Will pricing a Bermudan option default to a value of a European option?

Question

I have a call option with 2 expiry in two years. For the first 9 months I cannot excercise the option. After that the I can exercise at any time. I am pricing this option using a binomial tree using the traditional back-propagation method. However at node 9 (i.e. month 9) I have to switch from an American option node i.e. $= \max(\max(S-K,0)$, discounted exercise) to $\max(S-K,0)$ for a european call. Hence it almost does not matter what happens after month 9 and the value is a european option. Is my thinking here correct?

Answer 1

With timesteps of one month, and the ability to exercise after 9 months, you still need to consider early exercise at node 9 (representing the start of month 10, just as node 0 represents start of month 1).

Before the American part kicks in, the tree methodology will revert back to European style, in that you do not have to consider early exercise at that node. However, you are not valuing max(S-K,0) from node 8 to node 0 -- you are valuing the discounted payoff of potential early exercise between nodes 9 and 24. For node $(i,j)$, with $i\le8$, you are valuing:

\begin{eqnarray} V_{i,j}&=&E\left(\displaystyle\sup_{9\le \tau \le 24}max\left(S_{\tau}-K,0\right)B_\tau^{-1}\left| \right.\mathcal{F}_{i,j}\right) \, B_i\\ &=&e^{-r_i\delta_i}\, \left(V_{i+1,j+i} \, p_{i,j} + V_{i+1,j} \, (1-p_{i,j})\right), \end{eqnarray} where $r_i$ is the risk-free rate, $\delta_i$ is your timestep, $p_{i,j}$ is the risk-neutral up probability at node $(i,j)$ and $V_{i+1,j+i}$ and $V_{i+1,j}$ are the values at nodes representing an up jump and down jump respectively.