Bermudan Option Valuation Before and After Exercise Becomes Available
Summary
The document explains how to value a call option that cannot be exercised during an initial period but becomes exercisable afterward. Its central point is that the tree switches to exercise-versus-continuation comparisons only at nodes where early exercise is allowed. Before that point, values are propagated backward from the later exercise window using discounted risk-neutral expectations.
The example uses monthly steps, with exercise available from month nine through month twenty-four, and clarifies the node timing convention: node nine marks the start of month ten. The option is therefore not equivalent to a European option expiring at the exercise-window boundary; future opportunities to exercise still affect earlier node values. The explanation gives a pricing recursion but does not provide numerical results or discuss implementation details such as dividends, rate assumptions, or convergence across tree step sizes.
Key ideas
- At nodes where exercise is permitted, compare immediate exercise value with continuation value.
- Before the exercise window begins, discount the expected values of the following tree nodes.
- The node numbering convention matters when identifying when exercise first becomes available.
- An initial restriction on exercise does not make the option equivalent to a European option ending at that restriction.
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# Will pricing a Bermudan option default to a value of a European option?
# Will pricing a Bermudan option default to a value of a European option?
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 by ajc3 (score 1, accepted)
https://quant.stackexchange.com/a/11077
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.