Pricing Bermudan Swaptions

A European swaption gives you one shot: exercise on a single date or don't. An American option, the other extreme, lets you exercise at literally any instant. A Bermudan swaption sits between the two islands — named for Bermuda sitting between Europe and America — and lets you exercise on any one of a specific list of dates, say the start of each year for the next ten years, but no others. The pricing challenge isn't finding the payoff on any one date, which is just a European swaption calculation; it's figuring out, standing at an early exercise date with the option still alive, whether the immediate payoff beats the expected value of holding on for a better date later. That comparison, repeated backward through every possible exercise date, is what makes Bermudans genuinely hard.

The optimal-stopping logic

At each exercise date $t_i$, the holder compares two numbers:

$$V(t_i) = \max\big(\text{Exercise value}(t_i),\ \text{Continuation value}(t_i)\big),$$

where the continuation value is the expected, discounted value of holding the option unexercised and behaving optimally at every future date after $t_i$. In plain English: you exercise today only if today's payoff beats what you'd expect to get, on average, by holding out for the best of every future chance — and because continuation value itself depends on future optimal decisions, the whole problem has to be solved backward from the final date, where continuation value is zero (nothing left to wait for), toward today.

Worked example 1 — a two-date toy version

Simplify to just two exercise dates. At the final date, the swaption is worth $\max(S_T, 0)$ where $S_T$ is the intrinsic swap value — say it works out to an expected value of $40,000 today (that expectation is the continuation value seen from the earlier date). At the earlier date, the immediate exercise value is $35,000. Since $35{,}000 < 40{,}000$, the holder should not exercise early — holding out for the second date is worth more on average, so V(\text{earlier date}) = \40{,}000$, and the early-exercise right this month is simply unused.

Worked example 2 — when early exercise wins

Now suppose rates have moved sharply, and the immediate exercise value at the earlier date jumps to $52,000, while the continuation value (expected value of waiting for the later, still-uncertain date) remains $40,000 because the underlying rate is expected to mean-revert back down. Since $52{,}000 > 40{,}000$, the holder exercises now: V(\text{earlier date}) = \52{,}000$. This is exactly the situation a Bermudan swaption holder in a deeply in-the-money position with a mean-reverting curve outlook actually faces — locking in a large certain gain rather than gambling that the market drifts even further their way.

Think of backward induction as related to the stepping shown above, but running in the opposite direction and over decision dates instead of a loss surface: solve the last date first, then step backward one date at a time, at each step deciding whether the value there beats what was just solved for the date after it.

What this means in practice

The continuation value at each date isn't a clean formula in realistic models with several rate factors — it's usually estimated by simulating many future paths and running a regression of realized continuation payoffs against the current state (Longstaff-Schwartz), effectively teaching the model what "holding on" is worth from each simulated point without solving the problem analytically. Callable bonds, mortgage prepayment options, and convertible bonds all embed this same backward-induction logic.