Bermudan Options

A European option is like a train ticket you can only use on the day printed on it. An American option is a ticket valid for travel on any day up to the expiry date, whenever you like. A Bermudan option sits between the two, literally named for the island chain between Europe and America, valid only on a specific, pre-agreed list of dates, say the first of every quarter, rather than every day or only the last day. The holder faces a real decision at each allowed date: exercise now and take the payoff, or wait for the next date and hope conditions improve.

The exercise decision

At each allowed exercise date $t_i$, the holder compares the payoff from exercising immediately against the value of continuing to hold the option:

In words: $\text{Payoff}(t_i)$ is what the holder gets by exercising right now, at date $t_i$. $\mathbb{E}[V(t_{i+1}) \mid \mathcal{F}_{t_i}]$ is the expected value of holding through to the next allowed date $t_{i+1}$, given everything known at $t_i$, discounted back to today's dollars by $e^{-r\Delta t}$. The option's value at $t_i$ is whichever of the two is larger. Work this comparison backward from the final allowed date to today, and you've priced the Bermudan option, the same backward-induction logic used for American options, just evaluated only at the specific allowed dates instead of continuously.

Worked example 1, a two-date Bermudan on a simple tree

A stock at $100 can move up 20% or down 15% each period (two periods total), risk-free rate 0%. A Bermudan put, strike $100, is exercisable at the end of period 1 or period 2 (not continuously). At the end of period 1, if the stock is at $85 (the down move), exercising now pays $\max(100-85,0)=$ $15. Continuing to period 2, the stock could go to $102 (up) or $72.25 (down again); the put's period-2 payoffs are $0 and $27.75, and with equal risk-neutral-style weights the expected continuation value is roughly $13.88, less than the $15 available by exercising now. So the holder exercises early, at the period-1 allowed date, banking $15 rather than waiting.

Worked example 2, why the allowed dates matter

Take the same setup, but now the only allowed exercise date is the final period (effectively European). The holder at the $85 node has no choice this time and must wait to period 2, where the put's payoff is either $0 or $27.75, averaging to something below the $15 they'd have banked mid-tree. Restricting exercise from "period 1 or period 2" down to "period 2 only" strictly cannot increase the option's value, because it removes an opportunity that was worth taking. General rule: a Bermudan is worth at least as much as a European with the same final maturity, and at most as much as an American with continuous exercise, more allowed dates can only add value, never subtract it.

Imagine each point where the curve crosses a grid tick as one allowed exercise date on the Bermudan's schedule, a European option only gets the far-right point, an American gets every point on the curve, and a Bermudan gets whichever subset of tick marks the contract names. More ticks (denser exercise dates) can only push the option's value up, never down.

What this means in practice

Bermudan-style exercise shows up constantly outside plain vanilla options: callable and puttable bonds, mortgage prepayment options, and, most heavily traded, swaptions embedded in cancellable interest rate swaps, all giving one side the right to act on a fixed schedule. Because a clean closed-form formula usually doesn't exist, Bermudans are priced with binomial/trinomial trees for simple cases or Least-Squares Monte Carlo (Longstaff-Schwartz) for path-dependent or multi-underlying cases, since both handle the backward-induction "exercise or continue" comparison directly.