Curve Building

Explore how interpolation method choice affects forward rates, and learn to work with different curve types. This guide builds a calibrated DiscountCurve and then reconstructs it with three interpolation methods to compare their forward rate profiles.

Market Data and Calibration

Set up a USD SOFR market dataset and calibrate a discount curve. The dataset covers tenors from 1-month deposits through 10-year swaps.

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

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

nodes = {
    effective: 1.0,
    datetime.date(2025, 7, 16): 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,
    datetime.date(2032, 6, 16): 1.0,
    datetime.date(2035, 6, 16): 1.0,
}

curve = DiscountCurve(
    nodes, interpolation="log_linear", convention="act360", id="sofr"
)

dep_1m = Deposit(effective=effective, termination="1m", rate=0.0, convention="act360")
fra_3m = FRA(effective=effective, termination="3m", fixed_rate=0.0, convention="act360")
fra_6m = FRA(effective=effective, termination="6m", fixed_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")
irs_7y = IRS(effective=effective, termination="7y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")
irs_10y = IRS(effective=effective, termination="10y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")

instruments = [
    (dep_1m, 0.0430),
    (fra_3m, 0.0425),
    (fra_6m, 0.0415),
    (irs_1y, 4.00),
    (irs_2y, 3.85),
    (irs_3y, 3.75),
    (irs_5y, 3.70),
    (irs_7y, 3.75),
    (irs_10y, 3.85),
]

solver = Solver(curves=[curve], instruments=instruments)
result = solver.iterate()
assert result.converged

calibrated = solver.get_curve(0)
float(calibrated.discount_factor(datetime.date(2026, 6, 16)))  # 0.9610212208394634
float(calibrated.discount_factor(datetime.date(2030, 6, 16)))  # 0.8319869245862279

Querying the Curve

A calibrated curve supports three core queries: discount factors, zero rates, and forward rates. All rates use the act360 day count convention specified at construction.

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

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

nodes = {
    effective: 1.0,
    datetime.date(2025, 7, 16): 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,
    datetime.date(2032, 6, 16): 1.0,
    datetime.date(2035, 6, 16): 1.0,
}

curve = DiscountCurve(nodes, interpolation="log_linear", convention="act360", id="sofr")

dep_1m = Deposit(effective=effective, termination="1m", rate=0.0, convention="act360")
fra_3m = FRA(effective=effective, termination="3m", fixed_rate=0.0, convention="act360")
fra_6m = FRA(effective=effective, termination="6m", fixed_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")
irs_7y = IRS(effective=effective, termination="7y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")
irs_10y = IRS(effective=effective, termination="10y", frequency="a", fixed_rate=0.0, convention="act360", float_convention="act360")

instruments = [
    (dep_1m, 0.0430), (fra_3m, 0.0425), (fra_6m, 0.0415),
    (irs_1y, 4.00), (irs_2y, 3.85), (irs_3y, 3.75),
    (irs_5y, 3.70), (irs_7y, 3.75), (irs_10y, 3.85),
]

solver = Solver(curves=[curve], instruments=instruments)
result = solver.iterate()
calibrated = solver.get_curve(0)

# Discount factors
float(calibrated.discount_factor(datetime.date(2026, 6, 16)))  # 0.9610212208394634
float(calibrated.discount_factor(datetime.date(2030, 6, 16)))  # 0.8319869245862279
float(calibrated.discount_factor(datetime.date(2035, 6, 16)))  # 0.680542109248269

# Zero rates (continuously compounded, annualized)
float(calibrated.zero_rate(effective, datetime.date(2026, 6, 16)))  # 0.039214150340492254
float(calibrated.zero_rate(effective, datetime.date(2030, 6, 16)))  # 0.03626390009498523

# Forward rates between future dates
float(calibrated.forward_rate(datetime.date(2026, 6, 16), datetime.date(2027, 6, 16)))  # 0.03627210043804068
float(calibrated.forward_rate(datetime.date(2027, 6, 16), datetime.date(2028, 6, 16)))  # 0.03477612714625993
float(calibrated.forward_rate(datetime.date(2030, 6, 16), datetime.date(2032, 6, 16)))  # 0.038181993768805886

Interpolation Comparison

Rebuild the curve from calibrated nodes using three different interpolation methods to see how they affect forward rates between nodes. The calibrated discount factors at node dates are the same regardless of interpolation -- differences appear only between nodes.

Log-linear interpolation produces piecewise constant forward rates between nodes. Linear-zero-rate interpolation linearly interpolates in zero-rate space, producing smoother forward rates. Flat-forward is a step function that holds the forward rate constant within each node interval.

import datetime
from vade import DiscountCurve

cal_nodes = {
    datetime.date(2025, 6, 16): 1.0,
    datetime.date(2025, 7, 16): 0.9964294610977332,
    datetime.date(2025, 9, 16): 0.9891976578024332,
    datetime.date(2025, 12, 16): 0.9791251319654892,
    datetime.date(2026, 6, 16): 0.9610212208394634,
    datetime.date(2027, 6, 16): 0.9263207970596119,
    datetime.date(2028, 6, 16): 0.89414224902702,
    datetime.date(2030, 6, 16): 0.8319869245862279,
    datetime.date(2032, 6, 16): 0.7699195851637843,
    datetime.date(2035, 6, 16): 0.680542109248269,
}

curve_ll = DiscountCurve(cal_nodes, interpolation="log_linear", convention="act360")
curve_lz = DiscountCurve(cal_nodes, interpolation="linear_zero_rate", convention="act360")
curve_ff = DiscountCurve(cal_nodes, interpolation="flat_forward", convention="act360")

# Forward rates between 18M and 2Y (between the 1Y and 2Y nodes)
start_1 = datetime.date(2026, 12, 16)
end_1 = datetime.date(2027, 6, 16)

float(curve_ll.forward_rate(start_1, end_1))  # 0.03627210043804068
float(curve_lz.forward_rate(start_1, end_1))  # 0.035534583782283175
float(curve_ff.forward_rate(start_1, end_1))  # 0.07274350093003831

# Forward rates between 4Y and 5Y (between the 3Y and 5Y nodes)
start_2 = datetime.date(2029, 6, 16)
end_2 = datetime.date(2030, 6, 16)

float(curve_ll.forward_rate(start_2, end_2))  # 0.035530567199720826
float(curve_lz.forward_rate(start_2, end_2))  # 0.035285822519282804
float(curve_ff.forward_rate(start_2, end_2))  # 0.07106113439944166

At node dates, all three methods agree because the underlying discount factors are identical. Between nodes, the methods diverge significantly -- log-linear produces smooth constant forward rates within each interval, linear-zero-rate varies gradually, and flat-forward creates large jumps at node boundaries.

LineCurve

LineCurve stores zero rates directly instead of discount factors. This is useful for representing rate curves, spreads, or any quantity that should be interpolated linearly in its native space.

import datetime
from vade import LineCurve

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

line_curve = LineCurve(
    {
        effective: 0.040,
        datetime.date(2026, 6, 16): 0.039,
        datetime.date(2027, 6, 16): 0.038,
        datetime.date(2030, 6, 16): 0.037,
    },
    interpolation="linear",
    convention="act360",
)

float(line_curve.discount_factor(datetime.date(2026, 6, 16)))  # 0.9612299019061498
float(line_curve.zero_rate(effective, datetime.date(2026, 6, 16)))  # 0.039
float(line_curve.zero_rate(effective, datetime.date(2027, 6, 16)))  # 0.038

ForwardCurve

ForwardCurve stores instantaneous forward rates. Nodes represent continuously compounded rates on a per-day basis -- to express an annualized 4% rate, divide by 365. Discount factors are computed by integrating the forward rate function.

import datetime
from vade import ForwardCurve

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

fwd_curve = ForwardCurve(
    {
        effective: 0.04 / 365,
        datetime.date(2026, 6, 16): 0.038 / 365,
        datetime.date(2027, 6, 16): 0.037 / 365,
        datetime.date(2030, 6, 16): 0.036 / 365,
    },
    interpolation="flat_forward",
    convention="act365f",
)

float(fwd_curve.discount_factor(datetime.date(2026, 6, 16)))  # 0.9607894391523232
float(fwd_curve.forward_rate(effective, datetime.date(2026, 6, 16)))  # 0.04
float(fwd_curve.forward_rate(datetime.date(2026, 6, 16), datetime.date(2027, 6, 16)))  # 0.038

Next Steps

Pricing -- price instruments against calibrated curves
Calibration -- multi-curve frameworks and solver tuning