Quiz: Interest Rate Options

Last reviewed: September 25, 2026

This is a draft, shown for review before publication.

4 questions to check your understanding of the chapter “Interest Rate Options”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.

Question 1
As a hypothetical example, Company A has borrowed 5 billion yen at a floating rate, with interest paid once a year and every period one year long. Company A has bought an interest rate cap (cap) with a notional amount of 5 billion yen and a strike rate of 2.0%. When the floating rate for a period is set at 3.0% at the start of the period, how much is the interest on the loan for that period after subtracting what the cap pays?
1. A.150 million yen
2. B.50 million yen
3. C.100 million yen
4. D.200 million yen
Answer and explanation
Answer: C
The interest on the loan is 5 billion yen × 3.0% = 150 million yen. With $M$ as the notional amount, $\delta$ the length of the period in years, $L$ the floating rate set at the start of the period, and $K$ the strike rate, the cap pays $M \times \delta \times \max(L - K, 0)$ , which here is 5 billion yen × 1 × (3.0% − 2.0%) = 50 million yen. The net interest is 150 − 50 = 100 million yen, which is capped at the amount for the 2.0% strike rate (5 billion yen × 2.0%). However high the floating rate goes, the amount above the strike rate is paid by the cap and can go toward the interest. This is why buying a cap puts a ceiling on the interest cost.

The choice "150 million yen" is the interest on the loan without a cap. The choice "50 million yen" is the cap payment on its own. The choice "200 million yen" mistakes the cap payment for something Company A pays and adds it to the interest.

In a period where the floating rate is set at 1.5%, the difference is zero or below, so the cap pays nothing. The interest is 5 billion yen × 1.5% = 75 million yen, lower than in the 3.0% period and the same as for a loan without a cap. The amounts so far leave out the premium paid to buy the cap. If rates do not rise, the premium paid is an extra cost compared with not buying the cap. The cap seller's profit and loss is the opposite of the cap buyer's: the seller gains the premium if rates do not rise, and pays more the higher rates go. Because a cap is an option, the buyer's loss is limited to the premium paid.
Learn more: Caps and Floors
Question 2
As a hypothetical example, a cap and an interest rate floor (floor) with the same strike rate, term, payment dates, and notional amount are priced at 90 million yen and 150 million yen. What is the value of an interest rate swap that pays that strike rate as the fixed rate and receives the floating rate?
1. A.60 million yen
2. B.−60 million yen
3. C.240 million yen
4. D.It cannot be determined because it depends on the model
Answer and explanation
Answer: B
When the strike rate, term, payment dates, and notional amount are the same, the price of the cap equals the price of the floor plus the value of the interest rate swap. This swap pays the strike rate as the fixed rate and receives the floating rate. The value of the swap is therefore the cap minus the floor: 90 million yen − 150 million yen = −60 million yen. This is because buying the cap and selling the floor means that in every period you receive the difference when the floating rate $L$ is above the strike rate $K$ , and pay the difference when it is below. The value is negative for the side paying the fixed rate; for the other side, which receives the fixed rate, it is a positive 60 million yen.
$\max(L - K, 0) - \max(K - L, 0) = L - K$
This relationship is the put-call parity between caps and floors. Because it follows from the payments themselves, it holds whether the Black model or the Bachelier model (normal model) is used. Even if the two models give different prices for the cap and the floor, the cap minus the floor is the same. The last choice is therefore wrong.

The choice "60 million yen" is the floor minus the cap, which is the value of the same swap seen from the side receiving the fixed rate. The choice "240 million yen" is the cap and floor prices added together.
Learn more: Caps and Floors
Question 3
As a hypothetical example, Company A has bought a payer swaption with a notional amount of 5 billion yen, a swap term of 4 years, and a strike rate of 1.00%. The swap pays once a year and every period is one year long. At expiry, the swap rate at which the same 4-year swap can be entered into in the market is 1.40%. Compared with entering into the same swap at the market swap rate, how much less fixed interest does Company A pay each year on the swap it holds after exercising?
1. A.70 million yen
2. B.50 million yen
3. C.80 million yen
4. D.20 million yen
Answer and explanation
Answer: D
A payer swaption (the payer) is the right to enter into a swap paying the fixed rate. If the market swap rate at expiry is above the strike rate, the buyer can exercise and become the side paying a fixed rate lower than the market rate. In this example, the fixed rate is 1.40% − 1.00% = 0.40% below the market, so the difference is 5 billion yen × 0.40% = 20 million yen each year.

The choice "70 million yen" is the yearly fixed interest at the market swap rate (5 billion yen × 1.40%), and "50 million yen" is the yearly fixed interest at the strike rate (5 billion yen × 1.00%). Neither is the difference. The choice "80 million yen" is the yearly difference added up over 4 years, before discounting.

This difference is money received after expiry, at different times from the premium paid before expiry. If the yearly differences, discounted to today and added up, exceed the premium, the buyer gains; if they fall short, the buyer loses. If the swap rate at expiry is 1.00% or lower, the buyer of the payer does not exercise and loses the premium paid. The buyer of a receiver swaption, by contrast, exercises when the swap rate is below the strike rate. The seller's profit and loss mirrors the buyer's: in return for receiving the premium, the seller holds a swap at a fixed rate worse than the market if the option is exercised.
Learn more: Swaptions
Question 4
As a hypothetical example, a payer swaption (the payer) and a receiver swaption (the receiver) have the same notional amount, expiry, swap term, and strike rate. Under the Black model with a volatility of 30%, the prices are 48,100,402 yen for the payer and 93,946,937 yen for the receiver. If only the volatility is raised to 40%, what happens to the price difference (the payer price minus the receiver price)?
1. A.Both prices rise by the same amount, so the difference does not change
2. B.Only the payer price rises, so the difference becomes less negative
3. C.Only the receiver price rises, so the difference becomes more negative
4. D.Both prices approach 0, so the difference also approaches 0
Answer and explanation
Answer: A
If you buy the payer and sell the receiver 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; if it is below, the other party exercises the receiver. The difference between the two prices therefore equals the value of a forward-starting pay-fixed swap with the strike rate as its fixed rate. With $N$ as the notional amount, $A$ the annuity (the sum of the discount factors for each payment date of the swap that starts on exercise), $F$ the forward swap rate (the fixed rate that makes today's value of the swap starting at expiry zero) and $K$ the strike rate:
$V_{\text{payer}} - V_{\text{receiver}} = N A (F - K)$
Volatility does not appear on the right-hand side, so the difference does not change with volatility. At 30%, the difference is 48,100,402 − 93,946,937 = −45,846,535 yen. In the article's example (notional amount 10 billion yen, expiry 2 years, swap term 3 years, strike rate 1.50%), at 40% the payer is 69,848,655 yen and the receiver 115,695,190 yen, and the difference is the same −45,846,535 yen.

When volatility rises, the prices of both the payer and the receiver rise. The two choices in which only one price rises, and the choice in which the prices approach 0, get this direction wrong. When volatility rises, the buyer of either the payer or the receiver gains and the seller loses.
Learn more: Swaptions

Question 1

Question 2

Question 3

Question 4