
t w , ...,St w }, w = u or d
i.
Suppose that there are 2NT states and that each combination of
prices identifies a state. This means that at each date t the information
structure It partitions the set of states into 2Nt subsets. Each set of the
partition is partitioned into 2N subsets at the next time step. This yields
2 N(t+1) subsets at time t + 1.
Note that this partitioning is compatible with any correlation struc-
ture between the random variables that represent prices. In fact, correla-
tions depend on the value of the probability assigned to each state while
the partitioning we assume depends on how different prices are assigned
to different states.
Risk-neutral probabilities qi, i = 1,2,...,2N can be determined solving
the following system of martingale conditions:
2 N
i j i

∑qjStw

i()= S

t

j = 1

2 N

∑qj =^1

j = 1

j = 1,2,...,2N , i = 1,2,...,N

which becomes, after dividing each equation by Sti , the following:

2 N

∑qjw

i()j = 1

j = 1

2 N

∑qj =^1

j = 1

where wi(j) = ui or di for asset i in state j.