Caps & Floors

Price interest rate caps and floors using CapFloor with Black-76 (lognormal) and Bachelier (normal) volatility models. Caps pay when the floating rate exceeds the strike; floors pay when it falls below.

Setup

Cap and floor pricing requires a discount curve for present-valuing cashflows and a forecast curve for projecting forward rates. Here we calibrate a single curve and use it for both roles.

import datetime
from vade import CapFloor, DiscountCurve, Deposit, IRS, Solver

effective = datetime.date(2025, 6, 16)

curve = DiscountCurve(
    {
        effective: 1.0,
        datetime.date(2025, 9, 16): 1.0,
        datetime.date(2025, 12, 16): 1.0,
        datetime.date(2026, 6, 16): 1.0,
        datetime.date(2027, 6, 16): 1.0,
        datetime.date(2028, 6, 16): 1.0,
        datetime.date(2030, 6, 16): 1.0,
    },
    interpolation="log_linear",
    convention="act360",
    id="sofr",
)

dep_3m = Deposit(effective=effective, termination="3m", rate=0.0, convention="act360")
dep_6m = Deposit(effective=effective, termination="6m", rate=0.0, convention="act360")
irs_1y = IRS(effective=effective, termination="1y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")
irs_2y = IRS(effective=effective, termination="2y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")
irs_3y = IRS(effective=effective, termination="3y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")
irs_5y = IRS(effective=effective, termination="5y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")

solver = Solver(
    curves=[curve],
    instruments=[
        (dep_3m, 0.0430), (dep_6m, 0.0425),
        (irs_1y, 4.10), (irs_2y, 3.95),
        (irs_3y, 3.85), (irs_5y, 3.80),
    ],
)
result = solver.iterate()
assert result.converged

disc_curve = solver.get_curve(0)
forecast_curve = disc_curve

Cap Pricing with Black-76

The Black-76 model prices each caplet using a lognormal volatility assumption. Set model="black76" and provide an implied volatility in lognormal terms (e.g., 0.20 for 20%).

cap = CapFloor(
    effective=effective,
    termination="5y",
    strike=0.04,
    vol=0.20,
    notional=1_000_000.0,
    frequency="q",
    cap_floor_type="cap",
    model="black76",
    convention="act360",
)

float(cap.npv(disc_curve, forecast_curve))  # 169497.62285527054

Floor Pricing

A floor is the mirror of a cap -- it pays when the floating rate falls below the strike. With forward rates near 4% and a strike of 4%, the floor is deeply out of the money and has negligible value.

floor = CapFloor(
    effective=effective,
    termination="5y",
    strike=0.04,
    vol=0.20,
    notional=1_000_000.0,
    frequency="q",
    cap_floor_type="floor",
    model="black76",
    convention="act360",
)

floor_npv = float(floor.npv(disc_curve, forecast_curve))
floor_npv < 1.0  # True -- floor is deep OTM

cap_npv = float(cap.npv(disc_curve, forecast_curve))
cap_minus_floor = cap_npv - floor_npv
cap_minus_floor > 0  # True -- put-call parity: cap - floor approximates swap value

Bachelier Model

The Bachelier (normal) model uses an absolute volatility in rate terms rather than a lognormal percentage. A typical normal vol is around 50 basis points (0.005), compared to 20% (0.20) for Black-76. Set model="bachelier".

cap_bach = CapFloor(
    effective=effective,
    termination="5y",
    strike=0.04,
    vol=0.005,
    notional=1_000_000.0,
    frequency="q",
    cap_floor_type="cap",
    model="bachelier",
    convention="act360",
)

float(cap_bach.npv(disc_curve, forecast_curve))  # 169498.43835327454

Strike Sensitivity

Cap value decreases as the strike increases -- a higher strike means fewer caplets finish in the money. This relationship is visible across a range of strikes.

for strike in [0.03, 0.035, 0.04, 0.045, 0.05]:
    c = CapFloor(
        effective=effective,
        termination="5y",
        strike=strike,
        vol=0.20,
        notional=1_000_000.0,
        frequency="q",
        cap_floor_type="cap",
        model="black76",
        convention="act360",
    )
    npv = float(c.npv(disc_curve, forecast_curve))

# Strike 3.0%: higher cap value
# Strike 5.0%: lower cap value
assert float(
    CapFloor(effective=effective, termination="5y", strike=0.03, vol=0.20,
             notional=1_000_000.0, frequency="q", cap_floor_type="cap",
             model="black76", convention="act360").npv(disc_curve, forecast_curve)
) > float(
    CapFloor(effective=effective, termination="5y", strike=0.05, vol=0.20,
             notional=1_000_000.0, frequency="q", cap_floor_type="cap",
             model="black76", convention="act360").npv(disc_curve, forecast_curve)
)

Next Steps

FX Rates & Forwards -- FX rate triangulation and forward construction
Calibration -- Multi-curve calibration with Solver