Swaptions

Swaptions are used when you want to buy only the right to start an interest rate swap, rather than the swap itself. They are also used to analyze the prices of interest rate options and implied volatility.

The rights in payer and receiver swaptions

A swaption is an option on an interest rate swap. Financial institutions use swaptions to hedge their interest rate risk, among other purposes¹. The buyer pays the seller a premium and gets the right to enter into an interest rate swap at expiry at a fixed rate agreed in advance. This fixed rate is called the strike rate. The right to enter into a swap paying the fixed rate is a payer swaption, and the right to enter into a swap receiving the fixed rate is a receiver swaption².

Suppose Company A buys from Bank B a payer swaption with a notional amount of 10 billion yen, an expiry of 2 years, a swap term of 3 years, and a strike rate of 1.50%. Company A pays Bank B the premium now. If Company A exercises the right in 2 years, then for the next 3 years it pays Bank B interest at a fixed rate of 1.50% and receives interest at a floating rate from Bank B. If Company A does not exercise, the swap never starts, and Bank B keeps the premium it received.

Figure description

This figure shows the payments when Company A, the payer swaption buyer, exercises the right in 2 years. The premium (about 48 million yen, the payer's value found in the pricing example later) is paid only now, and after exercise, a fixed-rate payment and a floating-rate receipt continue each year from year 3 to year 5. The floating-rate receipt is not fixed when the trade is made, so the length of its arrow is only a guide. If Company A does not exercise, the only payment is the premium.

In the market, the expiry and the swap term are quoted in that order, as in "1 month into 10 year" (an expiry of 1 month on a 10-year interest rate swap)³. The example above is "2 year into 3 year," also written 2-year × 3-year. The fixed rate of the underlying swap depends on whether its floating rate is TONA (Tokyo Overnight Average Rate) or another rate, so the price of the swaption also differs⁴.

Receiving the fixed rate on an interest rate swap is similar to buying a bond, and paying the fixed rate is similar to selling a bond. For this reason, a receiver swaption can be classified as a bond call option and a payer swaption as a put option². If you instead take the fixed rate of the interest rate swap (the swap rate) as the underlying asset, the direction is reversed: a payer swaption gains value as the swap rate rises. The pricing formulas treat the swap rate as the underlying.

Exercise at expiry, settlement methods, and exercise styles

Whether to exercise at expiry depends on how the strike rate compares with the swap rate at which a swap of the same term can be entered into in the market at that time. The buyer of a payer swaption exercises if the market swap rate is above the strike rate, because exercising lets the buyer pay a fixed rate lower than the market rate. The buyer of a receiver swaption exercises if the market swap rate is below the strike rate, because exercising lets the buyer receive a fixed rate higher than the market rate⁵. In the example above (strike rate 1.50%, swap term 3 years, notional amount 10 billion yen), the comparison for each 3-year swap rate at expiry is as follows.

For the payer swaption at a swap rate of 2.00% (or the receiver swaption at 1.00%), exercising gains 50 million yen a year (10 billion yen × 0.50%) for 3 years, or 150 million yen in total before discounting. This money, however, arrives 3 to 5 years from now, while the premium is paid today. If the yearly differences, discounted to today and added up, exceed the premium calculated in the worked example (48,100,402 yen for the payer swaption under the Black model), the buyer gains; if they fall short, the buyer loses. If the swap rate is 1.50% or below, the buyer of the payer swaption does not exercise and loses the premium paid. The seller's profit and loss is the mirror image: the seller receives the premium and, if the option is exercised, holds a swap at a fixed rate worse than the market.

A swaption whose strike rate equals the forward swap rate (the future swap rate implied by today's yield curve, defined in the formula below) is called at-the-money forward (ATMF). For a payer swaption, a strike rate below the forward swap rate is in the money, and a strike rate above it is out of the money⁶.

Settlement can be physical, in which the swap actually starts on exercise, or cash, in which only a net amount based on the market value of the swap at exercise is paid⁷.

Swaptions are OTC derivatives, so the exercise style can be chosen for each trade. Styles include the European option, which can be exercised only at expiry, the American option, which can be exercised at any time, and the Bermudan option, which lies between the two. Swaption data available from information vendors is generally for European swaptions⁸. The formulas and worked example below cover European swaptions.

How the price is calculated (formulas using the annuity)

The price of a European swaption is the price of an option with the forward swap rate as the underlying, multiplied by the annuity. The annuity is the sum of the discount factors for each payment date of the swap that starts on exercise (when every payment period is one year). Exercising leads to payments over several years, so the annuity is used to discount each of those payments⁹.

Let the expiry be $T$ years and the swap term $n$ years, with annual payments and every period one year long. With $DF(t)$ as the discount factor for $t$ years from now, the annuity $A$ is:

$$A = \sum_{t=T+1}^{T+n} DF(t)$$

When a payment period is not one year long, each discount factor is multiplied by the length of its period (in years) before adding. This is the same form as the par rate formula of a swap. The par rate formula is covered in Valuing an Interest Rate Swap.

The forward swap rate $F$ is the fixed rate that makes the value today of an $n$-year swap starting in $T$ years (a forward-starting swap) equal to zero¹⁰.

$$F = \frac{DF(T) - DF(T+n)}{A}$$

Setting $T = 0$ gives the formula for the par rate of a swap starting today.

Let $N$ be the notional amount, $K$ the strike rate, and $\Phi$ the cumulative distribution function of the standard normal distribution. Under the Black model, the price $V$ of a payer swaption is given by the formulas below. $\sigma_B$ is the Black volatility, the standard deviation (annualized) of the rate of change of the swap rate⁹.

$$V = N A \left[ F \, \Phi(d_1) - K \, \Phi(d_2) \right]$$ $$d_1 = \frac{\ln(F/K) + \sigma_B^2 T / 2}{\sigma_B \sqrt{T}}$$ $$d_2 = d_1 - \sigma_B \sqrt{T}$$

The Bachelier model (normal model) assumes that the change in the swap rate itself follows a normal distribution. The price of a payer swaption is then given by the formulas below. $\sigma_N$ is the normal volatility, the standard deviation (annualized, in bp) of the change in the swap rate. $\phi$ is the probability density function of the standard normal distribution⁹.

$$V = N A \, \sigma_N \sqrt{T} \left[ d \, \Phi(d) + \phi(d) \right]$$ $$d = \frac{F - K}{\sigma_N \sqrt{T}}$$

The Black model assumes that the swap rate does not fall to zero or below. When $F$ is zero or below, $\ln(F/K)$ cannot be calculated, so neither can the price¹¹. How the Bachelier model came into wide use after rates turned negative, and how the assumptions of the two models differ, are covered in Caps and Floors.

Consider a hypothetical example in which the overnight rate is 0.50% and the OIS (overnight index swap) fixed rates are 0.60% for 1 year, 0.75% for 2 years, 1.10% for 5 years, and 1.50% for 10 years. Value a 2-year × 3-year swaption with a strike rate of 1.50%. The discount factors are 0.985156 for 2 years, 0.972123 for 3 years, 0.959262 for 4 years, and 0.946571 for 5 years. The annuity is the sum of the discount factors for 3 to 5 years, 2.877956, and the forward swap rate is (0.985156 − 0.946571) ÷ 2.877956 = 1.3407%. With a Black volatility of 30% and a normal volatility of 40bp, the prices are as in the table below¹².

The strike rate of 1.50% is above the forward swap rate, so the payer swaption is out of the money and cheaper than the receiver swaption. The payer price of 48,100,402 yen is about 0.48% (48bp) of the notional amount. The two volatilities were set to match through an approximate relationship in which the normal volatility is roughly the forward swap rate times the Black volatility¹³: 1.3407% × 30% ≈ 0.40% (40bp). At a strike rate of 1.35%, close to the forward swap rate, the two models give almost the same price. The further the strike rate moves from the forward swap rate, as at 1.50%, the wider the gap.

The difference between payer and receiver

If you buy a payer swaption and sell a receiver swaption with the same expiry, swap term, and strike rate, you end up holding a swap paying the strike rate as the fixed rate, whatever the swap rate at expiry. If the swap rate is above the strike rate, you exercise the payer swaption; if it is below, the other party exercises the receiver swaption. The price of the payer swaption minus the price of the receiver swaption therefore equals the value of a forward-starting pay-fixed swap with the strike rate as its fixed rate. This follows the same reasoning as put-call parity.

$$V_{\text{payer}} - V_{\text{receiver}} = N A (F - K)$$

In the worked example, this is 10 billion yen × 2.877956 × (1.3407% − 1.50%) = −45,846,535 yen. In the table, the Black model gives 48,100,402 − 93,946,937 and the Bachelier model gives 44,583,700 − 90,430,235, and both come to the same −45,846,535 yen. The relationship depends on neither the volatility nor the model. When the strike rate equals the forward swap rate, the right-hand side is zero, and the ATMF payer and receiver swaptions have the same price.

Deep diveChecking the difference formula from the payoffs at expiry

Let $S$ be the swap rate at expiry. The yearly gain from the swap obtained by exercise is $\max(S - K, 0)$ for the payer swaption and $\max(K - S, 0)$ for the receiver swaption. Whether $S$ is above or below $K$, the following holds.

$$\max(S - K, 0) - \max(K - S, 0) = S - K$$

Receiving $S - K$ every year on the right-hand side is the same as holding a pay-fixed swap at the fixed rate $K$ at expiry. If you add a receive-fixed swap entered into in the market at expiry at the fixed rate $S$, the floating payments cancel out and $S - K$ remains every year. Seen from today, its value is the value of the forward-starting swap, $N A (F - K)$. The value is $N A (F - K)$ because the forward swap rate formula gives $N [DF(T) - DF(T+n)] = N A F$, so the present value of the floating payments is $N A F$ and the present value of the fixed payments is $N A K$.

What moves the value: rates and volatility

A payer swaption gains value when rates rise, and a receiver swaption gains value when rates fall. If all the market rates for the curve in the worked example rise by 0.5 percentage points, the forward swap rate becomes 1.8437%, and under the Black model the payer price rises to 136,749,755 yen while the receiver price falls to 39,773,556 yen. If all the rates fall by 0.5 percentage points, the forward swap rate becomes 0.8377%, the payer price falls to 5,346,209 yen, and the receiver price rises to 199,770,571 yen. The buyer gains if the market moves in the buyer's favor, and if it moves the other way, the loss is limited to the premium. The seller is in the opposite position, and the loss can grow large when the market moves against the seller.

When volatility rises, the prices of both payer and receiver swaptions rise. Raising the Black volatility from 30% to 40% makes the payer price 69,848,655 yen and the receiver price 115,695,190 yen. Higher volatility benefits the buyer and hurts the seller. The difference between the payer and receiver prices stays equal to the value of the forward-starting swap and does not change with volatility.

What swaptions are used for

Swaptions are used to hedge against losses when rates move. Someone holding a position that loses when rates rise (such as a bond holding) can hedge with a swap paying the fixed rate or by buying a payer swaption. With the swap, if rates rise, the gain on the swap offsets the loss on the position, but if rates fall, the loss on the swap cancels out the gain on the position. With a payer swaption, there is the cost of the premium, but if rates rise, the buyer can exercise to offset the loss on the position, and if rates fall, the buyer does not exercise and keeps the gain on the position minus the premium¹⁴.

According to a 2021 article in the Ministry of Finance's Finance, Japanese life insurers buy swaptions as a tool for ALM, to limit the mismatch in maturity and amount between their assets and liabilities (interest rate risk)¹⁵. The duration of a life insurer's liabilities is longer than that of its assets, so when rates fall sharply, the interest rate risk of the liabilities grows faster than that of the assets. For example, consider an insurer buying a receiver swaption with a strike rate of 0.5% when the 30-year swap rate is 1%. If rates fall below 0.5%, the insurer exercises and holds a swap receiving the fixed rate, which adds interest rate risk on the asset side to match the increase on the liability side. If rates do not fall below 0.5%, the insurer does not exercise. In that case the interest rate risk of the liabilities has not grown much either, so the premium is the cost of protection that was not needed.

Conversely, some products, such as single-premium whole life insurance, see more policyholders surrender their policies when rates rise sharply, so the interest rate risk of the liabilities drops suddenly. According to the same article, buying an out-of-the-money payer swaption creates a position that reduces interest rate risk on the asset side when rates rise, which hedges the risk that the interest rate risk of the liabilities changes with the level of rates¹⁵.

The volatility backed out from a swaption price is called implied volatility. Implied volatility is also used to analyze how market participants see the movement of rates at each maturity. Swaptions cover a wide range of underlying swap terms, from 1 to 30 years, and the expiry is set for each trade, so data for the same expiry and term can be collected continuously¹⁶.

Risks and trading practice

The seller's loss can grow beyond the premium received (see the figures in the section "What moves the value: rates and volatility"). Because swaptions are OTC derivatives, there is also the credit risk of the other party to the trade (counterparty risk). A view reported in a 2021 article in the Ministry of Finance's Finance is that, to reduce this credit risk, cash settlement has been increasing for swaptions in foreign currencies⁷.

The low liquidity of yen swaptions has long been seen as a problem¹⁷. Although their liquidity is higher than that of caps and floors, when trading is thin, market prices may reflect the views of only a few participants. The same care about liquidity applies when using implied volatility for analysis.

Differences from other interest rate options

Besides swaptions, interest rate options include the interest rate cap and the interest rate floor (caps and floors), traded over the counter, and options on JGB futures listed on an exchange (JGB futures options). The differences are shown in the table below²¹.

Swaptions are more liquid than caps and floors and their data is easier to obtain. In the experience of the author of the 2021 Finance article, swaptions account for almost all analysis of OTC interest rate options. A cap is an option that pays the difference for each period in which the rate is above the strike rate. JGB futures options have expiries aligned by contract month, so trading tends to concentrate in them, and they are suited to seeing how much rates around 7 years are expected to move in the near future, less than a month ahead. Swaptions have an expiry set for each trade, so they also show views more than a year ahead.

Notes

1. Takahiro Hattori, "債券（金利）オプション入門―スワップションについて―" (Introduction to bond (interest rate) options: swaptions), Ministry of Finance, "ファイナンス" (Finance), August 2021 issue (in Japanese), page 49. It describes swaptions as options on interest rate swaps and says they are widely used, among other purposes, to hedge the interest rate risk held by financial institutions. ↩

2. The same document, page 50. It says a receiver swaption corresponds to a call option and a payer swaption to a put option, and that exercising the right creates a position receiving (paying) on a swap. ↩ ↩²

3. The same document, page 51. The example "1 month into 10 year（原資産は10年金利スワップ、満期は1か月後）" (our translation: the underlying is a 10-year interest rate swap and the expiry is one month later), the tendency for yen rates to be quoted in premium, the 35bp example, and the reason for not trading in volatility. ↩ ↩²

4. The same document, page 50. It says the option price also differs depending on which floating rate the underlying interest rate swap is linked to. ↩

5. The same document, page 57. As an example of a receiver swaption, it says that if the market swap rate is 1%, there is no benefit in exercising an option to receive at a strike rate of 0.5%, and that if it is 0.4%, there is a benefit in exercising. A payer swaption pays the fixed rate in the opposite direction, so it is exercised when the swap rate is above the strike rate. ↩

6. The same document, page 52. ↩

7. The same document, page 52. It describes physical settlement and cash settlement, and says cash settlement pays a net amount based on the market value at exercise. ↩ ↩²

8. The same document, page 51. The document uses the terms "ヨーロピアン・タイプ" (European type), "アメリカン・タイプ" (American type), and "バミューダ・タイプ" (Bermuda type), and says the Bermuda type lies between the other two. ↩

9. Takahiro Hattori, "スワップション入門（モデル編）―ノーマル（ブラック）・モデルおよびSABRモデルについて―" (Introduction to swaptions (models): the normal (Black) model and the SABR model), Ministry of Finance, "ファイナンス" (Finance), October 2021 issue (in Japanese), Section 3.2 (pages 60–61), formulas (4) and (5). The document's $N$ corresponds to $\Phi$, and $d_1$ is rewritten for an expiry of $T$ years. ↩ ↩² ↩³

10. Society of Actuaries, "Interest Rate Swaps" (May 2017), Section 4 "Special Formula," formula (4.1) (page 12) and Example 7 (page 13). The document's $P_{t_0}$ and $P_{t_n}$ correspond to $DF(T)$ and $DF(T+n)$. ↩

11. Takahiro Hattori, "スワップション入門（モデル編）" (Introduction to swaptions (models)), October 2021 issue (in Japanese), Section 2.1 (page 57) and the explanation after formula (5) (page 61). ↩

12. Prices are calculated in the same way as the simulator embedded on this page (rate-options) and rounded to the nearest yen. Discount factors and the annuity are shown rounded to 6 decimal places, and the forward swap rate to 4 decimal places (in %). ↩

13. Takahiro Hattori, "スワップション入門（モデル編）" (Introduction to swaptions (models)), October 2021 issue (in Japanese), Section 3.1, formula (3) (pages 59–60). The document says practitioners gauge the difference between the two volatilities with the approximation $\sigma_N \approx F \times \sigma_B$, and cautions that it is only an approximation. ↩

14. Takahiro Hattori, "債券（金利）オプション入門―スワップションについて―" (Introduction to bond (interest rate) options: swaptions), August 2021 issue (in Japanese), page 50. ↩

15. The same document, Section 5 (pages 56–57). ↩ ↩²

16. The same document, page 49 and pages 52–53. It notes that options are traded on swaps from 1 year to 30 years, and that trades set the period from the trade date to the expiry date. ↩

17. The same document, page 53. ↩

18. The same document, the note to Figure 1 on page 50. ↩

19. Japan Securities Clearing Corporation, "清算対象取引（金利スワップ）" (Transactions eligible for clearing: interest rate swaps, in Japanese), the note to the table of eligible transactions. An English version is available: "Clearing Products." Checked on September 25, 2026. ↩

20. Takahiro Hattori, "債券（金利）オプション入門―スワップションについて―" (Introduction to bond (interest rate) options: swaptions), August 2021 issue (in Japanese), pages 55–56. ↩

21. The same document, pages 52–53 (JGB futures options) and page 58 (caps and floors). The view that JGB futures options have low liquidity is on page 54. ↩