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:

$$V(t_i) = \max\Big(\text{Payoff}(t_i),\ \ \mathbb{E}\big[V(t_{i+1})\,\big|\,\mathcal{F}_{t_i}\big] \cdot e^{-r\Delta t}\Big).$$

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.