CFA Level II Exam · Valuation of Contingent Claims

Black Model for Futures and Interest Rate Options

Updated 7 October 2026 · Fact-checked

The Black model is Black-Scholes-Merton rewritten with a forward or futures price instead of a spot price. You discount the payoff at the risk-free rate and use the volatility of the forward price. For caps, floors and swaptions, the underlying is a forward rate or forward swap rate, and you also multiply by notional, period length and a discount factor.

Understand Black Model for Futures and Interest Rate Options

The Black-Scholes-Merton model prices an option on a spot asset. The Black model does the same job when the underlying is a futures or forward price. A forward price already contains the cost of carry, so the formula has no spot price, no dividend yield and no carry term. You only need the forward price F, the strike X, the time to expiry T, the volatility of the forward price σ and the risk-free rate r.

The structure is the same as BSM. A call is a discounted expected payoff: e^(-rT) × [F × N(d1) − X × N(d2)]. The forward price replaces the spot price compounded at the carry rate. That is why the model is a special case of BSM.

Interest rate options use the same formula with a different underlying. A caplet is a call on a forward rate (the FRA rate). A floorlet is a put on a forward rate. A cap is a series of caplets and a floor is a series of floorlets. Each caplet or floorlet covers one reset period. The option expires at the reset date, but the payment is made at the end of the period (in arrears). So you discount from the payment date, not the expiry date, and you scale by the period length (days/360) and the notional.

A swaption is an option to enter a swap at a fixed rate. A payer swaption gives the right to pay fixed and receive floating, so it is a call on the forward swap rate. A receiver swaption is a put on the forward swap rate. The payoff lasts over all the swap's payments, so you multiply by the present value of an annuity of the swap's payment periods, called the PV annuity factor.

The exam tests whether you can pick the right inputs from the vignette. The usual traps are using time to payment instead of time to expiry inside d1 and d2, using the wrong rate as F, and mixing up calls and puts.

Key formulas to remember

Black call on a futures or forward price: c = e^(-rT) × [F0 × N(d1) − X × N(d2)]; F0 is the futures or forward price for delivery at the underlying's date. T is time to option expiry. r is the continuously compounded risk-free rate.
Black put on a futures or forward price: p = e^(-rT) × [X × N(−d2) − F0 × N(−d1)]; Put-call parity for futures options: c − p = e^(-rT) × (F0 − X).
d1 and d2: d1 = [ln(F0 ÷ X) + (σ² ÷ 2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; σ is the volatility of the futures or forward price. There is no r or carry term inside d1.
Caplet value: Caplet = AD × NP × (days ÷ 360) × [FRA × N(d1) − X × N(d2)]; AD is the discount factor to the payment date (end of the period). FRA is the forward rate for the period that starts at option expiry. In d1 and d2 use t = time to option expiry.
Floorlet value: Floorlet = AD × NP × (days ÷ 360) × [X × N(−d2) − FRA × N(−d1)]; Same inputs as the caplet. A cap is the sum of its caplets and a floor is the sum of its floorlets.
Payer swaption: Payer = PVA × NP × [FSR × N(d1) − X × N(d2)]; FSR is the forward swap rate at option expiry. PVA is the present value annuity factor: the sum of discount factors times period fractions over the swap's payment dates. Use time to option expiry in d1 and d2.
Receiver swaption: Receiver = PVA × NP × [X × N(−d2) − FSR × N(−d1)]; A put on the forward swap rate. Payer − receiver for the same strike equals the PV of a forward swap paying fixed X against floating.
Interest rate option d1 and d2: d1 = [ln(F ÷ X) + (σ² ÷ 2) × t] ÷ (σ × √t); d2 = d1 − σ × √t; F is the FRA rate or forward swap rate. σ is the volatility of that rate.

How to solve Black Model for Futures and Interest Rate Options questions

Use this order for any Black model question, whether the underlying is a futures price, a forward rate or a swap rate.

1Identify the underlying and the option type. A futures option uses F0. A caplet or floorlet uses the FRA rate. A swaption uses the forward swap rate. Decide whether you need a call-type or put-type formula. Payer swaption and caplet are calls. Receiver swaption and floorlet are puts.
2Pull the inputs from the vignette: F (or FRA or FSR), X, volatility, time to option expiry, risk-free rate or discount factors, notional and day count. Note which time is the expiry and which is the payment date.
3Compute d1 = [ln(F ÷ X) + 0.5 × σ² × t] ÷ (σ × √t), then d2 = d1 − σ × √t. Always use time to option expiry here.
4Look up N(d1) and N(d2) in the normal table. For put-type formulas use N(−d1) = 1 − N(d1) and N(−d2) = 1 − N(d2).
5Plug into the call or put formula. For futures options, discount with e^(-rT). For caplets and floorlets, multiply by the discount factor to the payment date, the period fraction and the notional. For swaptions, multiply by the PV annuity factor and the notional.
6For a cap or floor, repeat for each period and add the values. Different periods have different F, t and discount factors.
7Sanity check with put-call parity. For futures options, c − p = e^(-rT) × (F0 − X). For caplet and floorlet with the same strike, the difference is the discounted value of (FRA − X) × period fraction × notional.

Quickest way: Parity and moneyness shortcut

When to use it: Use when the vignette gives you one of the call or put values and asks for the other, or when you must pick between three answers that differ in size.

If the call is given and you need the put, use parity instead of recomputing d1 and d2. For futures: p = c − e^(-rT) × (F0 − X).
For caplet and floorlet: floorlet = caplet − AD × NP × (days ÷ 360) × (FRA − X). Same for payer and receiver swaptions with PVA × NP × (FSR − X).
If F equals X, d1 is small and positive and the call equals the put. Both sit near 0.4 × F × σ × √T discounted. Use this to reject wildly wrong answers.
If F is above X, the call must be worth at least the discounted intrinsic value e^(-rT) × (F − X). Eliminate any option that is below that.
Keep the call positive. If your call comes out negative, you probably used the wrong sign in d2 or the wrong time.

Common mistakes in Black Model for Futures and Interest Rate Options

Using a spot price or adding a carry term in the Black model.
Students carry over the BSM formula and forget that the forward price already contains carry.
Fix: In Black, F0 is the input and nothing else adds a carry or dividend term. Discount only by e^(-rT).
Using time to payment instead of time to expiry inside d1 and d2 for caplets and floorlets.
The vignette gives both dates, and the loan period end feels like the relevant date.
Fix: Volatility accrues until the option expires. Use that t in d1 and d2. Use the payment date only for the discount factor AD.
Forgetting to scale by the period fraction (days ÷ 360) and notional.
The Black formula output looks like a final value, but it is a rate difference.
Fix: For caplets, floorlets and swaptions, the bracket is in rate terms. Multiply by notional and by the period length or the PV annuity factor.
Mixing up payer and receiver swaptions, or caplets and floorlets.
Students think in terms of bond prices rather than rates.
Fix: Anything that pays off when rates rise is call-like on the rate: caplet, payer swaption. Anything that pays when rates fall is put-like: floorlet, receiver swaption.
Treating the volatility as the volatility of the spot asset or of the bond price.
The word volatility is used loosely in vignettes.
Fix: Use the volatility the vignette gives for the futures price, forward rate or forward swap rate, which is the variable in the formula.
Treating a cap as one option and using one volatility, one F and one discount factor.
The word cap sounds like a single contract.
Fix: A cap is a sum of caplets. Each caplet has its own FRA rate, expiry, payment date and discount factor. Sum the values.

Worked examples

Example 1

Vignette: An analyst values a 6-month European option on a futures contract. The futures price is 100, the strike is 100, the volatility of the futures price is 20% and the continuously compounded risk-free rate is 4%. Questions: (1) Calculate d1 and d2. (2) Value the call. (3) What is the value of the put with the same strike?

Show the solution

Inputs: F0 = 100, X = 100, T = 0.5, σ = 0.20, r = 0.04.
d1 = [ln(100 ÷ 100) + 0.5 × 0.04 × 0.5] ÷ (0.20 × √0.5) = 0.01 ÷ 0.141421 = 0.0707.
d2 = 0.0707 − 0.1414 = −0.0707.
N(d1) ≈ 0.5282 and N(d2) ≈ 0.4718.
Bracket = 100 × 0.5282 − 100 × 0.4718 = 5.64.
Discount factor = e^(-0.04 × 0.5) = e^(-0.02) = 0.9802.
Call = 0.9802 × 5.64 ≈ 5.53.
Put by parity: p = c − e^(-rT) × (F0 − X) = 5.53 − 0 = 5.53.

Answer: d1 ≈ 0.0707 and d2 ≈ −0.0707. The call is worth about 5.53. The put is also about 5.53 because the futures price equals the strike. Small differences come from table rounding.

Example 2

Vignette: A 10 million notional interest rate cap includes a caplet with strike 3.0%. The caplet's option expires in 6 months and covers the following 90-day period, with payment at the end of that period. The FRA rate for that period is 3.2%. The volatility of the forward rate is 25%. The discount factor to the payment date (9 months) is 0.9750. Questions: (1) Calculate d1 and d2. (2) Value the caplet. (3) Value the floorlet with the same strike and inputs.

Show the solution

Inputs: FRA = 0.032, X = 0.030, t = 0.5, σ = 0.25, AD = 0.9750, days ÷ 360 = 0.25, NP = 10,000,000.
d1 = [ln(0.032 ÷ 0.030) + 0.5 × 0.0625 × 0.5] ÷ (0.25 × √0.5). ln(1.06667) = 0.06454. Add 0.015625 to get 0.08016. Denominator = 0.17678. d1 ≈ 0.4535.
d2 = 0.4535 − 0.1768 = 0.2767.
N(d1) ≈ 0.6749 and N(d2) ≈ 0.6090.
Bracket = 0.032 × 0.6749 − 0.030 × 0.6090 = 0.021597 − 0.018270 = 0.003327.
Caplet = 0.9750 × 10,000,000 × 0.25 × 0.003327 ≈ 8,110.
Floorlet: N(−d1) = 0.3251 and N(−d2) = 0.3910. Bracket = 0.030 × 0.3910 − 0.032 × 0.3251 = 0.011730 − 0.010403 = 0.001327.
Floorlet = 0.9750 × 10,000,000 × 0.25 × 0.001327 ≈ 3,234.
Parity check: caplet − floorlet = 0.9750 × 10,000,000 × 0.25 × (0.032 − 0.030) = 4,875. Our difference is 8,110 − 3,234 = 4,876, which matches within rounding.

Answer: d1 ≈ 0.4535 and d2 ≈ 0.2767. The caplet is worth about 8,110 and the floorlet about 3,234.

Exam tips

Black versus BSM is a favourite conceptual question. The key differences: Black uses the forward or futures price, has no carry or dividend term, and discounts at r. BSM uses the spot price.
In caplet and swaption questions, find the option expiry date first. That t goes into d1 and d2. The payment dates only affect discount factors and annuity factors.
Check which side you need. Payer swaption and caplet use N(d1) and N(d2). Receiver swaption and floorlet use N(−d1) and N(−d2).
When two values are given, use put-call parity rather than computing both. It saves time and acts as a check on your answer.
Do not forget the notional and the period fraction. If an answer option looks like a tiny decimal, you have probably left out a scaling step.

Black Model for Futures and Interest Rate Options in other exams

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

Black Model for Futures and Interest Rate Options: frequently asked questions

What is the difference between the Black model and Black-Scholes-Merton?

BSM prices options on a spot asset and uses the spot price, with dividends or carry. The Black model prices options on a forward or futures price, which already includes carry, so it has no spot price or carry term. Both discount the payoff at the risk-free rate.

How do I value an interest rate cap using the Black model?

Treat the cap as a series of caplets. For each caplet, find the FRA rate, the expiry, the payment date and the discount factor. Apply the Black call formula to the FRA rate, scale by notional and period fraction, discount to the payment date and add the caplets.

How is a swaption valued with the Black model?

A payer swaption is a call on the forward swap rate and a receiver swaption is a put. The Black bracket is multiplied by the notional and the PV annuity factor of the underlying swap. Use time to option expiry in d1 and d2.

Why does a caplet use the payment date for discounting but the expiry date in d1?

The rate is fixed at the expiry date, so uncertainty about the rate stops then. The cash is paid later, at the end of the period, so you discount from that payment date.