0
$\begingroup$

I wonder how to solve this problem. Lets say we have a binomial tree with the following parameters:

$u=1.25,\ d = 1/u,\ T=15$.

How many distinct possible values are there for $X_{7}$?

$\endgroup$
  • $\begingroup$ $d=1/u$ tells you it is a recombining tree. So there is one choice for $S_0$, two choices for $S_1$ (up and down), three choices for $S_2$ (up up, middle and down down), ... looks to me like there are $N+1$ choices for $S_N$. But draw the diagram to be sure... $\endgroup$ – noob2 Sep 14 at 23:47
2
$\begingroup$

Noob2 has given the answer, a recombining tree has $N+1$ nodes after $N$ time points. In our case, you have $8$ distinct possible values after $7$ time steps. I will next illustrate the difference between a recombining and non-recombining tree.

For a binomial tree, if $u=\frac{1}{d}$, then an up-move followed by an down-move gives the same result as a down-move followed by an up-move (since $ud=1$): $S_0ud=S_0du=S_0$. This is a huge numerical advantange since the amount of potential nodes grows only linearly with the number of time steps.

If $ud\neq1$, then the order of up- and down-moves occurrring does matter and the amount of nodes grows exponentially. In particular, $S_0ud\neq S_0du\neq S_0$. After $N$ time steps, there are $2^N$ distinct nodes. So the difference between recombining and non-recombining is (for your example of $N=7$) the difference between $7+1=8$ and $2^7=128$. For numerical implementation, the former is of course much easier and less costy: just set $N=100$. Then, a recombining tree has $101$ nodes but a non-recombining tree has roughly $10^{30}$ nodes.

The bottom line ist that a recombining tree ``re-uses'' nodes it has already created and comes back to them. A non-recombining tree however creates for every existing node two new ones which leads to exponential grow. The same issue of (non-)recombining trees equally applies to the trinomial trees where you allow for an up/middle/down-move.

This image illustrates the amount of nodes of a recombining tree (the case you have since $ud=1$) enter image description here

This image illustrates the amount of nodes for a non-recombining tree enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.