Enter Values

$
Loan/deposit amount
%
Cap/floor strike interest rate (simple, annualized)
%
Assumed flat forward rate (simple, annualized)
%
Volatility of forward rate
%
Continuously compounded risk-free rate
years
Total cap/floor duration — = 19 caplets
Black's Model Formulas
Capletk = L × δ × P(0,tk+1) × [F × N(d1) − RK × N(d2)]
Floorletk = L × δ × P(0,tk+1) × [RK × N(−d2) − F × N(−d1)]
L = Notional | δ = Period fraction | P = Discount factor | F = Forward rate | RK = Strike
Ryan O'Connell, CFA
Calculator by Ryan O'Connell, CFA

Pricing Results

Total Cap Premium --
Premium % of Notional --
Number of Caplets --
Average Price --
Cap-Floor Parity
Cap Price --
Floor Price --
Cap − Floor (Swap PV) --

Per-Period Breakdown

Period tk (yrs) tk+1 (yrs) Fk d1 d2 P(0,tk+1) Price ($)

Formula Breakdown (First Caplet)

Capletk = L × δ × P(0,tk+1) × [F × N(d1) − RK × N(d2)]
Step-by-step calculation for the first period
Model Assumptions
  • Black's model applied independently to each caplet/floorlet
  • Lognormal forward rate distribution (standard pre-2008 convention)
  • Flat forward rate term structure (all forward rates equal F)
  • Flat volatility surface (single σ applies to all periods)
  • Day count simplified as 1/frequency (e.g., 0.25 for quarterly)
  • Continuous discounting: P(0,t) = e−rt
  • First period excluded (rate already known at inception)

For educational purposes. Not financial advice. Market conventions simplified.

Understanding Interest Rate Caps & Floors

What Are Interest Rate Caps and Floors?

An interest rate cap is a derivative that protects the holder against rising interest rates. It consists of a series of call options on a floating interest rate, called caplets. Each caplet pays the holder when the reference rate exceeds the cap rate (strike) for that period.

An interest rate floor is the opposite — it protects against falling rates. It consists of floorlets, which are put options on the floating rate. A collar combines a long cap with a short floor, often structured to be zero-cost.

Black's Model for Caplets (Hull Eq. 29.7)
Capletk = L × δk × P(0, tk+1) × [Fk × N(d1) − RK × N(d2)]
Where d1 = [ln(F/RK) + σ²t/2] / (σ√t) and d2 = d1 − σ√t

Cap-Floor Parity

Cap-floor parity is analogous to put-call parity for equity options:

Cap-Floor Parity
Cap − Floor = PV of Swap (receive floating, pay fixed at RK)
= L × Σ δk × P(0, tk+1) × (Fk − RK)

A long cap combined with a short floor produces the same cash flows as a pay-fixed interest rate swap at rate RK. This relationship can be used to verify pricing consistency and to construct synthetic instruments.

Practical Applications

  • Borrowers buy caps to limit maximum borrowing cost on floating-rate debt while retaining the benefit of lower rates
  • Investors buy floors to guarantee a minimum return on floating-rate investments
  • Collars (cap + short floor) provide cost-effective hedging by financing the cap premium with floor income
Note: This calculator uses Black's lognormal model, which is the standard pre-2008 convention. For negative interest rate environments, the Bachelier (normal) model is more appropriate.

Frequently Asked Questions

An interest rate cap is a derivative that protects the holder against rising interest rates. It consists of a series of call options on a floating interest rate, called caplets. Each caplet pays the holder when the reference rate exceeds the cap rate (strike) for that period. Companies with floating-rate debt commonly buy caps to limit their maximum borrowing cost while still benefiting if rates fall.

A cap protects against rising rates by paying when the floating rate exceeds the strike rate, while a floor protects against falling rates by paying when the floating rate drops below the strike rate. Caps are portfolios of call options (caplets) on the rate, and floors are portfolios of put options (floorlets). A collar combines a long cap with a short floor, often structured so the premiums offset each other (zero-cost collar).

Each caplet is priced independently using Black's model (Hull equation 29.7). The model treats the forward interest rate as lognormally distributed and applies the Black-Scholes framework. The caplet price equals the notional times the accrual fraction times the discount factor times [F × N(d1) − RK × N(d2)], where d1 and d2 are calculated using the forward rate, strike rate, volatility, and time to the reset date.

Cap-floor parity states that the value of a cap minus the value of a floor (with the same strike rate) equals the present value of a swap that receives floating and pays fixed at the strike rate. Mathematically: Cap − Floor = L × Σ δk × P(0, tk+1) × (Fk − RK). This is analogous to put-call parity for equity options and can be used to verify pricing consistency.

Companies with floating-rate debt buy caps to limit their maximum borrowing cost while retaining the benefit of lower rates if rates decrease. The cap premium functions like an insurance payment that guarantees the effective borrowing rate never exceeds the cap rate plus the amortized premium cost. This is preferred over swapping to a fixed rate when the borrower wants downside protection but also wants to benefit from potential rate decreases.

Higher volatility increases both cap and floor prices because it raises the probability and expected magnitude of payoffs. This is the same effect volatility has on standard option prices in the Black-Scholes model. In practice, cap/floor volatilities are quoted as either flat volatilities (one vol for all caplets in a cap) or spot volatilities (different vol per caplet), with flat volatilities being more common in the market.
Disclaimer

This calculator is for educational purposes only. It uses Black's lognormal model with a flat forward rate assumption, which simplifies the actual term structure. Real-world cap/floor pricing uses individual forward rates from the yield curve and may employ different volatility models. This tool should not be used for trading decisions.

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