CFA Level II Exam · Valuation of Contingent Claims

Valuing Swaptions and Interest Rate Derivative Options

Updated 7 October 2026 · Fact-checked

A swaption is an option to enter a swap at a fixed rate set today. A payer swaption gives the right to pay fixed and receive floating. A receiver swaption gives the right to receive fixed. At expiry, value is the present value of the annuity of rate differences, if positive. Before expiry, use the Black model.

Understand Valuing Swaptions and Interest Rate Derivative Options

An interest rate option pays off based on a reference rate, such as Libor-type or SOFR-based rates, instead of an asset price. A caplet is a call on a rate. A floorlet is a put on a rate. A cap is a series of caplets. A floor is a series of floorlets. Each caplet or floorlet is usually priced on its own, then summed.

A swaption is an option on a swap. The buyer pays a premium today. At expiry, the buyer can enter a swap at the exercise rate (the fixed rate fixed in the contract). A payer swaption is a call on the swap fixed rate. You pay fixed and receive floating. It gains when market swap rates rise above the exercise rate. A receiver swaption is a put on the swap fixed rate. You receive fixed and pay floating. It gains when market swap rates fall below the exercise rate.

The payoff at expiry is an annuity. If the payer swaption is in the money, you can enter the swap at the lower exercise rate while the market rate is higher. The gain is the rate gap, times notional, times the length of each period, paid at each swap payment date. Its value at expiry is that gap times the PV of the annuity of the underlying swap's payments. The gap is (market swap rate − exercise rate) for a payer and (exercise rate − market swap rate) for a receiver.

Before expiry you use the Black model. The underlying is the forward swap rate, not a price. You compute d1 and d2 using the forward swap rate, exercise rate, volatility and time to expiry. Then multiply by the PV of the annuity of the swap payments. In the exam you are normally given the N(d1) and N(d2) values or a table. The main work is to pick the right inputs and the right discount factors.

Swaptions link to other instruments. A long payer swaption plus a short receiver swaption with the same exercise rate and terms replicates a forward swap at that rate. This is put-call parity for swaptions. A cap is a series of calls on rates; a floor is a series of puts. A cap and a floor with the same strike combine like a swap: long cap and short floor equals paying fixed on a swap at the strike.

Key formulas to remember

Payer swaption payoff at expiry: Value at expiry = Notional × Max[0, S − X] × PV annuity, where PV annuity = Σ(period fraction × discount factor) over the swap payment dates; S is the market swap fixed rate at expiry. X is the exercise rate. The discount factors in the PV annuity already do the discounting, so do not discount the payoff again.
Receiver swaption payoff at expiry: Value at expiry = Notional × Max[0, X − S] × PV annuity, where PV annuity = Σ(period fraction × discount factor) over the swap payment dates; Receiver gains when market swap rate falls below exercise rate. The PV annuity turns the per-period rate gap into a value at expiry, as in the payer formula.
Black payer swaption value: Payer = PVA × [F × N(d1) − X × N(d2)]; PVA is the PV of the annuity of the swap payments (sum of period fraction × discount factors × notional). F is the forward swap rate at option expiry.
Black receiver swaption value: Receiver = PVA × [X × N(−d2) − F × N(−d1)]; N(−d) = 1 − N(d).
Black d1 and d2: d1 = [ln(F ÷ X) + (σ² ÷ 2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; T is time to option expiry in years, not the swap tenor.
Swaption put-call parity: Payer − Receiver = PVA × (F − X); Equals the value of a forward swap paying fixed X. Long payer plus short receiver equals paying fixed at X.
Cap and floor as a swap: Long cap + Short floor (same strike and dates) = pay-fixed swap at the strike; The swap's fixed rate is the strike rate.
Caplet payoff (paid in arrears): Notional × Max[0, (Reference rate − Strike)] × (days/360), paid at the end of the period; The rate is set at the start of the period. The payment is made at the end.

How to solve Valuing Swaptions and Interest Rate Derivative Options questions

Use this method for any swaption or cap and floor item in a vignette.

1Identify the instrument: payer or receiver, cap or floor. Payer and cap are calls on rates. Receiver and floor are puts on rates.
2Find the exercise or strike rate X and the notional. Note the swap tenor and payment frequency.
3Find the current market or forward swap rate. At expiry it is the observed swap rate; before expiry it is the forward swap rate F.
4Compute the annuity factor: the sum of the period fractions times the discount factors for each swap payment date. Note that the option's expiry is not the same as the swap's start for discounting.
5For expiry payoffs, compute the rate gap, apply the payoff max(0, ...), and multiply by notional and the annuity factor.
6For pre-expiry value, compute d1 and d2 using the swap rate and the option's time to expiry, then apply the Black formula and multiply by the annuity.
7Check consistency using put-call parity or the sign of the position. Payer value should rise when F rises.
8Answer the specific question: premium, payoff, hedge, or which position profits.

Quickest way: Rate gap times annuity shortcut

When to use it: Use when the question asks for the value at expiry or the sign of a payoff, and when N(d1) values are given.

Decide the direction: payer wins if market swap rate is above X; receiver wins if it is below X.
Compute the gap in decimals and multiply by notional.
Multiply by the annuity factor given in the vignette. If not given, sum the discount factors times the period fraction.
For Black valuation, plug in the supplied N(d1), N(d2) and multiply by the annuity.
For a parity question, compute payer minus receiver as annuity × (F − X) and solve for the missing option.

Common mistakes in Valuing Swaptions and Interest Rate Derivative Options

Mixing up payer and receiver swaptions.
Payer sounds like it should gain when rates fall, because you pay a rate.
Fix: A payer swaption pays fixed, so it gains when market rates rise above X. Think of it as a call on the swap rate.
Using the swap tenor as T in the Black d1 formula.
Both the option and the swap have a time length and the vignette quotes both, such as 2 into 5.
Fix: T is the time to option expiry only. The swap tenor only enters the annuity.
Forgetting the annuity factor and giving the rate gap as the value.
The Black bracket looks like the answer.
Fix: Multiply the bracket by the annuity factor and the notional. The rate gap is per period.
Treating a caplet payment as paid at the start of the period.
The rate is set at the start, so students discount from there.
Fix: The rate is set at the start but the payment is made in arrears at the end of the period. Discount from the payment date.
Using N(d1) for a receiver without converting to N(−d).
Students reuse the payer layout.
Fix: For the receiver, use N(−d2) and N(−d1), which are 1 − N(d2) and 1 − N(d1).
Applying the spot swap rate instead of the forward swap rate in Black.
Both are quoted in the vignette.
Fix: The underlying is the forward swap rate that starts at option expiry.

Worked examples

Example 1

A company holds a 1-year payer swaption on a 3-year annual-pay swap with notional ₹50,00,00,000 and exercise rate 4.00%. At expiry the 3-year market swap rate is 5.00%. The discount factors at expiry for the three swap payment dates are 0.9615, 0.9246 and 0.8890. (1) Should the company exercise? (2) What is the value of the swaption at expiry?

Show the solution

The company pays fixed, so it gains when the market rate is above the exercise rate. 5.00% > 4.00%, so it exercises.
Rate gap = 5.00% − 4.00% = 1.00%.
Annuity factor = 0.9615 + 0.9246 + 0.8890 = 2.7751 (period fraction is 1 for annual pay).
Value = 0.01 × 2.7751 × ₹50,00,00,000 = ₹1,38,75,500 (0.027751 × 50,00,00,000 = 1,38,75,500).

Answer: (1) Yes, exercise. (2) The swaption is worth ₹1,38,75,500 at expiry.

Example 2

An analyst values a 1-year payer swaption on a 3-year annual-pay swap. Forward swap rate F = 3.00%, exercise rate X = 3.00%, volatility 20%, notional ₹10,00,00,000. The annuity factor is 2.80. With T = 1, σ√T = 0.20, so d1 = 0.10 and d2 = −0.10, giving N(d1) = 0.5398 and N(d2) = 0.4602. (1) What is the payer swaption value? (2) What is the receiver swaption value? (3) What does payer minus receiver equal?

Show the solution

Check the inputs: F = X, so ln(F ÷ X) = 0. d1 = (σ² ÷ 2) × T ÷ (σ × √T) = 0.02 ÷ 0.20 = 0.10. d2 = 0.10 − 0.20 = −0.10. These match the given N values.
Payer = annuity × notional × [F × N(d1) − X × N(d2)].
Bracket = 0.03 × 0.5398 − 0.03 × 0.4602 = 0.03 × 0.0796 = 0.002388.
Payer = 2.80 × 0.002388 × ₹10,00,00,000 = ₹6,68,640.
Receiver = annuity × notional × [X × N(−d2) − F × N(−d1)]. N(−d2) = 1 − 0.4602 = 0.5398 and N(−d1) = 1 − 0.5398 = 0.4602. Bracket = X × N(−d2) − F × N(−d1) = 0.03 × 0.5398 − 0.03 × 0.4602 = 0.002388.
Receiver = 2.80 × 0.002388 × ₹10,00,00,000 = ₹6,68,640 (at the money, F = X, so payer and receiver are equal).
Parity check: payer − receiver = annuity × (F − X) × notional = 0, which matches.

Answer: (1) Payer = ₹6,68,640. (2) Receiver = ₹6,68,640. (3) Payer minus receiver is zero, because F = X, so the forward swap has zero value.

Exam tips

Read the vignette for the notation "2 into 5" or similar: the first number is option expiry, the second is swap tenor.
Write down payer = call and receiver = put on the swap rate before any calculation.
If the question gives the annuity factor, do not rebuild it. If it gives discount factors, sum them and apply the period fraction.
Use parity to cross-check: payer minus receiver must equal annuity × (F − X) × notional.
For caps and floors, check whether the payment is in arrears and which discount factor applies.

Valuing Swaptions and Interest Rate Derivative Options in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Valuing Swaptions and Interest Rate Derivative Options: frequently asked questions

What is the difference between a payer and receiver swaption?

A payer swaption gives the right to pay the fixed rate and receive floating. It gains when swap rates rise above the exercise rate. A receiver swaption gives the right to receive fixed and gains when swap rates fall below it.

How do I price a swaption on the CFA Level II exam?

Use the Black model with the forward swap rate as the underlying and multiply by the annuity factor of the swap. The exam normally supplies N(d1) and N(d2) or the inputs to compute them. Use time to option expiry for T.

How is a cap related to a floor and a swap?

A cap is a series of calls on a rate and a floor is a series of puts on a rate. A long cap and a short floor with the same strike and dates equals paying fixed at the strike on a swap.

Is a swaption the same as a series of caplets?

No. A cap is a series of options on each period's rate, each exercised separately. A swaption is a single option on the whole swap, so you exercise once and enter all the swap payments.