238 Mathematics for Finance
terms this means that they are strongly positively correlated.
11.1 Binomial Tree Model......................................
The shape of the tree will be similar to that in Section 3.2. However, to facilitate
the necessary level of sophistication of the model, it has to be more complex.
Namely, the probabilities and returns will depend on the position in the tree.
We need suitable notation to distinguish between different positions.
By astatewe mean a finite sequence of consecutive up or down movements.
The state depends, first of all, on time or, in other words, on the number of
steps. We shall use sequences of letters u and d of various lengths, the length
corresponding to the time elapsed (the number of steps from the root of the
tree). At time 1 we have just two statess 1 =uord,at time 2 four states
s 2 =ud,dd, du,or uu.We shall writes 2 =s 1 uors 1 d, meaning that we go
up or, respectively, down at time 2, having been ats 1 at time 1.In general,
sn+1=snuorsnd.
The probabilities will be allowed to depend on particular states. We write
p(sn) to denote the probability of going up at timen+1,having started at
statesnat timen. At the first step the probability of going up will be denoted
bypwithout an argument. In Figure 11.1 we havep=0. 3 ,p(u) = 0. 1 ,p(d) =
0. 4 ,p(uu) = 0. 4 ,p(ud) = 0. 2 ,p(du) = 0. 5 ,p(dd) = 0. 4.
Figure 11.1 States and probabilitiesLet us fix a natural numberNas the time horizon. It will be the upper
bound of the maturities of all the bonds considered. The statessNat timeN
represent the complete scenarios of bond price movements.