Bermudan Swaptions

A Bermudan swaption comes up when pricing and hedging interest rate trades that have several termination or exercise dates.

A swaption with several exercise dates

A Bermudan swaption is an option that lets the buyer exercise a swaption on any of several fixed dates. On exercise, an interest rate swap starts on that date and runs to a maturity agreed in advance¹. The later the exercise date, the shorter the swap that starts. The Swaptions page covers the European swaption, which can be exercised only once at expiry.

A trade is named by its maturity and its non-call period. "5NC1" means a 5-year maturity whose first year is non-call (it cannot be exercised), with exercise allowed afterward. The same form "3NC1" means the option cannot be exercised in the first year, and exercise afterward starts a swap that ends 3 years from the beginning².

Consider a hypothetical example in which Company G buys a 5NC1 payer swaption with a notional of 100 million yen and a strike rate of 2%. A payer swaption is the right to start a swap in which G pays the fixed rate and receives the floating rate. G can exercise on four dates, 1, 2, 3 and 4 years from now, and whichever date it picks, the swap ends in 5 years.

Figure 1 shows the payments of the swap that starts if G exercises in year 1. The lengths of the swaps for the other exercise dates are in the table above.

Figure description

If G exercises in year 1, a 4-year swap starts, and in years 2, 3, 4 and 5 G pays a 2% fixed rate (2 million yen each time, which is 2% of 100 million yen) and receives the floating rate. The floating amounts are not yet known, so the arrow lengths are only a guide.

If the market fixed rate (the swap rate) on an exercise date is above the 2% strike rate, G exercises and pays a fixed rate lower than the market. Consider a hypothetical example in which the market fixed rate in year 2 is 3%. G pays 2% on a 3-year swap, which is 1 million yen a year (1% of 100 million yen) less than at the market level (before discounting). At 2% or below, exercising on that date brings no gain. If the market fixed rate in year 1 is 1.5%, exercising would mean paying 500,000 yen a year more than the market, so G does not exercise. G paid a premium when it bought the option, so if it never exercises, it loses the premium it paid. In the simulator's model, the premium is 1,642,137 yen.

A callable swap, in which the fixed-rate payer holds a right to terminate, is a combination of a plain swap and a receiver swaption. One with several termination dates is Bermudan style³.

The value is at least the largest European and at most their sum

The value of a Bermudan swaption is at least the largest of the European swaptions for each exercise date, and at most their sum⁴. Here, a European swaption has the same strike rate and maturity as the Bermudan swaption and a single exercise date. This relationship can be derived from the definition of a Bermudan swaption.

The lower bound follows from the buyer being able to exercise on any date. By choosing the same date as the largest European swaption, the buyer receives the same amount, so the Bermudan swaption cannot be worth less. The upper bound follows from the fact that exercising once pays out only one exercise date's amount, which cannot exceed the sum of the amounts for all the dates.

Deep diveDeriving the lower and upper bounds

Let the exercise dates be $i = 1, \dots, n$, and write $D_i X_i$ for the present value of the payoff if exercised on date $i$. $D_i$ is the discount factor to date $i$. $X_i$ is the value of the swap that starts on that date if that value is positive. If it is negative, $X_i$ is 0, because the buyer does not exercise. So $X_i \ge 0$. The value of the European swaption for date $i$ is, as a risk-neutral expectation, $V_i = E[D_i X_i]$.

The Bermudan buyer chooses the exercise date $\tau$ (one of $1, \dots, n$, decided at each point using only the information known by then). The value is the maximum over these choices, $V_B = \sup_\tau E[D_\tau X_\tau]$.

The lower bound comes from the fact that the rule "always exercise on date $i$" ($\tau = i$) is just one of the possible choices. This rule is worth $V_i$ and the maximum is at least as large, so $V_B \ge V_i$ for every $i$, that is, $V_B \ge \max_i V_i$.

The upper bound comes from $X_i \ge 0$, which gives $D_\tau X_\tau \le \sum_{i=1}^{n} D_i X_i$ in every scenario. The left side is one term, and the right side is a sum of non-negative terms that includes it. Taking expectations of both sides gives $E[D_\tau X_\tau] \le \sum_i V_i$ for every $\tau$, and the maximum cannot exceed it either, so $V_B \le \sum_{i=1}^{n} V_i$.

Therefore $\max_i V_i \le V_B \le \sum_i V_i$.

The next table shows the results of calculating G's example with this site's simulator. Amounts are per 100 million yen of notional, and the curve is flat at 2% a year. The short-rate volatility (σ) is 80bp, which is different from the swaption volatility used in practice⁵. The European prices are shown two ways: "Tree" (a numerical calculation that moves the short rate on a binomial tree) and "Closed form" (a value from a formula). The difference between the tree and the closed form is largest for the 1-year date, at a relative difference of 0.26%. The "Bermudan" column is the price when exercise is allowed on all dates up to that exercise date.

The Bermudan swaption that can be exercised on all four dates is worth 1,642,137 yen (164.21bp of the notional). That is 366,256 yen more than the largest European swaption (year 2) at 1,275,881 yen, and less than the sum of the four European swaptions, 4,110,240 yen⁶. The 366,256 yen is the value, in this model, of being able to choose exercise dates other than year 2. If the premium G pays is 1,642,137 yen, then having four exercise dates means G pays 366,256 yen more than it would for the largest European swaption alone. The Bermudan swaption with only the 1-year date has a single exercise date, so it is worth the same 1,212,092 yen as the European swaption.

The difference moves as follows when the conditions change. The "Difference" column in the table is the Bermudan price minus the largest European price.

Raising σ from 60bp to 100bp increases the Bermudan price from 1,230,538 yen to 2,054,339 yen, and the difference rises from 274,125 yen to 458,747 yen. With a 2-year non-call period and three exercise dates, the difference shrinks to 173,667 yen.

Consider a receiver swaption too. The Bermudan receiver swaption with the same 2% strike rate (receiving the fixed rate) is worth 1,631,320 yen, which is 10,817 yen less than the payer. At the money, European payer and receiver swaptions have the same price, but Bermudan swaptions do not, because the best timing of exercise differs.

Assumptions of this simulator's calculation

The simulator's prices are values under a simplified setup for learning, and they do not match actual market prices. The main assumptions are as follows.

• The model is a Ho-Lee binomial tree for the short rate. At each step the short rate moves up or down by the same amount, each with probability 1/2. The level at each step is set so that the tree's zero-coupon bond prices match the discount factors of the flat curve.
• σ is the annualized normal volatility (bp) of the short rate, and it is not the normal volatility of swaptions traded in the market. It is a different quantity from the 80bp in "swaption volatility of 80bp" as used in practice.
• The curve is flat, and the annually compounded rate is the same for all terms. The forward rate and the swap rate take the same value, a single factor drives interest rates, and there is no volatility smile.
• Fixed and floating payments are made once a year, with each period treated as 1 year. The floating leg is valued at par.
• The tree has an error that depends on the number of steps. For the four exercise dates of the default setup, comparing the European tree prices with the closed form gives a relative difference of at most 0.26% at 48 steps per year.

Valuation methods

There is no closed-form pricing formula for Bermudan-style options, so they are calculated numerically under an interest rate model⁷. Pricing is harder than for ordinary European swaptions⁸.

The methods depend on the number of variables that describe interest rate movements. With few variables, trees and numerical solution of partial differential equations have long been used. With many variables these become hard to use, and the Monte Carlo method becomes a candidate⁷.

When Monte Carlo is used for a Bermudan swaption, the value of continuing to hold the option is approximated at each exercise date by least-squares regression. The price of the exercise policy found this way is at most the true price⁷. This method is computationally heavy, and the calculated values tend to be biased depending on how the regression equation is set up⁹.

With a Markov functional model, whose state variable is a low-dimensional Markov process, prices can be found efficiently by repeating a backward calculation of conditional expectations at each exercise date¹⁰.

A short-rate model describes the movement of interest rates over a very short period and sets zero-coupon bond prices from that movement. This site's simulator calculates with the Ho-Lee model, which moves the short rate on a binomial tree. The Ho-Lee model was proposed in 1986¹¹. The price of a Bermudan swaption is calculated backward from maturity to today, choosing at each exercise date the larger of the amount received by exercising and the value of continuing to hold¹². Trees in general are covered in Binomial Trees.

Calibration

Setting a model's parameters to match market prices is called calibration¹³. For Bermudan swaptions, one method uses the fact that the swaps starting on each exercise date end on the same date, and fits the model to European swaptions with the same end date (co-terminal swaptions).

In practice, parameters are sometimes fitted to the swaptions for which the exercise date (the paper's maturity) plus the swap length equals the term of the Bermudan swaption. Parameters are fitted to them because traders often hedge Bermudan swaptions with them¹⁴. For Hull-White type models, it is customary to fit the volatility curve to the implied volatilities of swaptions with the same end date¹⁵.

In G's example, the four swaptions to fit to are the 4-year swap from year 1, the 3-year swap from year 2, the 2-year swap from year 3, and the 1-year swap from year 4. It is natural to fit to the swaptions used to hedge the option being priced, and for a relatively simple Bermudan swaption one could choose swaptions as the fitting targets¹⁶.

The simulator on this page does not fit to market prices. You enter σ directly.

Issues in valuation and hedging

The result of a valuation depends on the model. Even when models are fitted to the same market data, the calculated prices of a Bermudan swaption do not necessarily agree across different models¹⁷. In one calculation example, the difference between the two models was small enough to be acceptable in practice¹⁷.

Hedging involves rebalancing errors. In a method that continuously changes the hedge holdings to follow the market, rebalancing can be done only at fixed intervals, so a sudden market move can produce a large error¹⁸. In research, a semi-static hedge has been proposed under affine multi-factor models. It replicates a Bermudan swaption with a portfolio of options on combinations of zero-coupon bonds and limits rebalancing to a finite number of times¹⁸.

Notes

1. Jori Hoencamp, Shashi Jain and Drona Kandhai, "A semi-static replication approach to efficient hedging and pricing of callable IR derivatives" (arXiv, February 2022), page 4, "2.2 The Bermudan swaption pricing problem". ↩

2. Haruyasu Ota, "マルコフ関数モデルによる金利オプションの価格付けの実用化" (Putting the Markov-functional model into practice for pricing interest rate options), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2010), page 162, "4 節（2）バミューダン・スワップション価格の算出" (Section 4 (2), Calculating Bermudan swaption prices) (in Japanese). The definition of a Bermudan-style option is in footnote 3 on page 133. ↩

3. Intercontinental Exchange, "SDX Interest Rates Help," "Callable Swap," opening description. The distinction between European and Bermudan types, and the combination of a plain swap and a receiver swaption. ↩

4. Derived from the definition of a Bermudan swaption. The derivation is in the deep dive. ↩

5. Amounts are the simulator's results (the tree uses 48 steps per year, and the closed form uses the Ho-Lee formula for options on zero-coupon bonds), rounded to the nearest yen. ↩

6. Adding the European (tree) prices for the four exercise dates in the table gives 4,110,241 yen, but the table values are rounded to the nearest yen, so there is a 1 yen difference. The total of 4,110,240 yen is the sum of the values before rounding. ↩

7. Hoencamp et al. (February 2022), page 1, "1. Introduction". No closed-form pricing formula, the traditional methods, Monte Carlo, and the price of a regression-based exercise policy being at most the true price. ↩ ↩² ↩³

8. Natsumi Ochiai and Masamitsu Ohnishi, "Pricing of the Bermudan Swaption under the Generalized Ho-Lee Model" (Research Institute for Mathematical Sciences, Kyoto University, "Kokyuroku" no. 1802, 2012), page 256, "1 Introduction". ↩

9. Ota (October 2010), page 132, "1. はじめに" (1. Introduction) (in Japanese). For options with an early exercise right, a technique such as the least-squares Monte Carlo method of Longstaff and Schwartz [2001] is needed, which makes the computational load heavier still, and the calculated values tend to be biased depending on how the regression equation is set. ↩

10. Ota (October 2010), page 134 (in "2. マルコフ関数モデル" (2. Markov functional models), "(1) マルコフ関数モデルの概要" (1) Overview of the Markov functional model) (in Japanese). Because the state variable is a low-dimensional Markov process, interest rate options, including Bermudan-style options with an early exercise right, can be valued efficiently by repeating a backward calculation of conditional expectations. ↩

11. Ochiai and Ohnishi (2012), page 256, "1 Introduction". The Ho-Lee model is the first arbitrage-free model of the term structure of interest rates, proposed by Ho and Lee (1986). The Generalized Ho-Lee model treated in the paper is an extension of it and is a binomial-tree interest rate model. ↩

12. Ochiai and Ohnishi (2012), page 256, "1 Introduction". The price of a Bermudan swaption is found by deriving the optimality equation with dynamic programming and solving it backward in time. ↩

13. Ota (October 2010), page 132, "1. はじめに" (1. Introduction) (in Japanese). Fitting a model's parameters to market prices is also called calibration. ↩

14. Tomohisa Yamakami and Yuki Takeuchi, "Pricing Bermudan Swaption under Two Factor Hull-White Model with Fast Gauss Transform" (arXiv, December 2022), page 5, end of "2.1 Two Factor Hull-White Model". ↩

15. K. E. Feldman, "Berms without Calibration" (arXiv, version of December 11, 2025), page 1, Introduction. The paper reflects the author's personal views, not those of the bank the author works for. ↩

16. Ota (October 2010), page 138, "2. マルコフ関数モデル" (2. Markov functional models), "(2) 価格付けおよびフィッティングの対象とするオプション" (2) Options to be priced and fitted to (explanation of Figure 2) (in Japanese). ↩

17. Ota (October 2010), page 163, "4 節（2）バミューダン・スワップション価格の算出" (Section 4 (2), Calculating Bermudan swaption prices), part イ (in Japanese). A comparison (Table 8) between the Markov functional model and the LIBOR market model with the least-squares Monte Carlo method. ↩ ↩²

18. Hoencamp et al. (February 2022), page 2 (continuation of "1. Introduction"), and the abstract. Dynamic hedging, which rebalances continuously, becomes costly when transaction costs are high, and sudden market moves combined with discrete rebalancing can cause large hedging errors. The semi-static hedge is a proposal to replicate with a portfolio of options on a basket of zero-coupon bonds, needing only a finite number of rebalancings. ↩ ↩²