Fixed Income Derivatives
Black-76 Engine
Zero-Cost Collar
Interest Rate Cap, Floor & Collar Pricing Lab
Price interest rate caps, floors, and structured collars across multi-period forward curves using the standard Black-76 framework. Simulate caplet strips, calculate zero-cost collar strikes, and decompose Greeks risk.
Caplet: C_i = Notional × τ × P(0, t_{i+1}) × [ F_i Φ(d_1) - K Φ(d_2) ] |
Floorlet: P_i = Notional × τ × P(0, t_{i+1}) × [ K Φ(-d_2) - F_i Φ(-d_1) ]
Market Presets:
Contract Parameters
$
%
%
%
%
bps
Total Cap Premium
$124,500
124.5 bps
Total Floor Premium
$78,200
78.2 bps
Net Collar Cost
+$46,300
Long Cap − Short Floor
Zero-Cost Floor Strike
3.32%
Par Swap: 4.45%
Institutional Structure Visualizer
Portfolio Sensitivities & Cap Greeks
Total Cap Delta
$4,250,000
Total Cap Gamma
$18,200
Total Cap Vega
$8,450 / 1% vol
Total Cap Theta
-$120 / day
Period-by-Period Caplet & Floorlet Amortization Schedule
| Period | Fixing (t_fix) | Payment (t_pay) | Forward F_i | Discount Factor | Caplet Price ($) | Floorlet Price ($) | Caplet Vega ($/1%) |
|---|
Cap Price: Strike K_cap vs Volatility σ ($)
Cap Price: Maturity Tenor vs Strike K_cap ($)
Interest Rate Cap & Collar Mastery Quiz
1. When does an interest rate caplet make a positive settlement payment to the buyer?
2. How is a zero-cost collar structured by a corporate borrower?
3. According to Cap-Floor Parity, what is equivalent to Long Cap(K) − Long Floor(K)?
Interest Rate Derivatives Engineering & FAQ
An interest rate cap is an OTC derivative contract consisting of a series of European call options on a floating reference rate (such as SOFR or EURIBOR), called caplets. A borrower holding floating-rate debt purchases a cap to set an absolute maximum ceiling on interest expense. If the reference fixing rises above the agreed cap strike, the cap seller pays the borrower the difference at each period end.
The Black-76 model prices each individual caplet on forward rate F as a call option:
Caplet = Notional × τ × P(0, t_{i+1}) × [ F Φ(d1) - K Φ(d2) ], where τ is the accrual period, P(0, t_{i+1}) is the discount factor to the payment date, K is the cap strike, and σ is the forward rate volatility. The total cap price equals the sum of all individual caplets in the contract schedule.
Cap-Floor Parity is the fixed-income analogue of Put-Call Parity. For identical notional, payment dates, and strike K, buying a cap and selling a floor creates an identical cash flow profile to paying a fixed rate K and receiving floating in an Interest Rate Swap:
Cap Price - Floor Price = Value of Payer Interest Rate Swap. When strike K equals the fair par swap rate, the cap and floor have identical market values.
A zero-cost collar is created by simultaneously purchasing an interest rate cap (at an upper ceiling strike) and selling an interest rate floor (at a lower floor strike) such that the premium collected from the floor exactly equals the premium paid for the cap. This eliminates upfront option expense ($0 net cash outlay) while locking the borrower's effective floating rate into a bounded band between the floor and cap strikes.
In an upward-sloping yield curve, forward rates F_i increase with maturity. Consequently, distant caplets are deeper in-the-money (or have higher forward moneyness), making long-dated caplets significantly more expensive than short-dated caplets. Conversely, in an inverted yield curve, distant caplets lose value while long-dated floorlets become more valuable.